Hot electron spectroscopy and microscopy

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Rep. Prog. Phys. 67 (2004) REPORTS ON PROGRESS IN PHYSICS PII: S (04) Hot electron spectroscopy and microscopy J Smoliner, D Rakoczy and M Kast Institut für Festkörperelektronik,TU-Wien, Floragasse 7, A-1040 Wien, Austria juergen.smoliner@tuwien.ac.at Received 28 June 2004 Published 27 August 2004 Online at stacks.iop.org/ropp/67/1863 doi: / /67/10/r04 Abstract Semiconductor heterostructures, such as double-barrier resonant tunnelling diodes and superlattices, are nowadays used for many applications. One very versatile and powerful method to study electronic transport in heterostructures is hot electron spectroscopy. Hot electron spectroscopy can be carried out in two complementary versions: device-based techniques usually employ so-called hot electron transistors (HETs), while ballistic electron emission microscopy (BEEM) uses a scanning tunnelling microscope (STM) as the source of ballistic electrons. In this review, spectroscopic results obtained by these two methods are compared and discussed. It is shown that BEEM results are strongly influenced by electron refraction effects, while the behaviour of HET devices is dominated by inelastic scattering effects in the base and drift region of the device. Thus, STM-based BEEM/S and HET-based spectroscopy are genuinely complementary methods, which yield supplementary results. (Some figures in this article are in colour only in the electronic version) /04/ $ IOP Publishing Ltd Printed in the UK 1863

2 1864 J Smoliner et al 1. Motivation Hot electron spectroscopy is a valuable tool for studying transport in heterostructures. It can be carried out in two different ways: first, as device-based hot electron spectroscopy and second, as a scanning tunnelling microscope (STM)-based method usually referred to as ballistic electron emission microscopy (BEEM). As the activities in both fields are extremely broad, this article cannot and does not intend to give a complete survey of the field. An excellent review on hot electron spectroscopy was written by Heiblum [1], and more general topics on hot electron transport can be found in the book by Balkan [2]. There is a recent review on BEEM by Narayanamurti [3] and two somewhat older review articles by Prietsch [4] and Bell [5] which, together, give an excellent overview about the developements in BEEM. However, STM-based BEEM/S and HET-based spectroscopy are usually performed by completely different research groups, with only a little communication between the two communities. Therefore, till now the available literature still lacks a critical comparison of both techniques in order to identify their specific strengths and weaknesses. To make such a comparison, we can take advantage of the unique situation that at our institute experimental activities in device-based hot electron spectroscopy as well as BEEM have been carried out for many years. During this time, many complementary device-based hot electron spectroscopy and BEEM/S measurements were performed on the same types of heterostructure samples, which has lead to the rare opportunity of bringing together the expertise of researchers in these two fields. In this review, therefore, we want to discuss and compare the basic features of BEEM/S and device-based hot electron spectroscopy, especially the properties of the respective hot electron injectors. To show the spectroscopic strengths and weaknesses of both methods we will discuss selected results on superlattice transport obtained by both techniques. To illustrate the possibilities of spatially resolved measurements, some BEEM results on single impurities embedded in AlAs barriers will be reviewed in the last section of this article. 2. Introduction: state of the art in hot electron spectroscopy on heterostructures 2.1. Semiconductor heterostructures In modern semiconductor physics, heterostructures play a key role in the development of new device concepts and in fundamental scientific research. In heterostructures two semiconductors with different material properties, i.e. different band gaps, are grown on each other. Depending on the alignment of valence and conduction bands, potential steps occur at the interface. Various parameters, such as the composition of the compound semiconductors involved, the layer thicknesses or the doping concentrations in principle allow the engineering of any desired potential profiles and band structure properties of a heterostructure sample [6]. With the invention of growth techniques such as molecular beam epitaxy (MBE) [7, 8] and metallorganic chemical vapour deposition (MOCVD) [9] in the 1970s, it became possible to grow semiconductor layers on an atomic scale. This was the starting point for the development of heterostructure-based quantum structures. Standard quantum mechanics textbook examples like potential barriers or quantum wells could now be fabricated simply by growing multilayers of alternating semiconductor materials. The first experimental demonstrations of quantum physics in heterostructures were published in the early 1970s. A clear manifestation of the quantum-size effect in the optical spectra of GaAs AlGaAs quantum wells was demonstrated by Dingle et al [10]. The group of

3 Hot electron spectroscopy and microscopy 1865 Esaki studied the tunnelling current in biased GaAs/AlGaAs double-barrier heterostructures [11] and the observed maxima in the current were associated with resonant tunnelling processes. In 1970, Esaki and Tsu [12] proposed the possibility of achieving a negative differential conductivity (NDC) in one-dimensional, man-made crystals fabricated by growing periodic structures with layers of alternating semiconductor materials. In such superlattices a parabolic band would break into so-called minibands, separated by small forbidden gaps and having Brillouin zones determined by the superperiod. This proposal opened up a new and quickly emerging research field; since up to then solid state physics was limited to conventional crystals in which the fundamental parameters like lattice constants, band gaps, effective masses of the charge carriers as well as their mobilities, etc are predefined by nature. The large number of design parameters now made it possible to fabricate superlattices with superperiods in the nanometer range featuring narrow minibands. In the last three decades a lot of effort has been put into the exploitation of semiconductor superlattices in order to study both optical and electron transport properties in great detail. Probably, the most sophisticated device which emerged from this research activity is the superlattice-based quantum cascade laser (QCL). Based on the idea of generating stimulated emission in superlattices, the group of Capasso developed a unipolar cascade laser in a superlattice structure, which operates in the mid-infrared regime [13]. Another interesting feature in heterostructure device technology is the possibility of reducing the dimensionality of quantum confined systems. Two-dimensional electron gases (2DEGs), quantum wires (1D-systems) or quantum dots (0D-systems) can be achieved by (i) special growth techniques like cleaved edge overgrowth, modulation doping and selfassembly of dots, by (ii) standard etch processes or by (iii) using gate electrodes which locally deplete a two-dimensional electron gas to achieve 1D and 0D structures. Based on such low-dimensional electron systems, fundamental physical effects like the quantum hall effect (QHE) [14, 15] have been discovered which have had a great impact on modern solid state physics Ballistic electron transport in heterostructures Since device dimensions have reached the size of typical mean-free-paths (mfps) of electrons in crystals, ballistic electron transport effects have gained importance in semiconductor device physics. The development of heterostructure-based hot electron transistors (HET) as perfectly designable playgrounds for ballistic electrons facilitated the investigation of nonequilibrium electron transport in semiconductor bulk and heterostructure materials. In 1981, Heiblum [16] proposed a family of novel three-terminal devices, which are based on the injection of a quasimonoenergetic hot electron beam used to study electron transport in the transit region of the devices. This proposal was supported by Hesto et al [17], who presented numerical simulations ofan + nn + device for ballistic and quasi-ballistic electron spectroscopy. Somewhat later, Hayes et al [18] studied hot electron transport through heavily doped GaAs layers utilizing a HET based on Shannon s camel transistor [19]. Supported by Monte Carlo calculations, the mfp of hot electrons in the base layer was determined to be around 400 Å. This led to the conclusion that in such devices electron electron scattering is comparable in strength to the very efficient longitudinal optical phonon scattering process. In a subsequent work of this group, Levi et al [20, 21] provided evidence of quasi-ballistic hot electron transport through heavily doped GaAs layers. As a substantial step forward in hot electron spectroscopy, Heiblum et al [22] introduced the tunnelling hot electron transfer amplifier (THETA), which allows a systematic study of

4 1866 J Smoliner et al hot electron effects in GaAs/AlGaAs-based heterostructure devices [23]. The energetic width of the ballistic electron beam was measured to be about 60 mev for hot electrons with excess energies of about 300 mev above the thermal electrons. In the following, hot electron transfer from the Ɣ valley into the L valleys of GaAs in the presence of hydrostatic pressure was also reported [24]. In 1987, the THETA device was used to study electron interference effects [25] in the wide quantum well formed by the emitter barrier and the collector barrier. The strong modulation in the emitter current provides information about the nonparabolicity coefficient of bulk GaAs. In the same year, Levi et al [26] presented results of electron transport dynamics in wide GaAs quantum wells in two-terminal structures. They found that the lifetime of Ɣ electron states in double-barrier resonant tunnelling structures depends on the well width, longitudinal optical (LO) phonon scattering and the transfer into the subsidiary X and L minima. The THETA device was also used to demonstrate single optical phonon emission in the GaAs transit region as well as in the AlGaAs collector barrier [27]. The LO phonon scattering times were estimated to be 200 fs for 85 mev electrons in n + -type GaAs and 550 fs for 40 mev electrons in undoped AlGaAs. In the same year, Bending et al [28,29] published a quantitative study of the behaviour of an injected electron distribution in a magnetic field. They showed that the dependence of the distribution on a perpendicular magnetic field can be described quasi-classically, but with an effective mass 2 3 times larger than one would expect. The discrepancy was supposed to be due to a large degree of momentum randomization in the electron distribution incident at the collector barrier of the THETA device. Brill and co-workers used the THETA device to study electron heating effects in highly doped GaAs. The injected hot electrons heat the cold electrons which are confined in the thin, doped GaAs layer of the device. This manifests itself in a collector current which is larger than the input current [30, 31]. Later, they also reported on ballistic electron transport studies in high-purity GaAs at low temperatures [32, 33]. The mfp was determined to be several microns long for electrons with energies just below the LO phonon emission threshold in GaAs. The dominant scattering mechanism in this energy regime was suggested to be due to impact ionization of neutral impurities. It was also found that the mfp scales roughly inversely with the impurity concentration. In 1990, Choi et al [34] presented a study of quantum transport and phonon emission of nonequilibrium hot electrons utilizing a HET. The energy distribution of the injected electron beam was measured using both single and double-barrier analysers. The experimental results provided evidence of single-particle interference in current transport and the existence of phonon replicas in the hot electron distribution. Ballistic hole transport was also demonstrated utilizing a hot-hole transistor [35]. The energetic width of the ballistic hole beam was determined to be 35 mev and the mfp for ballistic holes was found to be 14 nm. The resonances in the injected current were used to support the light nature of the holes. In a subsequent paper the mfp of light holes in slightly doped GaAs was measured to lie in the range of nm, provided that the energy was below the LO phonon threshold. The dominant scattering mechanism in this energy regime was stated to be interband elastic scattering via ionized impurities [36]. Another application for hot electron transport is the investigation of higher valleys in heterostructures. For AlAs single barrier structures it was shown by hot electron spectroscopy, that the transferred electrons undergo strong inelastic scattering within the AlAs barrier and relax down through the ladder of X point subbands before being re-emitted into the base layer [37]. Note that lateral hot electron transport and hot electron noise measurements as well as hot electron electroluminescence spectroscopy can also be used to study intervalley scattering rates [38, 39].

5 Hot electron spectroscopy and microscopy 1867 Finally, Lyon et al [40] introduced ballistic electron luminescence spectroscopy (BELS), which involves the injection of ballistic electrons into a p-type semiconductor at low temperature together with the measurement of the luminescence produced when the electrons recombine with neutral acceptors. This technique was successfully utilized for ballistic electron transport in GaAs, in order to observe multi-phonon scattering of high energy electrons, and to determine band offsets of heterojunctions. Later, similar experiments were also realized using a STM as an emitter for ballistic electrons [41, 42] Hot electron transistors Besides the aspect of utilizing hot carrier transistors to study hot carrier effects in their base and transit regions, the technique of hot electron spectroscopy gained much interest in the late 1980s to exploit the knowledge about quasi-monoenergetic, ballistic electron transport in order to study tunnelling structures like resonant tunnelling diodes and superlattices. The first attempt to incorporate a resonant tunnelling structure into a HET as an analyser was reported by Capasso et al [43] in In this work, a double-barrier heterostructure is implemented in a p-i-n heterojunction where the emitter consists of a wide-gap p + layer. This device concept allows information on the hot electron energy distribution to be obtained directly from the measured resonant tunnelling collector current, without requiring the use of derivative techniques as requested in conventional hot electron spectroscopy. Later, Choi et al [44] realized this device concept on the basis of the unipolar THETA device. The differential conductance G = di/dv E showed features which were attributed to coherent tunnelling through the double-barrier structure. A second device, incorporating a single barrier, was used to measure the shape of the hot electron energy distribution. In a subsequent paper the authors presented a model which allows one to calculate the energy distribution of the hot electron beam, taking optical phonon emission and plasmon emission into account [45]. They found that the relative contribution of these two mechanisms depends on the base doping density N D. England et al [46, 47] presented the first HET incorporating a GaAs/AlGaAs-superlattice between the electron injection barrier and the collector barrier. The superlattice was heavily doped in order to be used as the base contact. Using this device the authors observed weak features in the injector tunnelling characteristic which were attributed to hot electron transport through superlattice minibands. In 1990, Vengurlekar et al [48] presented a study of hot electron transport through shortperiod InP/InGaAs-superlattices using an n-p-n bipolar transistor. Using the same device, Beltram et al [49] reported scattering induced tunnelling of hot electrons through biased superlattices in 1990, too. At a fixed injection energy of the hot electron beam, current resonances appear for all alignments of Wannier Stark states with the electron injection energy. In 1992, Kuan et al [50] demonstrated that the hot electron distribution of an electron beam can be used as a probing tool to determine the band structure of a semiconductor superlattice. The results fit quite accurately to miniband positions calculated using a transfer matrix method. Inspired by this work, Rauch et al [51] introduced a three-terminal device, based on the THETA device of Heiblum et al [27], which performed hot electron spectroscopy with an energetic resolution of 25 mev. Using this device, several experiments have been carried out successfully, like mapping superlattice miniband positions and widths [51], quenching of miniband transport due to external electric fields [52, 53], the observation of coherent and incoherent electron transport in undoped GaAs/AlGaAs-superlattices [54, 55], and ballistic transport in magnetic fields [56].

6 1868 J Smoliner et al In a complementary experiment, Petrov et al [57] used BELS [58] to map the energy positions of minibands in short-period superlattices. The luminescence results provide precise measurements of the ballistic electron energies, while the electrical transfer ratio is sensitive to the peaks and valleys in the transmission through the miniband. The combination of optical and hot electron spectroscopy eliminates the uncertainty in translating applied voltages to electron energies, which occurs in semiconductor tunnelling experiments. HETs are also interesting for practical applications. On InP, Miyamoto et al [59] presented HETs with a buried metal gate for high speed applications. The highest estimated cutoff frequency of their devices is approximately 1 THz. Magnetic applications can also be realized with HET devices. The unique properties of magnetic tunnel junction (MTJ) devices have led to the development of an advanced, high performance, non-volatile magnet random access memory (RAM) with a memory cell density approaching that of dynamic RAM and read write speeds comparable to static RAM. Both, giant magnetoresistance (GMR) and MTJ devices are examples of spintronic materials, in which the flow of spin-polarized electrons is manipulated by controlling the orientation of magnetic moments in magnetic thin film systems via external magnetic fields. Latest results on three-terminal hot electron magnetic tunnel transistors suggest that there are also other promising applications of spintronic materials [60 62]. Finally, infrared detectors can also be realized on the basis of HETs. For instance, Yao and co-workers [63] reported an infrared HET (IHET) with extremely high detectivity, D > cm 2 Hz/W at 4.2 K STM-based hot electron microscopy/spectroscopy of heterostructures Possibly motivated by the promising first results of device-based hot electron spectroscopy, it was also tried to combine the benefits of hot electron spectroscopy with the outstanding spatial resolution of a tunnelling microscope. As a result of these attempts, ballistic electron emission microscopy/spectroscopy (BEEM/S) was introduced by Bell and Kaiser in 1988 [64, 65]. BEEM/S is a three-terminal extension of STM, and utilizes an STM tip to inject hot electrons into a semiconductor via a thin metallic base layer. In this way, an energetically tunable electron injector is obtained, which in principle can be used on any semiconductor sample for spectroscopic purposes. On specially designed samples incorporating an avalanche diode, even single electron sensitivity can be achieved [66]. The functionality of STM-based BEES is the same as in device-based hot electron spectroscopy, but with higher flexibility in the material systems used, and including imaging capabilities. However, the advantage of spatial resolution has to be paid with a reduced energetic resolution, as we will show later. Initially, BEEM has been used to conduct Schottky barrier and band structure characterization in various material systems, including technologically important semiconductors such as Ga(Al)As [67 69], Ga(In)P [70 72], Ga(As)N [73 75], Si [76, 77] and SiC [78]. A recent example for a band structure investigation with BEEM was given by Reddy et al [79]. They investigated GaN x P 1 x samples for various nitrogen concentrations between zero and 3.1%. For x 0, they observed a fine structure in the ballistic spectra, which depended on the nitrogen concentration. This phenomenon was interpreted as a splitting in the degeneracy of the X valley due to the nitrogen induced, intense perturbation of the GaP lattice. Another huge area of applications for BEEM was opened up by the investigation of semiconductor heterojunctions. For instance, the band offsets of various semiconductor semiconductor interfaces have been measured [69, 80]. Additionally, the transport

7 Hot electron spectroscopy and microscopy 1869 behaviour through single barriers [81, 82], resonant tunnelling double barriers [83, 84] and superlattices [85, 86] have been studied extensively. The first report of the direct detection of quantum confined states in buried AlGaAs GaAs AlGaAs double-barrier resonant tunnelling diodes (DBRTDs) was given by Sajoto and co-workers [83]. Quasi-bound states and band structure effects were described in dependence of the temperature using cryogenic BEEM for temperatures in the range of K. The measured BEEM thresholds were found to be in good agreement with the energetically favoured transmission levels yielded by calculations. However, theoretical studies (e.g. [88]) as well as tunnelling and BEES experiments on Al(Ga)As-based DBRTDs show, that a simple description using merely the conduction band edge of the heterostructure is not always sufficient to explain fully the transport properties in BEEM experiments. As a consequence, intensive theoretical studies were carried out. The latest ones can be found in [89 92]. Especially on GaAs-based samples it was found, that the influence of higher valleys cannot always be neglected. The first tunnelling experiments on this topic were carried out by Mendez and co-workers [93, 94], who observed resonant tunnelling via X point states in AlAs GaAs AlAs and AlGaAs GaAs AlGaAs heterostructures. Later, a DBRTD in the X valley band profile of an AlAs GaAs AlAs GaAs AlAs structure was demonstrated by Shieh and Lee [95]. With BEEM, the positions of higher conduction band minima had already been determined on bulk GaAs as well as on single AlAs barriers, before the general interest focused on the ballistic transport behaviour of RTDs on the GaAs/AlGaAs [64, 69, 96 99], but also on the InAs/AlSb material system [100]. With further progress in the fabrication of nano-structured devices also one- and zerodimensional structures became available, i.e. quantum wires and quantum dots. For the investigation of such structures it is even more important to have a local probe technique with nanometer resolution. For instance, Eder et al [101] and Rakoczy et al [102] studied quantum wires fabricated by optical lithography. To demonstrate that BEEM is a powerful tool for the investigation of laterally patterned structures, they used GaAs/AlGaAs heterostructures and superimposed a lateral pattern of 200 nm wide wires with a pitch width of 800 nm. Finally, the semiconductor was partially removed by wet-chemical etching to form the protruding wires. BEEM images depicting the wires in good quality could be obtained in this way. A different source for low-dimensional structures is provided by self-assembled quantum dots. A quite elegant way of producing such quantum dots is provided by high strain epitaxy, e.g. on InAs/GaAs [103,104] and on Ge/Si(001) [105]. Under certain growth conditions, a thin layer of material deposited on a substrate with a high lattice mismatch will undergo a process of self-organization and form so-called self-assembled quantum dots (SAQDs). This method is capable of delivering large arrays of quantum dots with a quite narrow size distribution and a reproducible (opto)electronic behaviour an essential prerequisite for the development of quantum dot lasers [106, 107]. To mention another example, (self-assembled) quantum dots can be used to fabricate single electron transistors (see, e.g. [108]), which can, for instance, be exploited in high-density computer memories. As quantum dots are currently a field of high common interest, they are also studied by BEEM. On the InAs system Rubin et al [109] took BEEM images of InAs dots buried under a 75 Å thick GaAs cap layer and obtained a good contrast between the dots and off-dot regions. Rakoczy et al [110, 111] studied the band offset between InAs on surface dots and the GaAs underneath. For GaSb self-assembled quantum dots on GaAs [ ], the type II band lineup results in a barrier in the conduction band instead of a potential well as in InAs dots. As this results in spatially indirect transitions in optical experiments, reliable information on the band offset cannot be obtained by optical methods. Traditional transport experiments can

8 1870 J Smoliner et al also not be used. Therefore, BEEM is the only experiment providing local information on the conduction band profile of GaSb dots. Other material systems of interest are phosphide related quantum dots. They are especially interesting, since phosphide materials are widely used for optical applications in the visible spectrum [ ]. Again, BEEM is extremely useful to determine the band offsets on the dots as well as higher energy levels within the dots. Therefore, BEEM yields information on the expected optical properties of these dots. Quantum dots on Si were also studied by BEEM. As an example the work of Klemenc et al [118] shall be mentioned here, which is concerned with BEEM on Ge self-assembled dots embedded in Si. Klemenc and co-workers used MBE grown Ge dots on a Si(100) substrate, which were capped with a 10 nm thick epitaxial Si layer and an additional 3 nm thick layer of CoSi 2. They found that the inhomogeneous strain causes a change in the electronic surface structure, resulting in a lowering of the surface band-gap on top of the buried Ge dots. Detection of these dots by BEEM was possible, although the STM images yielded no visible surface contrast for the two-fold buried dots. Another field, where BEEM shows its strength in spatial resolution is the investigation of buried dislocations and defects. Here, several material systems were investigated, e.g. InGaAs/GaAs heterostructures [119, 115], which are interesting for pseudomorphic high electron mobility transistors, but also silicide systems like NiSi 2 [120, 121] and CoSi 2 [ ]. BEEM was also applied successfully to metal insulator semiconductor (MIS) structures, to study the transport related insulator properties on a microscopic scale, for instance, in buried CaF 2 /Si [130], SiO 2 /Si [131] and also in aluminium oxide junctions [132, 133]. BEEM was further used to shed light on quantum interference effects and the distribution of trapped charges in SiO 2 -based metal-oxide-semiconductor structures (MOS) [ ]. Another interesting application of BEEM was presented recently by Kurnosikov et al [137], who performed ballistic transport studies of the barrier properties of tunnel junctions grown without any auxiliary Schottky barrier. They used Co Al 2 O 3 Ru tunnel junctions as BEEM samples and measured an effective barrier height of 1.7 V. Further they observed, that a continuous current injection into a single point of the junction increased the local barrier height. As possible causes for this phenomenon charging effects and degradation of the barrier structure were suggested. Additionally, they found first evidence of pinholes directly showing in a BEEM image by increased transport through these areas. With the increasing interest in the field of spintronics, the magnetic properties of Co Cu thin films [138] and magnetic nanostructures [139, 140] were studied by BEEM. In the literature, this method is sometimes referred to as ballistic electron magnetic microscopy (BEMM). Under ultrahigh vacuum conditions it was demonstrated by Lu et al [142] that it is possible to image magnetic domains with nanometer resolution In the following, other groups used this technique to determine hot electron attenuation lengths in magnetic films [143]. Magnetic multilayers [144] and embedded ferromagnetic films were studied by BEEM, too. For a broader survey of the widespread activities in BEEM/BEES, the review by Narayanamurti and Kozhevnikov [3] is strongly recommended. 3. Basic principles of hot electron spectroscopy In hot electron spectroscopy, electrons with kinetic energies high above the thermal energy k B T ( hot electrons) are used to probe the electronic states of heterostructure samples. If the electrons stay unscattered during this process, they are often referred to as ballistic

9 Hot electron spectroscopy and microscopy 1871 A It Vt STMtip e- z y x Ic A metal base n-type collector Figure 1. Sketch of a BEEM set-up with a Schottky diode as a sample (not to scale). The tunnel voltage V t is applied between the STM tip and the metal surface of the sample, the tunnel current I t in this circuit is kept constant by the STM feedback loop. A second amperemeter measures the ballistic current I c between the metal base and the semiconductor collector. electrons. All sources of ballistic electrons reported so far utilize a three-terminal transistorlike configuration, where electrons tunnel from an emitter electrode into a thin base electrode. Between the base and the subsequent collector, another single barrier or a more complex barrier structure has to be placed in order to separate the base and the collector electrode. If the kinetic energy of the tunnelling electrons is high enough and the base is so thin that at least a certain percentage of the electrons can traverse the base without being scattered inelastically, those electrons are eligible to overcome the collector barrier and can be transferred ballistically into the collector region of the sample. Two configurations are typically used: one utilizes a STM to establish the injector structure (BEEM), while the other one uses MBE grown injectors, which are directly included within the sample under investigation (HET device). As already mentioned at the very beginning, the aim of this review is a critical comparison of both techniques in order to identify the strengths and weaknesses of hot electron spectroscopy using HET devices on the one hand and the STM based BEEM/S technique on the other hand. For this purpose, we have to discuss the basic principles of both techniques first BEEM/S The configuration using a STM as an emitter for ballistic electrons is frequently referred to as ballistic electron emission microscopy/spectroscopy (BEEM/BEES). STM with its spectroscopic and imaging capabilities has been a well-established measurement technique for many years now. Since a lot of textbooks exist on this topic, its principle is not described here. One can find an introduction to STM, for example, in [ ]. For BEEM/S, the sample has to have a conducting top (base) electrode. An additional contact on the sample backside acts as collector electrode for ballistic electrons. A typical sample is sketched in figure 1. Most BEEM samples are semiconductor structures with a well-conducting metal layer at the top acting as a base. A frequent choice for the base material is Au. To facilitate the measurement of the ballistic electrons, an n-type semiconductor has to be used as a collector. In this case, the natural band bending accelerates the ballistic electrons in the semiconductor away from the metal semiconductor interface and therefore prevents them from leaking back into the base. To record a BEEM spectrum, the STM tip is kept at a fixed xy-position while the tunnel voltage is varied within a chosen interval and the ballistic current in dependence of the tunnel voltage, I c (V t ), is measured at this position in constant current mode.

10 1872 J Smoliner et al vacuum barrier Vt Ez Ef evt STMtip evb base e- Ic collector Ef Vb Ic z Figure 2. Left part: schematic energy diagram of a BEEM experiment on a Schottky diode. Electrons tunnel from the STM tip into the base. If ev t >ev b, electrons will be able to surmount the Schottky barrier and enter the semiconductor. Right part: ballistic current spectrum I c (V t ) corresponding to the energy diagram on the left (for varying V t ). Defining zero by the Fermi level of the base, the current sets in at V t =V b. The conduction band profile of a simple Schottky diode sample is shown in figure 2, together with the corresponding ballistic current. Due to the applied tunnel voltage, with the tip under negative voltage with respect to the base, the potential for electrons lies much higher in the STM tip than in the base. Thus, electrons entering the base by tunnelling through the vacuum barrier have energies high above the Fermi level in the base metal, i.e. they are so-called hot electrons. As long as the energy provided by the tunnel voltage, ev t, is smaller than the Schottky barrier height, no electrons will be transferred into the semiconductor, thus the ballistic current is zero. Gradually increasing V t finally leads to a situation where some of the injected electrons can surmount the Schottky barrier and progress through the conduction band of the semiconductor. This causes a measurable ballistic current I c at the collector electrode. The higher the tunnel voltage, the more electrons will contribute to I c. Keeping the position of the STM tip constant in xy with respect to the sample surface and varying the tunnel voltage in a certain interval while recording the corresponding ballistic current represents the spectroscopic part of BEEM, often also explicitly denoted as BEES. The plot of I c (V t ) is usually called BEEM spectrum. Note that in the graph for the ballistic current depicted in figure 2 the axes are swapped compared to the usual representation, to illustrate more clearly the connection with the energy diagram on the left Hot electron transistors If the use of a STM as a hot electron emitter is not possible or desired, purely device-based hot electron emitters can also be realized. Such HET devices are powerful tools to study nonequilibrium electron transport in semiconductor bulk and heterostructure material systems. In figure 3 the conduction band profile of a tunnelling hot electron transfer amplifier (THETA) is shown. The picture was adapted from [22] and represents the standard design of an allsemiconductor unipolar HET. Between the emitter and the base a thin layer of a wide-gap material acts as a tunnelling barrier. By applying a negative emitter voltage electrons are

11 Hot electron spectroscopy and microscopy 1873 Energy (ev) E f Emitter Al x Ga 1-x As Injection Barrier I E Base I B Al x Ga 1-x As Collector Barrier Distance (nm) I C Collector Figure 3. Conduction band profile of a typical THETA device under forward bias. Quasimonoenergetic electrons are injected into the base region of the device. Depending on the collectorbase bias V CB and the emitter base bias V EB, electrons either reach the collector contact or relax down to the base contact. injected into the base region of the device to form a quasi-monoenergetic ballistic electron beam. The barrier placed between the base and the collector is used as a spectrometer, which also prevents thermal electrons in the base from flowing into the collector. While traversing the base region the injected electrons undergo different transport mechanisms. Electrons which have been scattered at least once and thus have suffered energy losses or direction changes modify the energy distribution of the hot electrons and can contribute as quasi-ballistic electrons to the collector current. By analysing the hot electron energy distribution, it is, therefore, possible to study ballistic and quasi-ballistic electron transport in the base region of the device. It should be pointed out, that the situation is completely different in the BEEM configuration. In the BEEM configuration with its metallic base layer, there is an additional criterion that only electrons under a very small angle of incidence will be allowed to enter the semiconductor (see section 4.1). Any scattering process of an electron within this acceptance cone will change its angle of incidence in a way so that it can no longer enter the collector electrode, and is therefore lost. As a consequence, the observed energetic distribution of electrons entering the collector will be close to the theoretically predicted distribution for genuinely ballistic electrons. However, the total amount of electrons contributing to I c will be very small. Unlike in BEEM measurements, the collector current in HET devices measured as a function of the emitter base bias V EB is not very useful for data analysis in hot electron spectroscopy. The reason for this lies in the nature of the STM, which keeps the tunnelling current constant, regardless of the applied bias. Thus, the emitter current stays always the same. In HET devices this is not the case. Here, the tunnelling current through the emitter barrier exponentially depends on the emitter bias. To get comparable results with BEEM data, a transfer ratio α = I c /I E has, therefore, to be defined. Further, it can be shown that the portion of electrons arriving at the collector with a vertical component of the kinetic energy (E z ) equal to the collector barrier height is given by the differential transfer ratio α = di c /di E. This technique, which appears in literature as hot electron spectroscopy or ballistic electron spectroscopy, was mainly used to demonstrate quasi-ballistic [20] and ballistic electron transport [23], as well as to probe scattering mechanisms [27], nonparabolicity [25] and intervalley transfer [24] in the base region of the device.

12 1874 J Smoliner et al (a) Au v Au xy e v Au z y Au-GaAs interface v GaAs xy v GaAs z e GaAs z (b) Au e e y Au-GaAs interface GaAs z Figure 4. (a) Electron refraction: by crossing the interface from Au to GaAs, the electron will gain velocity parallel to the interface and lose velocity perpendicular to it. (b) Total reflection: If the incident angle with respect to the z-axis is larger than the critical angle, the electron cannot enter the GaAs and is reflected. 4. Ballistic currents and transfer ratios in BEEM and HETs We now consider the ballistic collector currents in BEEM and HET devices in a more quantitative way. Most non-atomistic models used to calculate the ballistic current are based on the assumption, that during the transition through an interface the momentum parallel to the interface, k xy, is conserved. Already in the initial work of Bell and Kaiser [64, 65] the conservation of the momentum parallel to the interface was assumed in order to derive a model which described the experimental data. Although the parallel momentum conservation law, as it is often called, was initially derived for an idealised quantum mechanical system, it soon became clear, that this conservation law is in fact valid for the vast majority of actually conducted BEEM experiments [ ] as well as for most experiments with HET devices Electron refraction effects in BEEM The fact that the xy-momentum is conserved during the passage through the interface has farreaching implications in BEEM. Typical BEEM samples exhibit a huge potential step between the base and the collector (caused by the Schottky barrier) as well as strongly different effective masses in the involved areas. As a consequence, electron refraction effects are observed, which are described in detail below. Note that for refraction effects the influence of the potential step is usually even more significant than the mass difference (see (5)). In contrast to BEEM, the effective masses are more or less constant in HET devices and also the large potential step caused by the metal semiconductor interface is missing. Therefore, electron refraction effects are not an issue in HET devices. As a typical example for electron refraction effects, figure 4 illustrates the situation at the Au GaAs interface. The effective electron mass in the gold is equal to the free electron mass, m 0, while the effective electron mass in the semiconductor, m, is considerably smaller (for GaAs: m = 0.067m 0 ). To quantify the consequences of this situation, we first write down the energy of the motion parallel to the interface: E xy = E(k xy ) = h2 k 2 xy 2m(z) = h2 ( k 2 2m(z) x + ky 2 ). (1) Using the relation given in equation (1) for the energy associated with the motion in the xy-plane and the momentum conservation law for k xy, the energy component parallel to the

13 Hot electron spectroscopy and microscopy 1875 interface changes according to: E GaAs xy = E Au xy m 0 m. (2) Another limitation is the conservation of the total energy E, which can be written as: E = h2 k 2 2m(z) + E pot = h2 2m(z) (k2 x + k2 y + k2 z ) + E pot = E xy + E z + E pot. (3) Therefore, the energy associated with the movement vertical to the interface is given by: E GaAs z E GaAs z = E E GaAs xy = E Au z E b = E 0 + E Au z + E Au xy m 0 EAu xy m E b, (4) ( Exy Au m0 ) m 1 ev b0, (5) where ev b0 = E b E 0 is the height of the potential step at the interface, with E 0 the conduction band minimum in the metal and E b the conduction band minimum in the semiconductor. The first subtrahend in this expression represents the decrease in E z due to the change of effective mass, while the second one just originates from the potential step. For the particular case of an Au GaAs interface, where the effective mass in the GaAs is just 6.7% of the electron mass in the Au layer, Exy GaAs is almost 15 times higher than the energy parallel to the interface in the gold. Electrons crossing the Au GaAs interface will gain E xy according to equation (2) and lose E z according to equation (4), Therefore, they will be refracted away from the z-axis, as depicted in figure 4(a). Only those electrons which have exactly a perpendicular angle of incidence at the interface, i.e. k xy = 0, will not undergo any refraction. A direct consequence of this refraction away from the z-axis is, of course, the possibility of total reflection at the Au GaAs interface. Considering the configuration of a BEEM experiment, it is obvious, that only electrons with kz GaAs 0 can travel through the semiconductor and finally be collected at the backside of the heterostructure, i.e. contribute to the ballistic current. This automatically gives an upper limit for k xy and, therefore, a maximum angle of incidence (with respect to the z-axis) for an electron in the base. To calculate this critical angle for total reflection, one can use equation (3) to isolate E z and take this in turn to find an expression for the momentum component vertical to the interface, k z : kz Au 2m0 = h 2 (E E 0) kxy 2, (6) 2m kz GaAs = h 2 (E E b) kxy 2. (7) From kz GaAs 0 and equation (7) it follows directly, that: k 2 xy 2m h 2 (E E b). (8) The angle of incidence with respect to the z-axis can be described by sin(θ Au ) = k xy k with k = k = 2m0 (E E 0 ) h 2. (9) By combining relation (8) and equation (9), the critical angle is determined by: sin 2 (θ crit ) = m m 0 E E b E E 0 = m m 0 E ev b0 E, (10)

14 1876 J Smoliner et al with E = E E 0, the total electron energy referring to the conduction band minimum in the gold base, and ev b0, as defined above, the height of the potential step. In a typical experimental situation where the material in front of the interface is represented by the base layer of the BEEM sample, it is more convenient to refer all energies to the Fermi level in the base layer rather than to the conduction band minimum. This can be achieved by the simple substitution of ev = E E f and ev b = ev b0 E f leading to a critical angle of: sin 2 (θ crit ) = m ev ev b, (11) m 0 ev + E f where ev b, now, is the usual Schottky barrier height and E f is the position of the Fermi level with respect to the conduction band minimum in the base layer. ev + E f is the total energy of the incident electron, e.g. provided by a tunnel voltage. Electrons are only able to cross the interface, if their angle of incidence, θ, fulfils the requirement: sin(θ) sin(θ crit ). (12) Electrons with a larger angle of incidence will be reflected at the interface, as depicted in figure 4(b). In a BEEM experiment, this condition defines the opening angle of the so-called acceptance cone for electrons in the metal film, which selects a fraction of those electrons, which have overcome the tunnel barrier, for further transmission into the semiconductor. Taking typical values for a BEEM experiment on an Au GaAs interface, one can see nicely the influence of the effective mass change: assuming a Schottky barrier height of 1 ev and an injection energy of 1.1 ev, the critical angle without considering the effective mass would be about 18. In contrast, taking the change in effective mass into account, the critical angle will be just 4.5. These small critical angles have a crucial effect on the lateral resolution of BEEM. Only electrons within the acceptance cone will determine the lateral resolution, because only those will be able to enter the semiconductor and therefore to contribute to the ballistic current. Assuming a point-like electron source at the top of the base layer, the minimal lateral resolution is determined by: x = 2d tan(θ crit ), (13) where d is the thickness of the base layer. Taking the critical angle calculated above, the lateral resolution will be 16 Å for a 100 Å thick base layer and 11 Å for a 70 Å thick one. For injection energies closer to the Schottky barrier height, the critical angle and, therefore, the lateral resolution will be even smaller. However, these numbers are, of course, only valid for the lateral resolution at the metal semiconductor interface. For buried structures, the lateral resolution will decrease with the depth of the buried feature. Note that the determination of the lateral resolution by the acceptance cone further has the consequence that scattering in the base layer will usually not deteriorate the lateral resolution. Scattering in the base will just lead to a general decrease of the ballistic current, because it will remove electrons from the acceptance cone by changing their angle of incidence. Finally, because it will be needed later, also the restriction on the initial energy component parallel to the interface shall be written down here explicitly. = h2 kxy 2 /2m 0 it immediately follows, that: E Au xy E Au xy From equation (8) and m (E E b ). (14) m 0 Expressing this in terms of Ez Au rather than in terms of the total energy, by using E = Exy Au +EAu z +E 0, and, again further replacing E b directly by the barrier height, E b = ev b0 +E 0,

15 Hot electron spectroscopy and microscopy 1877 one obtains: E Au xy m (Exy Au m + EAu z + E 0 ev b0 E 0 ), (15) 0 Exy Au m m 0 m (EAu z ev b0 ). (16) Or, again referring to the more common Schottky barrier height with respect to the Fermi level in the Au base, by substituting ev b0 = ev b + E f : Exy Au m m 0 m (EAu z E f ev b ). (17) 4.2. The Bell Kaiser model When introducing BEEM, Bell and Kaiser presented a formula for the modelling of the ballistic current, which has been widely utilized in BEEM/BEES since then [64]. Their first step in developing their model was to use the well-known formalism for tunnelling between planar electrodes as an approximation. For simplicity, the STM tip and the base layer are assumed to be identical metals. Further, heterostructures incorporated into the collector, are not treated by the original Bell Kaiser Model. At T = 0, electrons tunnelling from the STM tip to the metal base occupy tip states within a half-shell (because of the restriction k z > 0) of the Fermi sphere between E = E f and E = E f ev t. Within the framework of this model, the tunnel current can be written as [64, 151]: I t (V t ) = 2eA (2π) 3 d 3 kt tb (E z ) hk z m 0 [f(e) f(e+ ev t )]. (18) For convenience, here as well as in the text that follows, all energies are referring to the STM tip conduction band minimum. The BEEM sample is assumed to be energetically lowered with respect to the tip by a positive tunnel voltage V t applied to the base layer of the sample. A is the effective tunnel area, f the Fermi function and T tb (E z ) the tunnelling probability. Note that the expression hk z /m 0 is the velocity component parallel to the z-axis. Provided that the barrier between tip and base at V t = 0 is a square barrier of height and width s, and will be distorted into a trapezoidal shape by applying a tunnel voltage, the tunnelling probability given by the WKB model can be approximated by [152]: T tb (E z ) = e αs E f + (ev t /2) E z (19) with α = 8m 0 / h = ev 1/2 Å 1. Substituting the integral over the wave vector by integrals over the energy components associated with k xy and k z yields: I t (V t ) = C 0 de z T tb (E z ) 0 de xy [f(e) f(e+ ev t )] (20) with the constant C = 4πAm 0 e/h 3. A similar expression can be directly obtained for the collector current. However, due to the conservation of the total energy and the xy-momentum at the metal semiconductor interface, additional restrictions apply to those tip states which can contribute to the collector current. The first one originates from the requirement, that the electrons must have enough energy to surmount the Schottky barrier between the base layer and the semiconductor ballistically. Because here, as mentioned above, E z refers to the tip conduction band minimum and the

16 1878 J Smoliner et al conduction band minimum in the base layer lies at ev t with respect to the one in the tip, the minimum E z which facilitates a ballistic entry of the semiconductor is: Ez min = E f ev t + ev b. (21) The second limit has its origin in the refraction at the metal semiconductor interface and is given by equation (17). Again, one has to bear in mind that the energies now refer to the tip, and therefore: Exy max m = m 0 m (E z E f + ev t ev b ). (22) Using these two restrictions as integration limits, one can express the collector current as: I c (V t ) = RC E min z de z T tb (E z ) E max xy 0 de xy [f(e) f(e+ ev t )]. (23) R is a measure of the attenuation due to scattering in the base layer. The attenuation length for ballistic electrons in metals is in very good approximation energy independent within the energy ranges used in BEEM experiments for the measurement of Schottky barrier heights, and, therefore, R shall be treated here as a constant. To eliminate the prefactor C, which is usually unknown in an experiment, Bell and Kaiser [64] suggested normalizing I c (s 0,V t ) by I t (s 0,V t ) for each voltage V t. The expression for I c then takes the following form, commonly known as the Bell Kaiser formula: E de z I c (V t ) = RI min z T tb (E z ) Exy max 0 de xy [f(e) f(e+ ev t )] t0 0 de z T tb (E z ), (24) 0 de xy [f(e) f(e+ ev t )] where I t0 is the constant tunnel current at which the ballistic spectrum is measured. The Bell Kaiser formula is well suited for fitting ballistic electron spectra on simple Schottky diodes in a tunnel voltage range of up to 200 mv above the onset, i.e. ev b. Many authors use the Bell Kaiser model even up to tunnel voltages of 600 mv above the onset. However, for such large ranges usually a noticeable deviation has to be taken into account. At higher tunnel voltages numerous scattering processes occur and energy dependent influences become important Heterostructure extension for the Bell Kaiser model As first shown by Sajoto et al [83], BEEM can also be used to investigate buried heterostructures. To calculate the ballistic current through such structures, Smith and Kogan [153] introduced an extension of the initial description of the ballistic current. Essentially, their model is a modification of the original Bell Kaiser model and results in the simple extension of the Bell Kaiser formula (equation (24)) by an additional transmission coefficient T hs which describes the properties of the heterostructure [153]: Ez I c (V t ) = RI min t0 de z T tb (E z )T hs (E z ) Exy max 0 de xy [f(e) f(e+ ev t )] 0 de z T tb (E z ) 0 de xy [f(e) f(e+ ev t )]. (25) Besides T hs, all variables are defined just as in equation (24). The integration limits are again given by equations (21) and (22). Note that T hs does not only describe the transmission behaviour of the buried heterostructure, but must also include the quantum mechanical reflections in the region between the metal semiconductor interface and the buried heterostructure. The total coefficient T hs is usually calculated by a transfer matrix method, as described in section 4.4. It must be pointed out that the above formula only accounts for the coupling of the transverse and the longitudinal energy via the upper integration limit Exy max. While this is

17 Hot electron spectroscopy and microscopy 1879 not a problem for simple Schottky barriers, the proper choice of Exy max for samples with burried tunnelling structures is not straightforward. In our opinion, a much more instructive way to include this coupling, is to calculate the transmission through the BEEM sample, starting at the metal semiconductor interface, in dependence of both, E xy and E z : Ez I c (V t ) = RI min t0 de z T tb (E z ) 0 de xy T hs (E z,e xy )[f(e) f(e+ ev t )] 0 de z T tb (E z ). (26) 0 de xy [f(e) f(e+ ev t )] In this expression Exy max no longer has to be declared explicitly, because this limit is now implicitly included via the dependence of T hs on E xy. Note that the transmission coefficient of the vacuum barrier and the transmission of the heterostructure should not be combined into an overall transmission coefficient since this leads to quantum interferences between the vacuum barrier and the collector barrier when calculating I c explicitly. In principle, such interferences could exist, but in practice they are not observed in BEEM, because of the scattering processes at the non-epitaxial Au GaAs interface and the electron electron scattering in the Au-base layer. Further details on the calculation of T hs can be found in section 4.5. It should be mentioned that the prefactor R, which is usually used as fitting parameter, can be calculated quantitatively when metal semiconductor interface induced scattering processes are taken into account (MSIS model) [98]. Including the anisotropy of the effective mass, the energy dependence of the electron mfp in the metal base, and finite temperatures, BEEM spectra were calculated quantitatively with surprising accuracy within this model and without needing further fitting parameters [154] Transfer ratio analysis for HET devices Like in BEEM/S the basic idea in device-based hot electron spectroscopy is to utilize an electron beam with a narrow vertical energy distribution as a probing tool to study band structure and transport properties of heterostructures. For the data analysis, however, slightly different procedures are applied. Usually only the ratio between the collector and the emitter current is of interest (see section 2.3). This so-called static transfer ratio of a semiconductor heterostructure is given by the convolution [50]: α(e z ) = 0 F(E z E z)t (E z ) de z, (27) where F(E z E z) denotes the vertical energy distribution of the injected hot electron beam and T(E z ) the energy dependent transmission function of the heterostructure. Although this looks much simpler than in BEEM/S, this is not the case since inelastic scattering processes complicate the analysis in device-based hot electron spectroscopy. In contrast to that, inelastic scattering by LO phonons is usually not an issue for typical BEEM samples. The main reason for this somewhat surprising behaviour can be found in the phonon scattering time, which has been determined to be 185 fs [27] in undoped GaAs. Looking at typical heterostructure samples investigated by BEEM, like buried double-barrier resonant tunnelling diodes and superlattices [86,155,156], one can see that the classical electron transfer time through the active region of the corresponding samples is usually only up to 100 fs and is, therefore, well below the phonon scattering time. Thus, LO phonon emission will not occur in the active region of the sample. Phonon absorption is suppressed by the low temperatures, at which the experiments are usually carried out.

18 1880 J Smoliner et al Energy (ev) E f 0.1 Emitter Al 0.3 Ga 0.7 As Injection barrier Drift region I E Superlattice GaAs/Al 0.3 Ga 0.7 As Minibands E 0.0 f AlAs Etch Base Collector α=i c /I stop layer E Distance (nm) Figure 5. Conduction band profile of a HET incorporating a semiconductor superlattice between the base and the collector. The etch stop layer has just technological purposes and is necessary to establish the base contact to the device without generating a shortcut to the collector layer. I C In contrast to that, phonon scattering in device-based hot electron spectroscopy is almost inevitable. The reason for this is twofold: First, the electron kinetic energies used in devicebased hot electron spectroscopy are normally much lower than in BEEM, since one usually tries to stay near the edge of the conduction band and in HET devices the huge potential step between the base and the collector is usually missing. Second, the so-called drift region between the emitter barrier and the collector barrier is usually relatively long to avoid the formation of quantized states and quantum resonances between those barriers. Thus, the transfer times through the active region of HET devices easily reach values comparable to the phonon emission times. For instance, for the sample shown in figure 5, the electron transfer time is approximately 500 fs. To illustrate the strong influence of inelastic scattering in device-based hot electron spectroscopy, we briefly consider the following three-terminal device: figure 5 shows the conduction band diagram of a typical HET [51] incorporating a short-period superlattice. The superlattice in the collector region has 5 periods consisting of 2.5 nm AlGaAs barriers and 6.5 nm GaAs wells (results taken from [157]). The mechanism of electron injection equals that of the THETA device: a ballistic electron beam is generated at a tunnelling barrier and reaches the superlattice after traversing a thin, highly doped n-gaas base layer and a slightly n-doped drift region. By tuning the energy of the ballistic electron beam with the emitter bias V EB it is possible to probe the band structure and the transport properties of the superlattice at a fixed collector base bias V CB. The probability of an injected electron to be transmitted through the superlattice is given by the static transfer ratio α = I c /I E. The corresponding transfer ratio is shown in figure 6. The calculated position of the first miniband is indicated with dotted lines. For energies below the first miniband the ballistic electrons are reflected at the superlattice and relax down to the base contact. These electrons do not contribute to the collector current I c. The onset at V E = 45 mv indicates the beginning of electron tunnelling through the first miniband. Increasing the emitter bias leads to an increase in the transfer ratio up to a critical emitter bias where the maximum overlap between the hot electron energy distribution and the miniband is achieved. At higher emitter bias the electron beam is reflected at the minigap, which separates the first and the second miniband, and thus, the transfer ratio decreases.

19 Hot electron spectroscopy and microscopy 1881 Transfer Ratio α=i c /I E st Miniband Phonon Replica V E (V) Figure 6. Measured transfer ratio of a 5-period superlattice consisting of 2.5 nm AlGaAs barriers and 6.5 nm GaAs wells. The calculated position of the first miniband is indicated with dotted lines. The first peak is attributed to ballistic electron tunnelling through the first miniband. At higher emitter bias phonon replicas contribute as inelastic background. The following peaks in the transfer ratio are due to electrons which have been scattered by LO phonons while traversing the drift region and have lost n hω LO = n 36 mev (n = 1, 2) during the scattering processes. The transfer ratio does not drop to zero in between these peaks, due to the overlap of the peaks. This is due to the fact that in HETs longitudinal optical phonon emission is the most efficient scattering process at liquid helium temperatures. At energies above the phonon threshold (36 mev in GaAs), electrons which are injected into the drift region emit LO phonons and therefore reduce their kinetic energies by the amount of the phonon energy. In the measured transfer ratio these quasi-ballistic electrons contribute as inelastic background. Consequently, inelastic scattering plays a major role in HET devices. Therefore, the transfer ratio has to be written as a superposition of a ballistic part and an inelastic background formed by phonon replicas. In equation (27) this can be included by writing the hot electron energy distribution as a sum consisting of ballistic electrons and electrons, which lose energy by sequential phonon emission. The index i denotes the index of the emitted phonon. F = F ball + Fph i. (28) i In hot electron spectroscopy F ball is utilized as a probing tool to study electron transport properties in heterostructure-based resonant tunnelling structures like superlattices. The phonon replicas contribute as unwanted inelastic background. Therefore, it is important to have detailed knowledge about the shape and the width of the hot electron energy distribution The transfer matrix method In the description of BEEM spectra as well as of HET transfer ratios the quantum mechanical transmission factor of the considered heterostructures has to be calculated. The method that is normally used for this purpose is the so-called transfer matrix method (TMM) [158, 159]. In the following we outline the principles of this method including the modifications necessary to account for the electron refraction effects mentioned earlier. The transfer matrix method can be used to calculate the transmission of an arbitrary potential shape by relatively simple means. Because it is based on the approximation of an

20 1882 J Smoliner et al Vj V(z) z 0 z j-1 z j z n-1 z Figure 7. Approximation of an arbitrary potential profile by a piecewise constant potential. The jth section is located between z j 1 and z j. Note that the width of the single steps does not need to be equal for all subdivisions, but can be adjusted according to requirements. arbitrary potential by a piecewise constant potential, it is especially well-suited to calculate the transmission through such heterostructures which exhibit a band profile consisting of rectangles, e.g. DBRTDs and superlattices. For the following, consider a one-dimensional, arbitrary potential profile V(z). The given potential shape can be subdivided into several sections, where the potential and the effective mass are assumed to be constant (see figure 7). Each potential change between two such sections is taken as an ideal step function. Under these conditions, the Schrödinger equation for each of these sections can be written down and solved easily. In the next step, those partial solutions are connected together by using the continuity of the wave function and its derivative to match the wave functions at each sampling point. First, a band profile which is subdivided into n sections can be written as: n 1 { const for zj 1 <z<z j, V(z)= V j with V j = (29) 0 elsewhere. j=0 The analogous description applies to the z-dependent effective mass, where m j is the (effective) mass in the jth region. Using a stationary approach, the one-dimensional Schrödinger equation within the region j (i.e. for a constant potential and a constant effective mass) is: [ h2 2m j d 2 dz 2 + V j E z ] ψ(z) = 0. (30) It must be pointed out that we first consider electrons with zero parallel momentum and therefore with zero components of kinetic energy due to a motion parallel to the barriers. As in this case the energy depends on k z only, it is, therefore, denoted as E z. Electrons with non-zero k xy and electron refraction effects will be considered later in this section. The general solution ψ j in the region z [z j 1,z j ] is written as: ψ j (z) = A j e ik j z + B j e ik j z, (31) with k j = 2mj (E z V j ) h 2. (32)

21 Hot electron spectroscopy and microscopy 1883 The boundary conditions at the sampling point z = z j between the two adjoining sections j and j + 1 are: ψ j (z j ) = ψ j+1 (z j ), (33) 1 m j ψ j (z) z = 1 z=zj m j+1 ψ j+1 (z) z. (34) z=zj For convenience, the following abbreviations are introduced: u j (z) = A j e ik j z and v j (z) = B j e ik j z. Now, the relations between u j and v j on the one hand and u j+1 and v j+1 on the other hand can be calculated from the boundary conditions (equation (33) and (34)), which read as: u j (z j ) + v j (z j ) = u j+1 (z j ) + v j+1 (z j ), (35) ik j u j (z j ) ik j v j (z j ) = ik j+1 u j+1 (z j ) ik j+1 v j+1 (z j ). (36) m j m j m j+1 m j+1 Before we continue to describe the transfer matrix method, we have to discuss the influence of non-zero k xy values on the relation between k j and k j+1. If k xy is zero, k j and k j+1 are simply calculated by equation (32). For non-zero k xy, the mass induced coupling between the parallel and vertical components of energy has to be taken into account. For better understanding, we first split the total energy E into a vertical component E z and an energy component parallel to the barriers, E xy. Thus, the total energy is: E = E xy + E z. While k xy is always conserved, E xy is not. Calculating E xy in the regions (j) and (j + 1) we get: E xy,j = h2 k 2 xy 2m j, (37) E xy,j+1 = h2 kxy 2. (38) 2m j+1 As the total energy has to be conserved, this simply means that E z is no longer the same in the regions (j) and (j + 1). The relation between E z,j+1 and E z,j is given by E z,j+1 = E z,j + (V j V j+1 ) + (E xy,j E xy,j+1 ). (39) From equation (39), the relation between k j and k j+1 for non-zero k xy can now be calculated as earlier: 2mj (E z,j V j ) k j = h 2, (40) 2mj+1 (E z,j+1 V j+1 ) k j+1 = h 2. (41) Having the relations between k j and k j+1, both for zero and non-zero parallel momentum, we can now write equation (36) in matrix form: ( ) ( ) uj (z j ) = M (j) uj+1 (z j ), (42) v j (z j ) v j+1 (z j ) where M (j) = k j+1m j k j m j+1 1 k j+1m j k j m j+1 1 k j+1m j k j m j+1 1+ k j+1m j. (43) k j m j+1

22 1884 J Smoliner et al Note that the matrix M (j) does not depend on z. This information is solely contained in the vectors consisting of u and v. Because the potential and the mass between two sampling points are constant, the wave function in this area can be written as: ( uj (z j ) v j (z j ) ) = N (j+1) ( uj (z j+1 ) v j (z j+1 ) ), (44) where ( e N (j+1) ik j+1 z j+1 ) 0 = 0 e ik j+1 z j+1. (45) The matrix N (j+1) depends only on the distance z j+1 = z j+1 z j between two sampling points, not on the location of a sampling point itself. The absolute position z is, again, only contained in the vectors consisting of u and v. For the whole sequence of potential steps one can write: ( ) ) u0 (z 0 ) = M, (46) v 0 (z 0 ) ( un (z n ) v n (z n ) where M is the product matrix of all M (j) and N (j+1) : M = M (0) N (1) M (n 1) N (n). (47) Equation (46) links the waves incoming on the potential V(z) to the outgoing ones by means of a series of (2 2) matrices (equation (47)). The matrices M j perform the connection of two parts of the global wave function across an interface, while the matrices N j+1 describe the propagation within a region of constant potential and constant (effective) mass. Finally, to calculate the global transmission through the structure, one should demand, that v n = 0, which means, that on the outgoing side of the structure just a transmitted wave exists. With this, the global transmission can be simply described by T(E)= k n m 0 A n 2 k 0 m n A 0 = k n m k 0 m n M 11. (48) 2 Note that the transfer ratio of a barrier is defined as the ratio of the impinging and the transmitted electron currents. As the current is a vector, and for non-zero k xy the current is not impinging vertically onto the barrier, the ratio of the absolute values of the k-vectors, k = kz 2 + k2 xy, has to be used in equation (48). Further reading on the TMM and especially on its application to double-barrier structures, can be found, for instance, in [160, 161]. 5. Measuring the hot electron energy distribution in HETs 5.1. Energy analysers in HET devices The resolution of hot electron spectroscopy is directly related to the energetic width of the hot electron beam generated at the emitter tunnelling barrier of a HET. In order to get a better insight into the electron transport mechanisms in superlattices by hot electron spectroscopy [51,54] it is essential to resolve the individual states in a miniband separately. Therefore, the determination and optimization of the energy distribution in a HET device as well as in BEEM is of greatest importance. To determine the energetic distribution in device-based hot electron spectroscopy, a special HET device was designed, the conduction band profile of which is shown in figure 8. The layer

23 Hot electron spectroscopy and microscopy 1885 Figure 8. Conduction band diagram of a HET incorporating a highly optimized injector design between emitter and base, and a triple barrier RTD between base and collector as energy filter. structure starts with a highly doped n + -GaAs collector contact layer (N D = cm 3 ) and the GaAs/AlGaAs-triple barrier, which is embedded between two drift regions. The drift regions are slightly doped (N D = cm 3 ) in order to avoid undesired band bending. As collector barrier, a triple barrier GaAlAs resonant tunnelling diode was chosen, which acts as band pass energy filter and analyser for the distribution of ballistic electrons. An advantage of band pass filters is that the transfer ratio α = I c /I E, which resembles the energy distribution, is obtained without using derivative techniques [1]. The layer structure continues with a 10 nm thin, highly doped n + -GaAs base contact layer (N D = cm 3 ), a 64 nm undoped GaAs spacer, and the AlGaAs injection barrier, which we also use as an etch stop layer. On top of the AlGaAs layer a 6 nm, undoped GaAs spacer is grown, followed by a 80 nm n -GaAs layer (N D = cm 3 ). The layer structure ends with a 300 nm thick, highly doped n + -GaAs emitter contact layer (N D = cm 3 ). In this optimized structure, an energetically narrow and tunable electron beam is generated at the tunnelling emitter barrier and reaches the energy filter after traversing the highly n-doped GaAs base layer and the slightly n-doped drift region. Due to the transmission properties of the band pass filter the static transfer ratio α = I c /I E is directly proportional to the energy distribution of the injected hot electrons. For a better understanding, the design of the energy filter has to be discussed in more detail. During our work on double-barrier resonant tunnelling diodes, which was carried out outside the field of this review, we have found that the transmission behaviour of resonant tunnelling structures can be shaped by adding an antireflection coating for electrons [162]. The simplest version of a resonant tunnelling structure with antireflection coating consists of a triple barrier structure, as it is shown in the left inset of figure 9. The shown TBRTD consists of three Al 0.3 Ga 0.7 As barriers and two GaAs wells. The two well widths are chosen to be 4.2 nm in order to get a first resonant state (for the isolated well) at E 0 = 100 mev. To optimize the transmission properties of the analyser the central barrier (8 nm) is chosen to be twice as thick as the neighbouring barriers (4 nm). To illustrate the functionality of this antireflection coating, figure 9 shows a comparison of the calculated transmission coefficients T(E) of the triple barrier RTD (solid line) and a double-barrier RTD (dotted line) with the same full width at half maximum of 1 mev centred at E 0 = 100 mev. The transmission coefficients T(E)were calculated using a simple transfer matrix method including nonparabolicity. This transmission function of the triple barrier RTD

24 1886 J Smoliner et al Figure 9. Transmission coefficient of a triple barrier RTD ( ) and a double-barrier RTD (- ---) calculated using a transfer matrix method. The energy window of the TBRTD shows much steeper edges compared to that of the double-barrier RTD. Figure 10. Bound states in the drift region of the HET calculated using a Schrödinger/Poisson solver. shows much steeper edges compared to that of the double-barrier RTD, which makes it more suitable for the use as an energy filter. The rectangular shape of the transmission coefficient T(E)arises from the coupling of the two neighbouring wells. Due to tunnel splitting two quasibound resonant states are obtained (at energies E 0,1 = mev and E 0,2 = mev), which are delocalized over the whole structure and which are separated by 0.03 mev. The superposition of the two corresponding transmission peaks leads to the special shape of the transmission coefficient [162, 163] Parasitic quantum states in the base region of the HET device One very inconvenient property of HET devices is the existence of quantum states in the drift region between the emitter barrier and the heterostructure between the base and the collector. The drift region embedded between the tunnelling barrier and the triple barrier RTD represents a wide AlGaAs GaAs AlGaAs-quantum well. Calculating the bound states of this quantum well gives the discrete spectrum shown in figure 10. Due to the epitaxial interface between the base and the collector region, and the fact that electron electron scattering in the base even at

25 Hot electron spectroscopy and microscopy 1887 Figure 11. I/V characteristics of the emitter base subdevice. The inset shows a magnified part of the emitter current curve. highest doping levels will be much lower compared to the scattering rates in the Au-base layers normally utilized for BEEM, parasitic quantum interferences between the emitter and the base are almost inevitable. At emitter biases where the quasi-monoenergetic hot electron beam is aligned with a bound state of the drift region the emitter current is expected to be enhanced, whereas at biases where the hot electron beam lies energetically between two bound states I E is expected to be reduced. In the case where the energy distribution of the ballistic electron beam is narrow enough, those states can actually be observed experimentally as oscillations in the emitter current. In figure 11 we have plotted the measured emitter current as a function of the applied emitter bias V E for a typical sample. The emitter current shows an exponential dependence on the emitter bias, which originates from the roughly symmetric structure of the emitter base subdevice, where an AlGaAs tunnelling barrier is sandwiched between two highly doped contact layers. The inset of figure 11 shows a magnified part of the emitter current curve. In this plot a clear modulation of the current is observed, which is attributed to quantum interference effects occurring in the drift region of the HET [25]. However, if only the transfer ratio is considered, these parasitic interferences cancel out almost completely provided they are not too large. Note that the problem of quantized states between the emitter barrier and the collector barrier does not exist in BEEM. First, possible quantum interferences within the base do not occur, because the base is polycrystalline and therefore the scattering rates are high. Second, possible quantum interferences between the Schottky barrier and the buried heterostructure are suppressed, because the interface between the base and the GaAs is not epitaxial. Thus, the observation of quantum interference effects in the base or in the drift region is quite unlikely in BEEM experiments Transfer ratio of the triple barrier RTD To determine the transfer ratio of our HET device, the emitter and collector currents are measured as a function of negative emitter bias at T = 4.2 K using an HP-semiconductor parameter analyser. The spectrometer, implemented by the triple barrier RTD, is used under flat band condition (e.g. zero bias between base and collector). The experimental results are plotted in figure 12. Again, the modulation of the emitter current arises from electron coherence effects occurring in the drift region of the device.

26 1888 J Smoliner et al Figure 12. Emitter and collector current as a function of negative emitter bias. Weak modulations are due to quantum interference effects occurring in the drift region of the device. Transport of ballistic electrons and phonon replicas through the triple barrier RTD is indicated with arrows. Below the energy of the first resonant state no collector current is observed, since the electrons that are injected into the drift region are reflected by the spectrometer and collected in the base contact. This also indicates that no significant leakage current occurs between base and collector under flat band condition. The first peak indicates resonant tunnelling of hot electrons through the TBRTD. The following peaks in the collector current are due to electrons which have been scattered by longitudinal polar optical phonons while traversing the drift region and which have, therefore, lost multiples of hω LO during the scattering processes. The collector current shows a similar modulation effect as the emitter current. Since the collector current is the portion of the emitter current which has been transported through the spectrometer, this modulation is just transferred through the TBRTD. The transfer ratio α = I c /I E is shown in figure 13. The onset of the transfer ratio at V E = 100 mv indicates the beginning of hot electron tunnelling through the first resonant state of the analyser. The first peak of the transfer ratio is proportional to the injected hot electron energy distribution, and its full width at half maximum was measured to be Ɣ = mv. The shape of the distributions is slightly asymmetric with its maximum at the high-energy side. The peak position (V E,max = 107 mev) indicates the maximum of the vertical energy distribution. Note that the oscillations in the emitter and collector current due to the quantized states in the base region do cancel out in the transfer ratio. Besides the width of the vertical energy distribution it is essential to study its shape. Dividing the first peak at the maximum into two parts one can define a high energy tail and a low energy tail. In the transfer ratio the high energy tail occurs at less negative emitter bias as it reaches the energy filter first while increasing the absolute value of the emitter bias. The experimental results show that the width of the high energy tail is smaller than the width of the low energy tail: Ɣ 1 = 5 and Ɣ 2 = mev. The shape of the high energy tail is due to electrons generated at the emitter tunnelling barrier whereas the shape of the low energy tail is due to elastic scattering processes such as charged impurity scattering and electron electron scattering in the base of the device. The low-energy tail is calculated to decay exponentially. This decay is not completely observed in the measured transfer ratios due to the superposition of the ballistic peak with the inelastic background formed by the phonon replicas.

27 Hot electron spectroscopy and microscopy 1889 Figure 13. Measured transfer ratios as a function of negative emitter bias. The first peak resembles the hot electron distribution of the injected hot electron beam. 6. Measuring the hot electron energy distribution in BEES Due to the special design of the emitter structure in a HET device, the ballistic collector current increases when the bias between the emitter and the base is increased, whereas the energetic distribution of ballistic electrons stays virtually unchanged. Thus, the energetic distribution of ballistic electrons can be mapped by sweeping the emitter energy over the energy window of the analyser. In BEEM, which is operated in constant I t mode, this is not the case and the energetic distribution of injected electrons changes strongly with emitter bias. Thus, for a mapping of the energy distribution the analysing collector electrode has to be swept in energy, while the emitter bias stays constant. The principle of using a buried double-barrier resonant tunnelling diode (DBRTD) as an analysing filter was first demonstrated by Sajoto et al [83]. They found, that a resonant level gives rise to a linear increase in the ballistic current I c (V t ). As first pointed out by Smoliner et al [86], this phenomenon originates in the electron refraction at the Au GaAs interface. This linear behaviour was also confirmed by other groups in various experiments and is since then regarded as typical for buried RTDs. For instance, Strahberger et al [84] presented a detailed experimental and theoretical study on AlGaAs GaAs DBRTDs, showing the effects of the k xy momentum conservation on the transmission through buried and sub-surface RTDs, respectively. Smoliner et al [86] also demonstrated the effect of the electron refraction on the transmission through buried AlGaAs GaAs superlattices. Later, the transmission behaviour of buried superlattices under the influence of an external collector bias was investigated and several data points in the I c (V c ) relation were deduced from conventionally measured BEEM spectra (i.e. I c (V t )) for different values of V c [87]. However, a direct measurement of the refraction, by mapping the energetic distribution of the ballistic electrons after their transfer through the interface, is extremely difficult in BEEM for several technical reasons. Rather severe restrictions exist on the leakage currents and the mechanical and electrical drift stability of the measurement set-up. A special sample design was utilized, which allows both the application of a bias voltage between the base and the collector and the performance of an energy spectroscopy by utilizing an AlAs GaAs-AlAs DBRTD as a narrow energy analyser. The decision to use AlAs instead of AlGaAs barriers (which are more common in BEEM on the Ga/Al/As material system) was based on the experience that at

28 1890 J Smoliner et al Figure 14. Sketch of the experimental principle together with the Ɣ conduction band profile of a typical flat band AlAs GaAs-DBRTD sample. V t denotes the tunnel voltage, V c the collector voltage, I t the tunnel current, I c the ballistic current, and E f the Fermi energy. The resonant level of the double-barrier structure is schematically indicated by the horizontal bar within the RTD. Also illustrated is the decrease in the velocity component parallel to the z-axis from v Au z much smaller v GaAs z and the consequential energy loss in E z. to the cryogenic temperatures the ballistic transmission behaviour of an AlAs GaAs heterostructure is mainly determined by the Ɣ valley. This leads to a large barrier height of the AlAs barriers and, therefore, to good energetic filtering capabilities of the RTD combined with the advantage of low leakage currents. In figure 14, a sketch of the experimental set-up together with the Ɣ valley conduction band profile of the sample is shown. The resonant level within the RTD is indicated by a horizontal bar. As usual, a tunnel voltage V t between the STM tip and the sample surface (i.e. the metallic base) is used to inject electrons into the sample, whereby the tunnel current I t is kept constant during the measurements. Those electrons, which travel ballistically into the semiconductor heterostructure and can either energetically surmount or tunnel through the RTD are measured via I c. Thus, the onset in the ballistic current on this sample will be determined by the energetic position, where the Fermi level in the STM tip is aligned with the first resonant level inside the DBRTD. For the experiments presented here, the tunnel voltage was always chosen in such a way that the Fermi level in the STM tip, representing the maximum possible energy E z of any injected electron, is still well below the barrier height of the RTD. Therefore, only electrons with an E z suitable for a transmission through the resonant level of the RTD will be able to reach the collector and to contribute to I c. Consequently, the RTD acts as an energy filter in E z. Although the sketch in figure 14 does show the additional voltage source which was used during the experiment for biasing the sample, the band structure depicted corresponds to a collector voltage of V c = 0 V. Actually applying a collector voltage V c 0 means to shift the Fermi level in the collector region and leads to a tilt in the band profile. With this tilt also the energetic position of the resonant level is shifted, and, therefore, the energy filter can be adjusted to select a certain E z. By varying V c continuously and recording the corresponding collector current I c, one can scan the electron distribution in dependence of E z. Also illustrated in figure 14 is the impact of the electron refraction on the energy E z. As discussed in section 4, during the transition of electrons through the Au GaAs interface the momentum component k xy is conserved while the effective mass is changing. Since the

29 Hot electron spectroscopy and microscopy 1891 Figure 15. (a) Curve 1 (left axis): Ɣ valley conduction band profile of the sample used, calculated for V c = 0 V. Curve 2 (right axis): effective mass profile. The sample surface is on the left, z-values <0 indicate the region of the Au base layer. The collector region (on the right side) is not shown in full depth. (b) Transmission through the RTD shown in (a), calculated for k xy = 0, i.e. for electrons at the Ɣ point. momentum is the product of effective mass and velocity, a constant momentum component with a changing mass will lead to a change in the velocity component parallel to the interface, as indicated in figure 14. Because the effective mass decreases at the Au GaAs interface, the electrons will gain velocity perpendicular to the growth axis. This, in turn, means a gain in the energy component associated with the xy-component of the velocity, E xy. Because the total energy is, of course, conserved, the velocity parallel to the growth axis and the energy associated with it, E z, must change accordingly, i.e. E z will decrease. One can show that the z-associated component of the energy in the GaAs is given by (see (5)): ( Ez GaAs = Ez Au Exy Au m0 ) m 1 ev b0. (49) Note that the first subtrahend in this expression represents the decrease in E z due to the change in effective mass, while the second one originates from the potential step between the Au and the GaAs, which, via this term, also contributes to the refraction. In summary this means, that a conversion of E z into E xy takes place at the Au GaAs interface. As a final consequence, those electrons which initially have too high a value of E z to be transmitted through the resonant level of the RTD can lose enough E z at the Au GaAs interface so that they finally match with the position of the resonant level. Of course, those electrons which have exactly a perpendicular angle of incidence at the interface, i.e. k xy = 0, will not undergo any refraction. To illustrate the sample design in a more quantitative way, curve 1 in figure 15(a) shows the conduction band profile in the Ɣ valley of the heterostructure, including also the region of the gold base layer. It was calculated using a program by Snider [164], which is based on a self-consistent solving of Poisson s equation. As one can see clearly, the region between the Au GaAs interface on the one hand and the δ-doping (at z = 750 Å) on the other hand is virtually flat. Behind the δ-doping starts a gradual potential decrease down to the level of the collector region. In curve 2 of figure 15(a), the size of the effective electron mass is plotted along the growth axis. In the metallic base layer the effective mass is equal to the free electron mass, while in the semiconductor region it is much smaller (0.067m 0 in GaAs, 0.162m 0 in AlAs). In figure 15(b) we have plotted the calculated transmission through the Ɣ band profile of the semiconductor heterostructure for electrons with k xy = 0, i.e. with an angle of incidence

30 1892 J Smoliner et al Figure 16. Ballistic spectra for different collector voltages. T = 10 K, I t = 2 na. Curves 1 5 were measured at V c = +50 mv, 0 mv, 50 mv, 100 mv, and 150 mv, respectively. A vertical offset is added for better viewing. Line a indicates the shift of the lowest resonant level in the Ɣ valley, line b the one of the L valley resonant levels. (Note that for technical reasons V t is the voltage at the sample base.) which is perpendicular to the interface. The calculation was carried out employing the transfer matrix method (TMM) within the parabolic band effective mass approximation, as described in section 4.5. The first resonant level of the RTD is represented by a quite sharp peak, which lies at an energy of 1.2 ev, that means 200 mev above the GaAs conduction band edge. The next resonant level has its maximum transmission approximately 900 mev above the GaAs conduction band edge, and is, therefore, well out of the range of interest for this experiment. Although this second peak is, mainly due to its close vicinity to the top of the AlAs barrier, noticeably broadened, a significant increase in the calculated transmission does not appear until 1.85 ev. This excellent energetic isolation of the first resonant level together with the possibility of applying an external bias voltage V c to the heterostructure makes this sample design very well suited as a tunable energy filter for ballistic electron emission spectroscopy. Note, as we have shown in our previous work [156], that the Ɣ band profile is indeed the relevant one for the BEEM experiment presented here and the amount of electrons coming from an overshoot of the AlAs X valley barrier can be neglected at low temperatures. The only visible feature in the measured BEEM spectra at higher V t is a second current onset which is due to the resonant levels existing in the L valley of the DBRTD. However, this feature is also energetically well isolated from the first resonant level in Ɣ and, moreover, less pronounced than the typical resonance of a level in the Ɣ valley Influence of V c on the BEEM Spectra We now study the dependence of the BEEM spectra on the collector voltage. As discussed earlier, applying an external voltage to the collector will tilt the band structure and therefore shift the energetic position of the resonant level. On the other hand, the onset of the ballistic spectrum is determined by the energetic position, where the Fermi level in the STM tip is aligned with the first resonant level inside the DBRTD. Therefore, the onset in the BEEM spectra will shift accordingly to the applied collector voltage. In figure 16, the influence of the external collector voltage V c is illustrated by showing BEEM spectra for different values of V c. Line a in the figure is a guide to the eye to illustrate the onset shift more clearly. For positive values of V c the onset voltage in the BEEM spectra is shifted to lower values of V t, because in this case the resonant level is lowered in energy.

31 Hot electron spectroscopy and microscopy 1893 Figure 17. Ballistic current measured as a function of collector voltage V c for a constant tunnel voltage V t. T = 10 K, I t = 2 na. Curves 1 4 were measured at V t = 1.05 V, 1.10 V, 1.15 V, and 1.20 V, respectively. A vertical offset is added for better viewing. The straight line is a guide to the eye to indicate the current onset positions. Because negative collector voltages correspond to higher energetic positions of the resonant level, the horizontal axis is plotted from positive to negative values of V c. For negative values of V c the energetic position of the resonant level rises and therefore the current onset is shifted to higher emitter bias values. Though it is more difficult to perceive, because the onset for the transport through resonant levels in the L valley RTD is only weakly pronounced, the resonant levels existing in the L band do shift parallel to the Ɣ resonant level, as expected (line b in figure 16). From the amount of the onset shift one can determine which sample regions are affected by the application of a collector bias V c. As described earlier, the total thickness between the sample surface and the collector is 1250 Å, while the DBRTD itself lies 600 Å below the surface. Thus, if the applied collector voltage dropped uniformly over the whole sample, the energy levels inside the RTD would be shifted by approximately ev c /2, where e is the electron charge. However, the data in figure 16 indicate that the onsets of the spectra measured at V c = +50 mv and V c = 150 mv are shifted by about 170 mv with respect to each other. This suggests, that the main part of the applied collector voltage does actually drop over the region between the sample surface and the DBRTD, while the region between the δ-doping and the collector plays just a minor role for the biasing Dependence of the ballistic current on V c at constant V t For a direct mapping of the energy distribution of the ballistic electrons the ballistic current is measured while the tunnel voltage V t is kept constant and the collector voltage V c is swept over a certain voltage range. As already mentioned above, varying V c shifts the resonant level in the DBRTD in its energetic position E z and, therefore, can be used to scan an energy range, detecting the number of transmitted electrons in dependence of E z. In contrast to superlattices, the transmission coefficient of an RTD remains almost unchanged when a collector bias is applied. Thus, for a constant tunnel voltage, the ballistic current measured as a function of collector voltage directly represents the energetic distribution in E z of the ballistic electrons in the GaAs. In figure 17 the result of this measurement is shown. As one can see, for high energetic positions of the resonant level, i.e. for a large negative collector bias, the tunnel voltage is not sufficient to exceed the resonant level and, therefore, to cause a signal in the I c (V c ) curves.

32 1894 J Smoliner et al Figure 18. Calculated ballistic current (more precisely: (di c /de z ) E z ) as a function of E z for a constant tunnel voltage V t = 1.15 V. (a) Energy distribution of the ballistic electrons in the Au base (within the acceptance cone). (b) Energy distribution of the ballistic electrons in the GaAs. Decreasing the energetic position of the resonant level, i.e. going to more positive values of V c, leads to a detectable ballistic current as soon as the resonant level is aligned with the Fermi level in the STM tip, which is defined by the tunnel voltage. Thus, the onset in an I c (V c ) curve depends on the specific tunnel voltage used for the recording of the curve. This is illustrated in figure 17 by a comparison of curves for different V t. As expected, higher values of V t shift the current onset to more negative values of V c. Following the line of a single I c (V c ) curve again, the ballistic current increases for more positive values of V c and finally saturates. Due to the limited range available in V c just the beginning of a saturation can be seen in the graph. To provide a comparison with the experimental data and to facilitate a more detailed discussion, the amount of the ballistic current which flows in a small, fixed energy interval, (di c /de z ) E z, was also determined by calculation. As already mentioned above, this quantity reflects the energetic distribution of the ballistic electrons as a function of E z. In the experiment, this corresponds directly to the I c (V c ) curves, where the energy interval E z is provided by the width of the resonant level of the RTD. For the calculation, an extended Bell Kaiser model [64, 65], which fully includes the description of the electron refraction [86, 153], was used (see section 4.3). To keep the computing time within reasonable limits, all scattering effects in the metal base as well as in the GaAs region in front of the DBRTD were neglected. To allow a comparison with the experiment over the whole range of interest, a tunnel voltage of 1.15 V and a scanned energy range from 0.9 to 1.2 ev were chosen as input parameters for the model. In figure 18 one can see the results of the calculation, both for the electron distribution in the gold base and for the one in the GaAs, just after crossing the Au GaAs interface. In figure 18(a) one can see the current distribution from those electrons in the gold base which are able to enter the semiconductor, that is, the electrons within the so-called acceptance cone. Due to the influence of the transmission coefficient of the vacuum barrier between the tip and the Au base layer and due to the large difference in effective mass at the Au GaAs interface, which leads to a total reflection of all electrons outside the rather small acceptance cone, this distribution is quite narrow. Most of the ballistic current flows close to the Fermi energy in the STM tip, which is at 1.15 ev for the chosen parameters. As figure 18(b) shows, the ballistic current distribution in the GaAs looks quite different from the one in the Au base. Because of the refraction at the Au GaAs interface, a large number of the ballistic electrons are transferred from high values of energy E z in the Au film into lower values of E z in the GaAs, just as discussed earlier. As a result, the ballistic current distribution in the GaAs becomes extremely broad, with a maximum at rather low energies

33 Hot electron spectroscopy and microscopy 1895 and just a long slope which decreases slowly with increasing E z, finally reaching zero at the Fermi energy. With decreasing energy, the ballistic current per energy interval first increases, then saturates at E z = 1.05 ev, and finally decays quickly for energies E z below the Schottky barrier height at the Au GaAs interface (ev b = 1.0 ev). The ballistic current below the Schottky barrier originates from electrons which initially have an E z >ev b (and, therefore, can surmount the Schottky barrier), but are transferred via refraction into energy regions below ev b when crossing the Au GaAs interface. As long as the band profile of the biased sample is tilted downward, the conduction band minimum after the interface lies below the Schottky barrier height, and thus provides the necessary free states at E z <ev b, where the electrons can be refracted into. A comparison of the measured I c (V c ) curve for V t = 1.15 V (curve 3, figure 17) with the calculated current distribution in the GaAs (figure 18(b)) shows that the long slope of the calculated distribution for increasing E z is appropriately reproduced by the experiment. So is the saturation behaviour for more positive V c, which corresponds to lower energies in E z.on the other hand, the calculated current decay for electron energies below the Au GaAs Schottky barrier height is not observed in the experiment. This discrepancy is most probably due to the simplicity of the used model. For instance, all scattering effects in the base as well as scattering in the 600 Å drift region between the Au GaAs interface and the DBRTD were neglected. However, scattering processes in the metal base were found to be important by several groups [ ]. Furthermore, V c does not drop linearly over the drift region for the whole bias range, which may also contribute to the observed deviation between the experimental and the calculated data. Although one can see from the shift of the current onsets in the ballistic spectra (figure 16) that V c drops almost completely over the drift region for 150 mv <V c < +50 mv, further experiments gave evidence to the fact that this is not the case for higher values of V c. For higher V c, indications of a nonlinear shift of the current onset position in dependence of V c are observed. This implies that the total amount of the applied collector voltage is no longer effective in lowering the energetic position of the resonant level and that, therefore, very low energies cannot be accessed by mapping the current distribution using the RTD, which acts as the analysing filter in this experiment. Further, due to the increased leakage current, the measured spectra rapidly become more noisy in this region, so that the data are no longer reliable enough. 7. Hot electron spectroscopy on superlattices 7.1. BEEM results To compare the spectroscopic capabilities of BEEM and hot electron spectroscopy with HET devices, we now discuss some spectroscopic results obtained on GaAs AlGaAs superlattices. In our group, STM based ballistic transport through the lowest miniband of a buried GaAs AlGaAs superlattice was studied already some time ago [155, 86]. In this work, we had shown that the miniband transport results in a BEEM current threshold clearly below the height of the AlGaAs barriers, and that the measured and calculated miniband position in the GaAs AlGaAs superlattice are in very good agreement [155]. Like the DBRTs discussed in the previous sections, the miniband in the superlattice also acts as tunable energy filter, where both the position and the transmission of the miniband are a function of the applied bias voltage V c. The data are in excellent agreement with the model of Smith and Kogan [153], which indicates that the electron transport through the superlattice is highly coherent. As a sample for this BEEM experiment we used a 10 period 25 Å (Al 0.4 Ga 0.6 As)/30 Å (GaAs) MBE

34 1896 J Smoliner et al Figure 19. Conduction band diagram of the sample and schematic view of the experimental set-up. E f is the Fermi energy, V c the collector voltage. Figure 20. (a) BEEM spectra of the superlattice sample recorded at V c values of 200 mv, 0 mv and +200 mv, respectively. A vertical offset was added for clarity. The arrows indicate the onset voltage of the BEEM spectrum. (b) Second derivative of the BEEM current plotted for V c voltages of +200 mv, +100 mv, 0 mv, 100 mv, and 200 mv, respectively. The second derivative was obtained numerically from the experimental data. grown superlattice on top of 600 Å of undoped GaAs and a highly doped n-type collector region. The superlattice was followed by 300 Å of undoped GaAs before finally capping it with an Au base layer. In order to provide flatband conditions at the Au GaAs interface, a p-type δ-doping (N A = cm 2 ) was inserted between the superlattice and the highly doped collector region. The flatband condition in the superlattice regime is essential for the formation of a miniband. If the electric field across the superlattice is too high, the miniband breaks up into localized states and ballistic transport through the miniband will be inhibited. This flatband approach was first used by Sajoto et al [83] as well as by O Shea et al [170] who studied single and double AlGaAs barriers buried in the semiconductor. A self consistently calculated conduction band profile of this sample is shown in figure 19 together with a sketch of the experimental set-up. Due to the sample design, only the lowest miniband lies energetically in the superlattice. All other minibands are energetically in the continuum above the AlGaAs barriers. By applying an additional voltage V c between the collector and the base contact, the electric field in the superlattice regime can be changed. In figure 20(a) we have plotted typical BEEM spectra measured at collector voltages of V c = 0 mv, +200 mv and 200 mv, respectively. Two features are evident: first, the onset

35 Hot electron spectroscopy and microscopy 1897 Figure 21. (a) Transmission coefficient of the superlattice calculated by the transfer matrix formalism, plotted on a semi-logarithmic scale. As typical examples we have plotted the transmission for collector voltages of +200 mv, 0 mv, and 200 mv, respectively (curve (1), (2), (3)). (b) Superlattice transmission integrated over the miniband width. For better comparison, the calculated curves were scaled with respect to the experimental data. The dashed line is just a guide to the eye. voltage for BEEM current detection is shifted with collector bias. For positive collector bias, the onset is shifted to lower STM bias voltages, for negative collector bias the onset is shifted to higher STM bias voltages. As a second feature, the BEEM current is reduced as soon as a collector bias voltage is applied, independent of the bias polarity. At low temperatures, where the Fermi distribution function is sharp enough, the onset voltage of the BEEM spectrum marks the energetic position, where the Fermi energy in the STM tip is aligned with the bottom of the superlattice miniband. Thus, the onset voltage is expected to change, if the miniband is shifted in energy by applying a collector voltage to the sample, as already dicussed in section 5. To analyse the data more quantitatively, we used the model of Smith and Kogan [83]. A detailed discussion of the spectral features of these BEEM data is published in [86]. As the main result of the Smith Kogan model, it is found that the second derivative of the BEEM current is directly proportional to the transmission coefficient of a sub-surface superlattice. This can be used for a direct determination of the miniband position from the transmission as a function of collector voltage. In figure 20(b) the second derivative of the BEEM current is plotted for collector voltages of V c = 200 mv, 100 mv, 0 mv, +100 mv and +200 mv, respectively. Close to the onset voltage of the BEEM current, in the region where the Fermi level in the STM tip is aligned with the superlattice miniband, the d 2 I BEEM /dv 2 curves show a well-resolved peak. As expected, the peak in the d 2 I BEEM /dv 2 curves shifts to lower voltages for positive collector bias and to higher voltages for negative collector bias. In addition, the peak amplitudes decrease for both collector bias polarities. Note that the d 2 I BEEM /dv 2 curves completely go to zero outside the energetic regime of the miniband. Besides the decreasing d 2 I BEEM /dv 2 amplitudes for both collector bias polarities, this is another indication that the energetic distribution of the ballistic electrons is close to ideality and that inelastic scattering by LO phonons can be neglected. In the equivalent experiment carried out on HET devices, this is not the case. As we show later, the transfer ratio of HET devices does not go to zero outside the energetic regime of the superlattice miniband due to the strong influence of phonon replicas on the energetic distribution of ballistic electrons. To analyse this behaviour quantitatively, we have calculated the miniband transmission as a function of electron energy using the transfer matrix method. The result is plotted on a semilogarithmic scale in figure 21(a) for V c values of 200 mv, 0 mv and +200 mv, respectively. In contrast to the experiment, the individual states of the miniband are clearly visible, which is due to the fact that scattering was not included into the calculation. The calculated miniband

36 1898 J Smoliner et al Figure 22. Calculated conduction band profile of a HET incorporating a 5-period superlattice. The hot electron energy distribution is used as a probing tool to map the energetic structure of the superlattice. The energy of the hot electron beam can be tuned independently of the superlattice bias condition. (a) Sample A old injector, (b) sample B optimized injector. positions are quantitatively in good agreement with the experimentally measured peaks in the d 2 I BEEM /dv 2 curves, which indicates that the self-consistent calculation of the conduction band profile is adequate and that flatband conditions are really achieved at zero collector bias. Although, due to the simplicity of the model, the measured decrease of the transmission is not reproduced, the behaviour of the total transmission, i.e. the transmission integrated over the miniband width, is nicely described by the model. This can be seen in figure 21(b). The solid line represents the calculation, the dashed line connecting the experimental data is just a guide to the eye. In the integrated miniband transmission, the calculated collector bias dependence is well-reproduced by the experimental data. The observed decrease in the total transmission for both collector voltage polarities and the good agreement with the calculated transmission is an indication that inelastic scattering in the superlattice by phonons can be neglected. In the case of scattering, for positive collector voltage the scattered electrons would be driven into the collector by the external electric field, while for negative collector voltage they would be driven back into the Au-base. Therefore, scattering would lead to much higher integrated transmissions for V c > 0 than for V c < 0. As this is not observed, we think that inelastic scattering within the finite superlattice plays a minor role in this experiment and that the transport through the superlattice is genuinely ballistic. This is further supported by the fact that the transfer time through the superlattice is below 0.1 ps, and thus below the typical timescale for LO phonon scattering Results obtained with the HET device To give a brief and concise demonstration of the strength of HET devices, we will restrict ourselves to the results obtained on short-period superlattices. In the following, three different samples are presented. The first one (sample A) represents the original injector designed by Rauch et al [51, 54] applied to a 5-period superlattice, which is placed between the base and the collector. Figure 22(a) shows the conduction band profile of this HET. The superlattice consists of 3.5nmAl 0.3 Ga 0.7 As barriers and 3 nm GaAs wells. In figure 23(a) the conduction band profile of the superlattice is shown in detail. The first miniband is positioned between 122 and 158 mev, which results in a miniband width of = 36 mev. The energetic position of the lower edge of the miniband is chosen to lie above 100 mev in order to avoid undesired leakage current through the structure. The spacing

37 Hot electron spectroscopy and microscopy 1899 Figure 23. Superlattices between the base and the collector of a HET device. The left sides in both figures show a schematic conduction band profile, the right sides the corresponding transmission coefficients in the energy range of the first miniband calculated using a transfer matrix method. Sharp transmission peaks indicate the energy positions of individual states. (a) 5-period superlattice (samples A and B), (b) 4-period superlattice (sample C). of the individual miniband states (6 11 mev) was designed in a way to be comparable to the energetic width of the hot electron beam generated by the improved electron injector. The second miniband already lies above the AlGaAs barrier height. In the course of our work the injector design was optimized for energetic resolution. The improvements achieved can be seen in the samples B and C. Sample B (see figure 22(b)) consists of the optimized injector design together with the the same 5-period superlattice as in sample A. Sample C consists of the optimized injector design and a 4-period superlattice, which is shown in figure 23(b). It consists of 4 nm Al 0.3 Ga 0.7 As barriers and 3.2 nm GaAs wells. The first miniband is positioned between 122 and 143 mev. As a result the miniband width equals = 21 mev and the spacings between the individual states (4 9 mev) are smaller than in the 5-period superlattice. Using two types of superlattices (samples B and C) providing different numbers of periods and different separation of the individual miniband states allows us to study the resolution of this spectroscopy method as well as the hot electron transport inside the superlattices. The conduction band profile of sample B is shown in figure 22(b). Using the superlattices under flatband conditions (V c = 0), the transfer ratios of samples A, B, and C are plotted in figure 24. Below the energies of the first miniband, no collector current is observed, since the ballistic electrons are reflected at the superlattice. The onsets at V E 120 mv indicate the beginning of hot electron tunnelling through the first miniband. Comparing the transfer ratio of sample A to the transfer ratios of samples B and C, the increase of the resolution is clearly evident. The transfer ratio of sample A does not show any feature in the energy range of the first miniband whereas, due to the optimized injector design, the transfer ratios of samples B and C show detailed features which can be attributed to hot electron transport through the individual miniband states. At energies above the first miniband, the ballistic electrons are reflected at the superlattice due to the minigap which separates the first and the second miniband. However, the transfer ratio does not drop to zero in this energy range due to the overlap with phonon replicas generated in the GaAs drift region. Table 1 presents the results for the 5- and 4-period superlattices. Comparing the calculated energies of the individual states with the measured peak positions shows that the experimental results exceed the calculated energies up to 25 mev. This can be explained by considering (i) the energy offset E = ev E ev E,max between the applied emitter bias and the maximum

38 1900 J Smoliner et al Figure 24. Measured transfer ratios of the 5- and 4-period superlattices using the original injector in sample A and the improved injector in samples B and C. Hot electron transport through the individual states is indicated with arrows. At higher emitter bias, the phonon replicas contribute as inelastic background. Table 1. Calculated (E) and measured ( ev E ) energy positions of the individual states in the first miniband of the 5- and 4-period superlattice. 5-period SL 4-period SL Index E (mev) ev E (mev) Index E (mev) ev E (mev) of the hot electron distribution and (ii) a possible deviation between the nominal and real superlattice parameters due to the growth accuracy of the molecular beam epitaxy (about one monolayer). Since E is supposed to be constant in the energy range of the first miniband, it cancels out when deriving the energy spacings E ij = (E i + E) (E j + E) = E i E j. (50) The measured energy spacings between the individual states fit best to calculated spacings using superlattice parameters of 3.3 nm AlGaAs barriers separated by 2.9 nm GaAs wells for the 5-period superlattice and 3.7 nm AlGaAs barriers with 3 nm GaAs wells for the 4-period superlattice, respectively. These deviations to the nominal superlattice parameters lie within one monolayer for GaAs and AlGaAs. The reduced barrier and well widths result in broader minibands which are shifted to higher energies. The calculated first miniband, using these modified parameters, is positioned between 123 mev and 166 mev ( = 43 mev) for the 5-period superlattice and between 124 and 156 mev ( = 32 mev) for the 4-period superlattice. The smallest energy spacing resolved in the experiments is E 21 = 8 mev in the 4-period superlattice. Thus, the resolution of the device is found to be E 8 mev. After the energetic structure of the superlattice miniband has been determined, we can now try to reconstruct the energetic distribution of those electrons which are transmitted through the superlattice coherently. For this purpose, we first split the transmission coefficient of the

39 Hot electron spectroscopy and microscopy 1901 superlattice into a coherent and an incoherent component: T(E z ) = T coh (E z ) + T incoh (E z ). (51) As is common in HET spectroscopy, here coherent means without any scattering while incoherent refers to electrons affected by scattering processes. In a HET the hot electron energy distribution in the emitter region is much broader compared to the very sharp transmission peaks of the resonant quantum states in the superlattice. Therefore, it is valid to replace the sharp transmission peaks of the superlattice states by a sum of weighted δ-functions N T(E z ) = (ccoh i + ci incoh )δ(e z Ez i ), (52) i=1 which are positioned at the eigenenergies Ez i. The differences in the peak areas between the δ-functions (peak area = 1) and the individual transmission peaks are taken into account by introducing the coefficients and c i coh = 0 c i incoh = 0 T i coh (E z) de z (53) T i incoh (E z) de z. (54) Inserting equations (52), (53), and (54) into (27) the following result is obtained: α(e z ) = = = 0 0 F(E z E z)t (E z ) de z F(E z E z) N (ccoh i + ci incoh )δ(e z Ei z ) de z i=1 N (ccoh i + ci incoh )F (Ei z E z). (55) i=1 This means, that the total transmission per state c i = ccoh i +ci incoh can be extracted from the measured transfer ratios in the form of the corresponding peak amplitudes. These are gained by applying multi-peak fit procedures using the measured energy distributions of the ballistic electron beam and of the phonon replicas. In figure 25 the measured transfer ratio of a 4-period superlattice is compared to the convolution of the vertical energy distribution of the hot electron beam with the coherent transmission coefficient. As one can see, the agreement between the transfer ratio and the calculated ballistic electron distribution is excellent in the range where the contribution of the phonon replica peaks can be neglected. As demonstrated in this section, HETs provide an excellent energetic resolution for hot electron spectroscopy. Furthermore, they also facilitate tuning the energy of the injected electrons independently of the electric field applied to the superlattice, providing a situation analogous to the BEEM experiments discussed in section 7.1. For the superlattices shown in figure 23, the individual Wannier Stark states in the first miniband have been resolved up to electric fields of 27.6 kv cm 1. The results on this are presented in [169]. The basic transport through the Wannier Stark states in such short-period superlattices was found to be coherent. By tuning the Wannier Stark state splitting into the optical phonon energy, the opening of LO phonon mediated transport paths was observed.

40 1902 J Smoliner et al Figure 25. Measured transfer ratio of the 4-period superlattice ( ) compared to the convolution of the vertical energy distribution of the hot electron beam with the coherent transmission coefficient of the 4-period superlattice. The convolution equals the superposition of 4 hot electron energy distributions of amplitudes ccoh i with i = 1, 2, 3, BEEM studies of single impurities in the regime of the Mott transition In the last section of this article, we would like to discuss some BEEM studies on single impurities embedded in an AlAs barrier [171], an experiment which benefits both from the spectroscopic capabilities of BEEM as well as its imaging capabilities. Doping studies of heterostructures have always been of interest, since in heterostructures, thresholds for the Mott transition for thin layers can vary [172] compared to those in bulk materials, and even small doping concentrations [173] or delta-doping layers [81, 174] can have a huge impact on the transport properties of such a system. With the devices becoming smaller and smaller, the local influence of doping atoms and unintentional impurities on the electrical behaviour of a device becomes more and more critical. This immediately leads to BEEM/BEES as a promising instrument for the investigation of doping in nanoscale devices. To investigate the influence of impurities on the transmission of ballistic electrons through a heterostructure in a most direct way, samples with a single, Si-doped AlAs barrier, embedded in a GaAs matrix, were used. According to values from the literature for bulk AlAs, isolated silicon atoms cause an impurity level 70 mev below the AlAs conduction band edge (X valley) [176]. In order to judge the feasibility of imaging single impurities with BEEM on such samples as well as to choose a suitable scan range for this purpose, a rough estimation of the expected size of the impurities was conducted using the simple hydrogen model [175]. In a material with a dielectric constant ɛ rel and an effective electron mass m the effective Bohr radius a is obtained from the Bohr radius a 0 by a = m 0 m ɛ rela 0. (56) This yields an effective Bohr radius of about 104 Å for bulk GaAs and a value of about 20 Å for bulk AlAs. Those values are within the limits of the spatial resolution of a BEEM experiment ( 10 Å) and indicate that, in principle, it should be possible to depict single impurities in GaAs/AlAs with BEEM. Furthermore, the estimated size of the impurities suggests a scan range of, e.g., nm 2 to be suitable for the search for impurities. Another key point is the impurity concentration level. In order to actually conduct measurements of impurities with a local probe, the choice of a useful doping concentration is

41 Hot electron spectroscopy and microscopy 1903 Figure 26. Schematic view of the experimental set-up together with the conduction band profile of the sample. The conduction band edge is determined by the Ɣ valley in GaAs and by the X valley in AlAs. (The sketch shows also the Ɣ valley in the AlAs barrier.) V t denotes the tunnel voltage, I t the tunnel current, I c the ballistic current and E f the Fermi energy. constrained by two limits: on the one hand, the doping should be high enough to provide at least one impurity within the chosen scan range ( nm 2 ); on the other hand it should be low enough to avoid an electrostatic distortion of the conduction band profile due to a charge buildup by the donor ions. Calculating the band profile by a self-consistent solving of Poisson s equation [164] for various doping concentrations shows, that a doping concentration of N D = cm 3 still causes no noticeable change in the conduction band profile, while for N D = cm 3 the band bending can no longer be neglected and therefore changes in the current onset directly due to the impurities cannot be distinguished from band bending effects. Besides a reference sample with no intentional doping in the barrier of the heterostructure, the doping concentrations actually chosen for the experiment were N D = cm 3 and N D = cm 3, respectively. Making the simplifying assumption that all doping atoms are electrically active, the mean distance between two impurities can quickly be estimated using the following consideration: for a doping concentration of N D, the average volume where exactly one impurity can be found will be 1/N D. Assuming this volume to be a sphere with the impurity in its centre, this leads to a radius of r = 3 3/4πND. In this simple model, the average distance between the centres of two impurities can be approximated by 2r. For N D = cm 3 this yields 400 Å, while for N D = cm 3 the average distance is 267 Å. Therefore, in both cases the probability of finding at least one impurity within the chosen scan range is sufficient to conduct this sort of measurements. All samples used in this project were fabricated by molecular beam epitaxy, using semiinsulating GaAs(100) wafers as a substrate. After a 1000 Å thick GaAs buffer layer and several smoothing layers, the actual heterostructure starts with a highly n-doped GaAs region (Si, N D cm 3 ), which provides the collector for the ballistic current. The thickness of this region is 6500 Å for the samples with the undoped barrier and 5000 Å for the samples with the doped barriers. The collector region is followed by 1500 Å of undoped GaAs, which serves as a spacer. On top of this layer a 100 Å thick AlAs barrier is grown, either nominally undoped (i.e. N D < cm 3 ) or with a silicon doping of N D = cm 3 or N D = cm 3, respectively. Finally, all barriers are covered with an undoped GaAs cap layer (100 Å) for protection purposes. In figure 26 a sketch of the band profile together with the principle of the experimental set-up is shown. The low temperature measurements were carried out in He exchange gas at temperatures of 180 K and 10 K, respectively. The

42 1904 J Smoliner et al Figure 27. Typical BEEM spectra measured at T = 180 K and I t = 2 na. For better comparison with the spectra of the doped samples, a reference spectrum obtained on the undoped barrier is shown in both, (a) and (b). (Note that, for technical reasons, V t is applied to the base and therefore positive in our experiment.) (a) Undoped barrier compared with a barrier doping of N D = cm 3 (vertical offset added for more clarity), (b) undoped barrier compared with a barrier doping of N D = cm 3. tunnel current used for the presented spectra as well as the BEEM images was I t = 2 na. Etched tungsten tips with a sputtered gold covering were employed as STM tips for good spatial resolution and spectroscopic stability The influence of doping in AlAs single barriers on the current onset in BEES To investigate the actual influence of a certain amount of doping on the transport behaviour of the barrier, a nominally undoped AlAs barrier (N D < cm 3 ) was used as a reference sample. For this sample, the current onset in the BEEM spectra is given by the position of the AlAs X valley. As can be seen from figure 27, at T = 180 K the current onset in a BEEM spectrum for an undoped AlAs barrier is typically at 1.19 V. Reference measurements on a GaAs bulk sample yielded an Au GaAs Schottky barrier height of 0.96 ev at T = 180 K. Taking into account that the AlAs barrier lies below a 100 Å thick GaAs cap layer and the band structure is slightly tilted (see figure 26), this means that the AlAs X valley lies about 280 mev above the conduction band edge in the GaAs (i.e. the GaAs Ɣ valley). This measured value is in good agreement with data from other BEEM experiments (see, e.g. [69,97]) as well as with results from tight-binding calculations, as, for instance, provided by Jancu et al [177]. One might notice that the size of the ballistic current in figure 27 is considerably less than the typical BEEM current known from measurements at room temperature (up to several pa, see, e.g. [82]). This suppression of Ɣ X transport at low temperatures was also observed in previous experiments [99] as well as by other groups [69]. The doping concentration in the AlAs barrier of the first doped sample was chosen to be N D = cm 3. As mentioned earlier, this leads to a mean distance of approximately 40 nm between two impurity centres, and gives just a sufficiently high probability of having at least one impurity within a scan range of nm 2. In figure 27(a) a typical BEEM spectrum measured on this sample is shown in comparison with one of the undoped barrier. The two spectra are almost identical, showing both the same onset position in I c (V t ) and about the same amount of transmitted ballistic current.

43 Hot electron spectroscopy and microscopy 1905 Figure 28. Histograms of I c current onsets extracted from BEEM spectra which were measured at arbitrarily distributed locations on a sample. (V t is applied to the base and therefore positive.) Data measured at T = 180 K: (a) undoped barrier, (b) barrier doping N D = cm 3,(c) barrier doping N D = cm 3. This fact is confirmed by further measurements of BEEM spectra taken at various positions on both samples. All spectra look very similar, whether they are acquired on the sample with the moderately doped barrier or on the undoped reference sample. To get an overview of the transmission behaviour, the minimum tunnel voltage necessary to detect a ballistic current (= onset voltage) is extracted from all spectra. In figure 28(a) we have plotted the distribution of these onsets for the nominally undoped barrier, while figure 28(b) shows the result for the barrier with N D = cm 3. To provide a convenient way for a quick comparison, a simple Gaussian fit to the histograms was utilized. This yields a mean current onset of 1.19 V for both barrier types, showing no significant influence of this amount of doping. The only visible difference between the two samples is that the width of the distribution for the undoped barrier is only 23 mv, while the medium doped barrier has a noticeably wider distribution of 38 mv. Apart from this broadening, the onset distribution of this medium doped barrier does not reveal any local effects of the impurities on the current onset.

44 1906 J Smoliner et al For the next type of sample the doping within the AlAs barrier was chosen to be N D = cm 3. A quick estimate yields a mean distance of approximately 27 nm between two impurity centres for this doping concentration. In figure 27(b) a typical BEEM spectrum of this sample in comparison with a spectrum of the undoped barrier is shown. As can be seen clearly, the doping now causes a significant lowering of the onset voltage for the ballistic current. For a more thorough comparison, several BEEM spectra were also measured at various positions on this highly doped sample and used for a systematic determination of the corresponding onset voltages. The result, which can be seen in figure 28(c), shows unambiguously a general lowering of the onset voltage compared to the undoped barrier. For a better comparison, again a Gaussian curve was fitted to the distribution. The mean onset voltage for N D = cm 3, as provided by the Gaussian fit, is 1.08 mv. This is 110 mv less than for the undoped barrier and clearly indicates that the doping opens an additional transmission channel which lies below the AlAs X valley. Although the onset distribution for the highly doped barrier is broader (31 mv) than the one for the undoped barrier (23 mv), it is still peaked around a single maximum. Again, as for the moderately doped sample, from the onset voltage distribution no local influence of the impurities on the current onset is apparent. This general decrease in the current onset voltage for the highly doped barrier indicates that in this case the regions influenced by individual impurities are actually overlapping, and thus cause not a punctual but a wide-ranging alteration of the transmission behaviour. Therefore, these measurement data lead to the conclusion that already a doping concentration of N D = cm 3 is sufficient to form a continuous impurity band. This can also explain the fact that the onset measured for this doped sample is 110 mev below the AlAs X valley, while the value found in the literature for the energetic position of a silicon impurity in AlAs is only 70 mev below the AlAs X valley. The latter is denoted to be valid for an isolated impurity with zero-dimensional characteristics, while in an impurity band the lower level of confinement is expected to lead to a decrease in the energetic position. Although the onset distributions of the doped barriers do not show any obvious pattern which would allow one to decide whether the data had been measured within or outside of the local range of influence of an impurity, the ballistic spectra do reveal an interesting detail at higher tunnel voltages. Figure 29 shows plots with a series of ballistic spectra for each sample. As expected, the spectra of the undoped barrier show just some dispersion around an average behaviour for the whole range of V t. On the contrary, the spectra of the doped barriers split into two groups when the tunnel voltage increases. Apparently, despite that the current onset for each sample is preferentially found around a single value of V t (see figure 28), in the region of higher tunnel voltages there are distinctive areas of enhanced current transmission on the doped samples. This effect is seen for the sample with a barrier doping of N D = cm 3 as well as for the one with N D = cm 3. Therefore, this observation suggests that the impurities diminish the effective barrier thickness in their wider surroundings, even in the regime where the doping concentration is too low to form a continuous impurity band and the barrier height is generally still determined by the AlAs X valley Imaging of impurities in AlAs single barriers The finding of a locally enhanced transmission at higher tunnel voltages for the doped barriers provided new encouragement to conduct a search for single impurities in the imaging mode of BEEM. In figure 30 BEEM images for all three barrier types can be seen. All images

45 Hot electron spectroscopy and microscopy 1907 Figure 29. Ballistic spectra taken at arbitrarily distributed locations on a sample. All data were measured at T = 180 K using a tunnel current of I t = 2 na: (a) undoped barrier, (b) barrier doping N D = cm 3,(c) barrier doping N D = cm 3. (Note that, for technical reasons, V t is applied to the base and therefore positive in our experiment.) were recorded with the same tunnelling conditions. To facilitate a direct comparison, the same colour code for the z-scale was used in all three BEEM images. In order to differentiate between topography related transmission variations and any features in the BEEM images which are not caused by topographic effects, simultaneously to the BEEM images also the corresponding STM images were measured. Those surface topographies are also depicted in figure 30 and show just the usual corrugation of the evaporated Au base layer for all samples. As can be seen from figure 30(a), the BEEM image of the undoped barrier displays a quite uniform transmission profile and reflects just the topographic features also visible in the STM image, primarily the granular structure of the gold base layer. Quite a different picture is exhibited for the barrier with the intermediate doping concentration, which can be seen in figure 30(b): superimposed on the typical granularity of the gold base layer is a larger structure in the transmission profile. In particular, two brighter spots protrude from a plane with a generally low transmission. The simultaneously measured STM surface topography shows just the usual corrugation of the evaporated gold base layer, as for the other samples. This confirms that those bright areas in the BEEM image of the medium doped sample are not caused by any topographic irregularities but do originate in a local modification of the transmission in the sub-surface heterostructure due to the impurities in the AlAs barrier.

46 1908 J Smoliner et al Figure 30. Left column: BEEM images of the three different sample types, recorded at T = 180 K, I t = 2 na, V t = 1.4 V. Right column: corresponding STM surface topographies (recorded simultaneously with the BEEM images). The maximum z-range in the topographies is 20 Å: (a) undoped barrier, (b) barrier doping N D = cm 3, (c) barrier doping N D = cm 3. Although the diffuse shape of these spots makes their size rather hard to determine, a rough estimate of the mean radius shows that it is about nm. Using the hydrogen model of impurities this size is more like the effective Bohr radius of an impurity in GaAs ( 10 nm) than the one of an impurity in AlAs ( 2 nm). Of course, the observed radius is only a very rough estimation and the experimental difficulties can have a distorting effect on the estimated size. On the other hand, one should not forget, that the AlAs barrier in the heterostructures is comparatively thin and embedded in a GaAs matrix. Therefore, although the doping was done only during the growth of the AlAs layer, the behaviour of impurities in this heterostructure can differ significantly from the behaviour of impurities in bulk AlAs. One should also keep in mind, that the distribution of the impurities within the barrier is stochastic. That means, an impurity might be located in the centre of the barrier as well as be sitting right at the AlAs GaAs interface. The actual position of an impurity will most probably