OPTICAL STARK EFFECT IN SEMICONDUCTOR QUANTUM WELLS: A COMPARATIVE STUDY

Transcription

1 U.P.B. Sci. Bull. Series A ol. 7 No. 8 ISSN -77 OPTICAL STARK EFFECT IN SEMICONUCTOR QUANTUM WELLS: A COMPARATIE STUY Ecaterina NICULESCU Adrian RAU Anca IORGA Nivelele de energie de suandă în gropi cuantice din GaAs-GaAlAs având formă triunghiulară paraolică şi rectangulară sunt calculate utliând metoda variaţională în aproximaţia masei efective. Se studiaă dependenţa coeficientului de asorţie corespunător traniţiilor interandă de amplitudinea undei laser şi de geometria structurilor cuantice. The suand energy levels in finite triangular paraolic and square GaAs- GaAlAs quantum wells under intense laser field are calculated y using a variational method in the effective mass approximation. The dependence of the asorption coefficient related to the interand transitions on the laser field amplitude and the geometric shape of the quantum wells is discussed. Key words: triangular quantum well paraolic quantum well square quantum well laser field optical asorption.. Introduction With the development of the molecular-eam epitaxy growth method quantum wells can e made into different forms. Such structures present desirale optical features for devices design and favorale Stark shift characteristics which can e used to control and modulate the intensity output of the devices. For instance the paraolic quantum wells (PQWs) have een used as the graded arrier part of the quantum-well lasers to improve the optical confinement factor and enhance the carrier collection into the thin quantum well so as to reduce the threshold current density []. PQWs are also used to design infrared detectors with low leakage currents [] and employed in resonant tunneling devices for their potential applications in high-speed circuits [4]. -shaped quantum wells (QWs) have een grown and experimentally studied those structures eing potentially interesting for designing millimeter wave communication devices [5] lightemitting devices [6] and MOSFET devices [7]. As a consequence many theoretical studies of the electronic states in graded composition quantum wells Prof. Physics epartment University "Politehnica" of Bucharest ROMANIA niculescu@physics.pu.ro Assist. Physics epartment University "Politehnica" of Bucharest ROMANIA radu@physics.pu.ro Stud. ETTI University "Politehnica" of Bucharest ROMANIA

2 5 Ecaterina Niculescu Adrian Radu Anca Iorga (GCQWs) with different profiles have een performed [8-]. The general picture indicates a sustantial dependence of the electronic states on the main features of the well profile. More recently these studies have een extended to quantum wells under intense electric fields created y high-intensity TH lasers. Neto et al. [] have derived the laser-dressed quantum well potential for an electron in a square quantum well (SQW) in the frame of a non-perturation theory and a variational approach. A simple scheme ased on the inclusion of the effect of the laser interaction with the semiconductor through the renormaliation of the effective mass has een proposed y Brandi and Jalert [4]. Oturk et al [5] have investigated the effect of the laser field on the intersuand optical transitions for a graded QW under external electric field. In this work we studied effects of the laser field on the energy spectra in finite QW PQW and SQW GaAs quantum wells. The asorption coefficient for the interand transitions is also discussed as a function of the laser parameter and geometric shapes of the wells. To our knowledge this is the first study of the effects of laser fields on the energy spectra in QWs and PQWs.. Theory For a particle moving in a quantum well under a high-frequency laser field the Hamiltonian is given ( y: ) p qa H (). () m Here q and m * are the particle electric charge and the effective mass respectively A is the vector potential of the laser field and is the quantum well potential energy. We use an isotropic single-and effective-mass Hamiltonian for the electron (hole) single-particle states. We ignore the effects of the effective-mass discontinuity at the well interfaces and we use the well masses throughout the entire structure. We assume that the laser field can e represented y a monochromatic plane wave of frequency. For a linear polariation along the - direction the vector potential of the laser field is A( t) A cos( t) u. For these Ψ single-particle Hamiltonians we find the ound electron and hole states e h with energies E k y solving: d d m d Ψ () E Ψ () k k k ()

3 Optical Stark effect in semiconductor quantum wells: a comparative study 5 where d is the laser dressed potential: [ ]dt t d / sin () and * m A q is laser dressing parameter. For a finite quantum well the carrier confinement potentials are given y: < n ) ( (4) where is the conduction (valence) and offset and L / is the half-width of the quantum well. We study the n cases corresponding to the - shaped quantum well (QW) paraolic quantum well (PQW) and square quantum well (SQW) respectively. It is significant to emphasie that the approach () of the laser-dressed potential is valid for the high-frequency regime e.g. >> τ where τ is the transit time of the electron in the quantum structure []. For a GaAs-Ga.7 Al. As SQW with a width of aout Å sujected to a monochromatic CO laser eam ( 4 s - ) this condition is very well satisfied []. Since for a given arrier height the suand energy levels increase with the decreasing of n one may expect that the high-frequency condition will e even etter verified for PQWs and QWs. For n and n the laser-dressed potential d as defined in Eq. () is given y the following expressions: () () () (). QW d (5)

4 Ecaterina Niculescu Adrian Radu Anca Iorga 54. PQW d (6) where min [ ) [ ] and. For the SQW we otain the same functional form of dressed potential as in Ref. [6]:. SQW d (7) The laser field determines a mixing of the eigenstates so that in Eq. () the suand wave functions are taken as linear cominations of the eigenfunctions φ k of the QW []: Ψ j j k j k c ) ( ) ( φ (8) where j denotes all ound states in the quantum well. The dressed ound-state energies k E and the coefficients k j c are determined using the system of linear equations: C S H E (9) where H and S represent respectively the matrices of the particle energy in the laser field and the norm with respect to the asis functions { } () j φ. C denotes the column matrix of the coefficients k j c.

5 Optical Stark effect in semiconductor quantum wells: a comparative study 55 Once all ound state energy levels are determined the asorption coefficient corresponding to interand transitions ( h j e j ) etween the states of heavy-hole and electron can e evaluated. Neglecting the population effects for oth initial and final states the asorption coefficient reads []: h e t A( E) A Φk Φ k Θ( E E j ). () j e E * * p m Here emh A E * * p is the Kane matrix element n is the nceml me mh t refractive index of the QW l is the total thickness of the structure and E j is the interand transition energy. By choosing A as the natural unit for A ( E) Eq. () descries the usual ladder profile for the asorption coefficient as expected from the joint density of states of the suands.. Results and discussion In the aove calculations we will consider an L Å GaAs-Ga -x Al x As * * quantum well. As in Ref [] we take m e.665m m h.6m and an aluminum concentration in the arrier material x.. For the electron (hole) states the arrier height is otained from the 6% (4%) rule of the and gap discontinuity Δ E g 47x (me). In order to show the effects of the potential-shape n the laser dressing parameter on single-particle spectra in QWs we have calculated energy levels of the electron and heavy-hole in GaAs-Al x Ga -x A QWs as a function of n and. Figs. (a)-(a) present the laser-dressed potential d as a function of the position for different values of the laser parameter and for n n and n. The corresponding ound-state energy levels versus the laser-dressing parameter are shown in Figs. ()-(). The numers on the curves represent the indices of the electron (hole) suands. The ero point of energy is situated at the edge of the conduction (valence) and in the finite arrier GaAs QW. In our calculations the energy of the hole is considered to e a positive value. It is seen from the figures that the ground state energy in the conduction and presents an increase with the laser intensity. Note that at a given value of the laser parameter the energy shift decreases with the increase of n as a result of a weaker localiation of the electron wave function in the structure.

6 56 Ecaterina Niculescu Adrian Radu Anca Iorga (a) () Fig.. QW: a) Laser-dressed potential versus position for different values of the laser parameter ) Bound-state energy dependence on the laser field parameter. (a) () Fig.. The same as Fig. for PQW. For the SQW in agreement with Ref. [] the energy levels are more laser-sensitive as the suand index increases. For the PQW and QW where the e geometrical confinement overwhelms the laser effect E is less dependent on the e. Furthermore in the QW case E is slowly decreasing with the increase of. The laser field less affects the energy of a heavy-hole than the energy of an electron this ehavior eing a common feature of the quantum structures [].

7 Optical Stark effect in semiconductor quantum wells: a comparative study 57 (a) () Fig.. The same as Fig. for SQW. We have also investigated the changes of the asorption coefficient A ( E) y the laser field. The staircase lineshape of the asorption coefficient in the e GaAs-Ga.7 Al. As QW PQW and SQW with L Å and and Å is shown in Fig. 4. As seen in this figure the laser field induced lue shift of the fundamental asorption edge for QW is greater than for PQW and SQW. Fig. 4. The asorption coefficient for L Å GaAs-Ga.7 Al. As SQW PQW and QW at e and Å. The optical Stark effect on the excited energy levels is stronger for the SQW and thus the shift induced y the laser field on the e-h interand transition is greater than for the PQW and QW.

8 58 Ecaterina Niculescu Adrian Radu Anca Iorga 4. Conclusions We have studied the effects of the high-frequency laser field on the intersuand transitions in different shaped quantum wells (QW PQW and SQW). The dependence of the asorption coefficient on the laser parameter geometric shape of the wells was discussed. The results we otained for SQW are in agreement with previous results. To our knowledge this is the first study of the effects of laser fields on the energy spectra in QW and PQW. We conclude that for the quantum wells the laser field amplitude have a significant effect on the electronic and optical properties. This gives a new degree of freedom in various device applications ased on the interand transition of electrons. R E F E R E N C E S [].. Kasamet C. S. Hong N. B. Patel and P.. apkus IEEE J. Quantum Electron QE []. L. J. Mawst M. E. Givens C. A. Zmudinski M. A. Emanuel and J. J. Coleman IEEE J. Quantum Electron QE []. R. P. G. Karunasiri and K. L. Wang Superlatt. Microstruct [4]. A. C. Gossard R. C. Miller and W. Wiegmann Surf. Sci [5]. St. Ciugni T. L. Tansley and G. L. Griffiths J. Cryst. Growth [6]. R. J. Choi H. W. Shim S. M. Jeong H. S. Yoon E. K. Suh C. H. Hong H. J. Lee and Y. W. Kim Phys. Stat. Solidi (a) 9 4. [7]. A. T. M. Fairus and. K. Arora Microelectronics Journal [8]. E. Kasapoglu H. Sari and I. Sokmen Physica B 9 7 idem Physica B 9 7. [9]. L. Zhang Supperlat. Microstructures []. L. Zhang and H. J. Xie Mod. Phys. Lett. B []. Wu-Pen Yuen Phys. Rev. B []. H. Akas C. ane K. Kasapoglu and N. Talip Physica E []. O. O.. Neto and Q. Fanyao Supperlat. Microstructures 5 4. [4]. H. S. Brandi and G. Jalert Solid State Commun. 7. [5]. E. Oturk H. Sari and I. J. Sokmen J. Phys [6]. F. Qu P. C. Morais Phys. Lett. A 46.