Operation of a Bloch oscillator

Transcription

1 Oeration of a Bloch oscillator K. F. Renk *, A. Meier, B. I. Stahl, A. Glukhovskoy, M. Jain, H. Ael, and W. Wegscheider Institut für Angewandte Physik, Universität Regensburg, Regensburg, Germany We reort the oeration of a Bloch oscillator. The active medium was a staticvoltage driven, doed GaAs/AlAs suerlattice which was electromagnetically couled to a resonator. The oscillator roduced tuneable microwave radiation (frequency ~ 60 GHz; ower ~ 0.5 mw; efficiency ~ 4 %). The gain (~ 0 4 cm - ) was due to the nonlinearity mediated by miniband electrons. We also resent a theory of the oscillator. The Bloch oscillator should in rincile be feasible for generation of radiation u to frequencies of 0 THz and more. * karl.renk@hysik.uni-regensburg.de Conduction electrons in a semiconductor suerlattice can undergo Bragg reflections at the suerlattice lanes and the energy of motion along the suerlattice axis can be confined to a miniband [, 2]. The confinement can cause a negative differential resistance [2] and Bloch oscillations [3] which have been observed by transort [4] and otical [5] studies, resectively. Ktitorov et al. [6] resented a theory indicating that a suerlattice in a negative-differential resistance state should be a ( Bloch ) gain medium for high frequency radiation from almost zero u to the Bloch frequency, which is determined by the suerlattice eriod and the strength of a static field alied to the suerlattice and can reach 0 THz or more. Gain has been concluded [7] from an anomalous THz transmissivity of an array of suerlattices, switched by a voltage ulse into a negative-differential resistance state. In this Letter, we reort the oeration of a Bloch oscillator, i.e. an oscillator based on Bloch gain, with a voltage-driven semiconductor suerlattice as the active medium couled to a resonator. We also resent a theoretical descrition of the oscillator roerties.

2 In the Bloch oscillator (Fig. a), a suerlattice which is art of a suerlattice electronic device (SLED) is connected to a static-voltage source delivering a current I. The suerlattice generates, by stimulated emission, radiation at the resonance frequency, ν, of the resonator. Fig.. (a) Princile of the Bloch oscillator; r, reflector and r2, artial reflector. (b) Arrangement. (c) Suerlattice electronic device (SLED). (d) Current-voltage curve of the SLED. By varying the resonator length, ν is changed. In our arrangement (Fig. b), a SLED was mounted in a metal-cavity resonator (height 2 mm, width ~ 4mm) with a moveable backshort. A gold whisker antenna couled the SLED to the resonator and 2

3 connected it to a voltage source. A filter avoided radiation loss to the bias circuit. By changing the diameter of the iris, we varied the quality factor of the resonator. The SLED (Fig. c) contained the suerlattice (30 eriods, each eriod consisting of 4 monolayers of GaAs and 2 monolayers of AlAs) embedded in gradual layers and n + GaAs layers, grown by molecular beam eitaxy on an n + GaAs substrate. To and bottom of the SLED were covered with ohmic contact layers. The currentvoltage curve of the SLED (Fig. d, solid line) was ohmic at small voltage. The current showed, with increasing voltage, a maximum at the eak current I (~ 2 ma; eak-current density j ~ 00 ka/cm 2 ) and then decreased slightly. Figure 2 exhibits an emission line (at a frequency near 60 GHz) of the oscillator. The line broadening was mainly due to an insufficient stabilization of our voltage source and to thermal fluctuations within the SLED; with a.5v battery, the halfwidth was about 200 khz. Outside the line center, the signal decreased strongly; the background was mainly due to the noise of the sectrum analyzer we used to monitor the line. A thermal ower meter indicated a ower (0.5 mw) which corresonded to an efficiency of about 4 ercent for conversion of electric to radiation ower. Fig. 2. Emission line of the Bloch oscillator. The ower was obtained for a reflectivity of the iris of about 0.7. From this we conclude that the gain coefficient of the active suerlattice was about 0 4 cm -. To 3

4 study the tuning behavior, we used an iris of smaller diameter. We found that mechanical tuning was ossible over a wide frequency range (Fig. 3a), however, P/P max (Fig. 3a, uer art), which is the ower relative to that of maximum emission, decreased strongly if the resonator length deviated from that of maximum ower P max. The current increased slightly (by few ercent). We observed that the oscillation frequency ν increased with increasing voltage (Fig. 3b). Within a voltage range (.2 V to 2.2 V) in which the oscillator oerated at one mode, ν increased by about ercent; the ower increased by a factor of two. An increase of current (Fig. d, dotted line) indicated a ositive differential resistance of the suerlattice in the oscillating state. The switching to an oscillating state was joint with an abrut decrease of current from a oint on the solid curve to the corresonding oint on the dotted curve. Fig. 3. (a) Frequency (and ower) of the Bloch oscillator for different resonator lengths; d, resonator length of maximum ower. (b) Frequency of the Bloch oscillator for different values of the voltage. 4

5 We attribute the oscillation to Bloch gain. For an analysis of our results, we use the 2 electron moving along the suerlattice axis, the miniband width and a the disersion relation ε = ( coska) where ε is the energy, k the wave vector of an suerlattice eriod. We describe the electron as a wave acket with its center given by the trajectory ξ. Under the action of a static field, E, the electron erforms a eriodic motion (Fig. 4a) between ˆ ξ ( ε = 0 ) and + ˆ ξ ( ε = ). The corresonding de Broglie wavelength changes from to 2a, resectively. When the de Broglie wavelength reaches the value 2a, the electron is Bragg reflected and reverses its direction. Accordingly, the electron oscillates with the Bloch frequency ν B = eae h where e is the elementary charge and h Planck s constant. Under the additional action of a high frequency field at the frequency ν = ν B, the electron transfers, within one half eriod, energy to the high frequency field. During the other half eriod, the same energy is transferred back to the electron. If relaxation is introduced, there is a net energy transfer via the Bloch oscillating electrons from the static to the high frequency field if ν < ν B and vice versa if ν > ν B. The corresonding solution of the Boltzmann equation delivers the time deendent drift velocity [8] t t dt t t0 ea v(t) = 2v ex sin E( t τ τ h t0 0 ) dt () where v = 4h a is the eak-drift velocity, τ the intraminiband relaxation time, ( U + Ûcos ωt) E(t) = the instantaneous field strength, and L the suerlattice length; L for simlicity, we neglect elastic scattering at defects. Without feedback (Û = 0), eq.() delivers the Esaki-Tsu characteristic [2] 2 2 (U / U c )( + U / U c ) I = 2I where is the critical voltage, i.e. the voltage across the suerlattice at the eak current I = NeAv where N is the free carrier concentration and A the suerlattice cross section area. Taking into account a series resistance 5

6 (Rs), our exerimental curve is reroduced by the theory (Fig. d, dashed) in the range of ositive differential resistance if we choose aroriate data (Rs ~ 30 Ω; Uc ~ 0.57 V; N ~ cm -3 ), and τ (~ s). The corresonding critical field E c = U c /L ~ 0 kv/cm and v (~ 0 7 cm/s) are in accordance with the microscoic arameters ( ~ 0.4 ev; L ~ 0.6 µm; a ~ 4.3 nm). Fig. 4. (a) Motion of an electron in a suerlattice along the suerlattice axis; ξ, trajectory and λ db, de Broglie wavelength. (b) Resistance of the suerlattice (uer art) and ower (lower art) for different amlitudes Û of the high frequency voltage. (c) Otimum-gain frequency, ν ot, for different values of the static field. 6

7 We determined, for a fixed static voltage (U = 2 Uc) and different values of Û, the amlitude vˆ of the drift velocity at the frequency ν and of the current ( Î I = vˆ v ). Then, we calculated the dynamic resistance R = Û Î and the corresonding ower 2 P = Û R. There is a range of negative values of R and P (Fig. 4b) indicating gain 2 (for ν < ν B) and another region of ositive values indicating absortion ( ν > ν B). R increases with increasing amlitude of the high frequency field. This shows that the suerlattice can match itself to the external circuit. There is an otimum oeration oint for Û (at R ~ Ω) where a maximum ower can be transferred to the high frequency field. The maximum ower ( 0. I U c ~ 0.6 mw) is almost equal to the exerimental value. The frequency, ν ot, of otimum gain increases with the static field strength (Fig. 4c) and can reach (for U ~ 8 U c ) a value of 0 THz. The observation of a frequency increase (Fig. 3b), though limited by the resonator, is in accordance to the theoretical result. We note that our results are in accordance with earlier gain calculations [6, 9]. Theoretical studies [0] redicted the ossibility that roagating sace charge domains are formed in a suerlattice in a negative resistance state. For a corresonding domain-mediated oscillator, the oscillation frequency ν dom would be equal to the recirocal transit time, a domain takes to form and to traverse a suerlattice. According to the Esaki-Tsu curve (Fig. d) it is exected that ν dom decreases with increasing voltage; this is confirmed by extended simulations [] and is also known for Gunn oscillators, which are domain oscillators. For our suerlattice, the criterion of domain formation, (NL) d 5 0 cm -2 [7, 2] was fulfilled, NL ~ 3 (NL) d. The observation of an increase of the oscillation frequency with the voltage (Fig. 3b) suggests however that the Bloch-gain state governed the oscillation. Further evidence follows from the good agreement between the calculated and measured ower. The absence of a jum like, strong current decrease near maximum current in the exerimental current-voltage curve for the suerlattice without feedback (Fig. d, solid line) is a direct sign of a minor role of domains. 7

8 The Bloch oscillator can be described as a arametric oscillator. The main rocess is the arametric interaction of a high frequency field with an elementary, singleelectron Bloch oscillator, which is driven by the static field. Due to relaxation, the Bloch frequency has an uncertainty, δν B, determined by the uncertainty relation 2πh δν B τ h. Accordingly, arametric interaction requires only the weak criterion nν ~ ν B where n is an integer number. The gain for the arametric interaction is of the same order of magnitude from almost zero frequency u to frequencies well above 2πντ ~, e.g. ~ 0 (or 0 THz), as can be concluded from results of gain calculations [6, 9]. An increase of the suerlattice volume together with aroriate samle cooling and resonator design should lead to higher ower levels. An increase of doing by an order of magnitude may increase the gain coefficient corresondingly. The higher doing would result in an uer cut-off frequency j (ηε 0 E c ) - ~ 0 THz (η, dielectric constant and ε 0, electric field constant). Materials with smaller effective masses, like InGaAs/InAlAs suerlattices, would allow reaching frequencies above 0 THz. The use of materials with a shorter relaxation time and high doing, like GaN/GaAlN suerlattices, would extend the cut off frequency to almost 30 THz. The develoment of a Bloch oscillator for the THz range may contribute, as the quantum cascade laser [3], which is oerated at a temerature near liquid nitrogen temerature, towards a develoment of the THz frequency range. In conclusion, we have reorted the oeration of a Bloch oscillator based on Bloch gain in a semiconductor suerlattice and resented a theoretical analysis of its roerties. The Bloch oscillator reresents a room-temerature, tuneable monochromatic radiation source suitable for generation of microwave and THz radiation. Acknowledgement The work has been suorted by the Deutsche Forschungsgemeinschaft. One of us (B.I.S.) would like to thank L. Esaki for discussions (about Bloch oscillations) during the 54 th Nobel Laureate Meeting in Lindau, June

9 References [] L. V. Keldysh, Sov. Phys. Solid State 4, 658 (962). [2] L. Esaki and R. Tsu, IBM J. Res. Develo. 4, 6 (970). [3] F. Bloch, Z. Physik 52, 555 (928); C. Zener, Proc. Roy. Soc. London Ser. A 45, 523 (934). [4] A. Sibille, J. F. Palmier, H. Wang, F. Mollot, Phys. Rev. Lett. 64, 52 (990). [5] J. Feldmann et al., Phys. Rev. B 46, R7252 (992); Ch. Waschke et al., Phys. Rev. Lett. 70, 339 (993). [6] S. A. Ktitorov, G. S. Simin, and V. Ya. Sindalovskii, Fiz. Tverd. Tela 3, (97) [Sov., Phys. Solid State 3, 872 (972)]. [7] P. G. Savvidis, B. Kolasa, G. Lee, and S. J. Allen, Phys. Rev. Lett. 92, (2004). [8] A. A.Ignatov, E. Schomburg, J. Grenzer, K. F. Renk, and E. P. Dodin, Z. Phys. B 98, 87 (995); A. A. Ignatov and Yu. A. Romanov, Phys. Stat. Sol. B 73, 327 (976); A. A. Ignatov, K. F. Renk, and E. P. Dodin, Phys. Rev. Lett. 70, 996 (993). [9] A. A. Ignatov and Yu. A. Romanov, Phys. Stat. Sol. B 73, 327 (976); A. A. Ignatov, K. F. Renk, and E. P. Dodin, Phys. Rev. Lett. 70, 996 (993); H. Kroemer, cond-mat/ (2000); E. Schomburg, N. V. Demarina, and K. F. Renk, Phys. Rev. B 67, (2003). [0] M. Büttiker and H. Thomas, Phys. Rev. Lett. 38, 78 (977); A. A. Ignatov, V. I. Piskarev, and V. I. Schashkin, Sov. Phys. Semicond. 9, 345 (985). [] B. Rieder, Semiclassical Transort in Semiconductor Suerlattices with Boundaries, PhD thesis, Regensburg 2004, unublished. [2] H. Kroemer, Proc. IEEE 58, 844 (970). [3] R. Köhler et al., Nature 47, 56 (2002); L. Mahler et al., Al. Phys. Lett. 84, 5446 (2004). 9