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Temporal stimulated intersubband emission of photoexcited electrons F.T. Vasko∗ Institute of Semiconductor Physics, NAS Ukraine, Pr. Nauki 41, Kiev, 03028, Ukraine A. Hernández-Cabrera† and P. Aceituno 5 Dpto. F´ısica Básica, Universidad de La Laguna, La Laguna, 38206-Tenerife, Spain 0 (Dated: February 2, 2008) 0 Wehavestudiedthetransientevolutionofelectronsdistributedovertwolevelsinawidequantum 2 well,withthetwolevelsbelowtheopticalphononenergy,afteranultrafastinterbandexcitationand n cascade emission of optical phonons. If electrons are distributed near thetop of thepassive region, a a temporal negative absorption appears to be dominant in the intersubband response. This is due J to the effective broadening of the upper level state under the optical phonon emission. We have 5 thenconsideredtheamplificationofthegroundmodeinaTHzwaveguidewithamultiquantumwell 2 placed at the center of the cavity. A huge increase of the probe signal is obtained, which permits thetemporal stimulated emission regime of thephotoexcited electrons in theTHz spectral region. ] i c PACSnumbers: 73.63.Hs,78.45.+h,78.47.+p s - l r I. INTRODUCTION t m . The temporal emission of THz radiation due to the (c) t a coherent oscillations of electrons under ultrafast inter- (a) m band excitation have been studied during last decades o 0 f t=0 - (see1 for review). The duration of the generated THz d ex A pulseisintheorderofpicosecondsduetoeffectiverelax- n ationofthe coherentresponse. Furthermore,the steady- o c state spontaneous and stimulated emissions from differ- o- 21 B [ entsemiconductorstructuresunderanelectricfieldpump have been demonstrated (see2 and Refs. therein). How- 21 2 p ever,the characterofthe temporalevolutionofphotoex- v 0 cited electrons during the non-coherent stage of the re- excitation 6 laxationis not completely understoodyet. After the ini- 3 tial photoexcitation and emission of the optical phonon (b) 6 cascade, which takes place in a picosecond time inter- 0 val, which have been considered in details (see1,3,4,5 and L THz MQW 4 Refs. therein), a non-equilibrium distribution appears 0 FIG. 1: Scheme of the intersubband transitions for electrons in the passive region, with the energy ε less than the / ex t optical phonon energy, ~ω 6. A temporal evolution of with energies near the top of the passive region (the stim- a o ulated emission and absorption are shown by the solid and m this distribution takes place during a nanosecond time dashed arrows, respectively) (a), the geometry of the THz interval. This evolution is caused by the quasi-elastic - waveguidewiththeMQWstructureinthecenter(b),andthe d scattering of electrons with acoustic phonons. Due to initial distributions over the passive region for the cases A n the partial inversion of this distribution, a set of pecu- and B (c). o liarities of the magnetotransportcoefficients, such as the c total negative conductivity, or the negative cyclotron ab- : v sorption, appears7. i X Forthe wide quantumwell(QW)caseunder consideration,withtwolevelsinthe passiveregion,thecharacter THz waveguide, as it is shown in Fig. 1b. In this pa- r a of the evolution appears to be more complicate due to per we consider the amplification of the probe ground the interlevel scattering. The regime of negative absorp- mode in the THz resonator caused by the temporal neg- tion is possible due to a moreeffective broadeningofthe ative absorptiondescribed above. We have found a huge absorption (dashed arrow in Fig.1a) in comparison with amplification of the probe signal, so that it may be con- theintersubbandemission(solidarrow). Suchregimeap- cluded that a temporal stimulated emission takes place pears because the absorption process involves the state in the THz waveguide with weak cavity losses and with in the active region for which the optical phonon emis- the MQW structure placed at the center. sion is allowed. Thus, a question arises about the temporal stimulated emission coming from a wide multiple Our study is based on the general kinetic equation for quantum well (MQW) structure placed at the center of the distribution function in the conduction c-band, f , αt 2 written in the following form: II. TEMPORAL EVOLUTION OF THE ELECTRONIC DISTRIBUTION ∂f αt =G +J(f αt), (1) ∂t αt | We shall now turn to consider the temporal evolution of the photoexcited electrons in the passive region caused by the quasielastic scattering with acoustic whereG andJ(f αt)arethe photogenerationrateand αt | phonons. Since the above-discussed initial distributions collisionintegralforthec-bandstateα,respectively. The generation rate in (1) is given by4: are isotropic over the 2D-plane, one has to consider the energy dependent distribution functions f governed 1,2εt by the system of kinetic equations: G =w2 g δ (ε ε ) (2) αt t Xl l ∆l α− l ∂f1εt = J(f 1εt) ν (f f ), ∂t | − ε−ε21 1εt− 2ε−ε21t where wt is the form-factor of the excitation field with ∂f2εt = J(f 2εt)+ν (f f ). (4) the pulse durationτp,the factorgl describesthe relative ∂t | ε+ε21 1ε+ε21t− 2εt contributions of different valence v-band states (heavy Moreover, these equations are written for the intervals and light holes confined in the wide QW). The shape (0,~ω )and(ε ~ω ,~ω )forthefirstandsecondsub- of the photoexcited distribution is given by the Gaus- o 21− o o bands, respectively. The interlevel relaxation frequency sian function δ (E) = exp[ (E/∆)2]/(√π∆) with the ∆ − is introduced here as width ∆ and centred at E = 0. The width of these function is determined by two different broadening pro- 2π 2π dϕ cesses: the spreadingdue to the energy-timeuncertainty νε = θ(ε) ~ Z 2π |CQ|2 relation and the anisotropy of the valence band. These 0 pX′,q⊥ values are estimated by the energy values ~/τp and βεl, 2e−iq⊥z 1 2(2N +1)δ(ε ε′), (5) respectively, where β 1 is the anisotropy parameter. × | | Q − ≪ (cid:12)(cid:10) (cid:11)(cid:12) Under the optical phonon emission an additional broad- where C is th(cid:12)e matrix ele(cid:12)ment for the bulk electron- Q ening appears. Such a broadening is caused by the weak phonon interaction with the acoustic phonons. The phonondispersion,andcharacterizedbytheenergyδεopt. quasielastic collision integral in the jth subband is Thus, the initial conditions in the passive region, af- reached as ter the picosecond stage of the evolution due to optical ∂ ∂f phonon emission, are obtained from Eqs. (1,2) in the J(f jεt)= D jεt +V f . (6) jε jε jεt | ∂ε(cid:18) ∂ε (cid:19) following manner: The diffusion and drift coefficients in (6), D and V , jε jε n(j) are determined as follows: f = l δ (ε ε(j)). (3) jεt=0 Xl ρ2D ∆l(j) − l Djε = πρ2D 2π dϕ ∞ dq⊥V C 2 (7) (cid:12)(cid:12) Vjε (cid:12)(cid:12) ~ Z0 2π Z−∞ 2π | Q| Heneerregρy2Disisdettheerm2Dindedenasisty∆o(jf)statems,axth(~e/tτot,alβbεro,adδeεnin)g, (cid:12)(cid:12) (cid:12)(cid:12) j e−iq⊥z j 2 (2NQ+1)(~ωQ)2/2 , l ∼ p l opt × | | (cid:12) ~ω (cid:12) and j =1,2 are the corresponding two levels in the pas- (cid:12)(cid:10) (cid:11)(cid:12) (cid:12) Q (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) sive region. Below, we will concentrate solely on two where N = [exp(~ω /T) 1](cid:12)−1. A similar descript(cid:12)ion models of the initial distribution: a narrow Gaussian Q Q − of the quasi-elastic relaxation in a two-level system was peak (case A) and a flat distribution over the interval (~ω ε ,~ω ) (case B), which are shifted from the discussed in Ref. 8. o 21 o − Since the elastic interlevel relaxation is the dominant boundary of the passive region, as Fig. 1c shows. Both process for the energy interval (ε ,~ω ), one obtains distributions are relaxed between the sub-bands and to 21 o f f f if t exceeds the interlevel relax- thebottomofthepassiveregion. Thetemporalintersub- a1tεiotn≈tim2eε.−εU21stin≃g thεte energy variable ξ = ε ~ω we band responseis determined by the interlevelredistribu- − o tion of the population. determine the distribution function fξ+~ωot ≡ fξt from the diffusion-drift equation for the two-level zone of the The analysis we will carry out next is divided in two passive region, 0>ξ >ε ~ω : 21 o sections. The temporal evolution of the distribution − over the two subbands is described in Sec. II, includ- ∂f ∂ ∂f ξt ξt = D +V f , (8) ing the consideration of the resonant intersubband re- ξ ξ ξt ∂t ∂ξ (cid:18) ∂ξ (cid:19) sponse. The results for the transient amplification of a probe mode in the THz waveguide are given in Sec. III, where D and V are the energy diffusion and the drift ξ ξ withourconclusionspresentedinSec. IV.TheAppendix coefficients determined by Eq. (7). These coefficients contains the microscopical evaluation of the broadening are shown in Fig. 2 for a GaAs-based QW which is 320 energy for the intersubband transitions. ˚A wide, with an interlevel distance ε = 15 meV and 21 3 1.0 0.15 0 V / V , 0 f 0 t .10 0.5 D / 0.05 D 0.0 0.00 0 10 20 30 40 10 15 20 25 30 (meV) (meV) FIG. 2: Normalized diffusion and drift coefficients versus en- FIG. 3: Evolution of the electron distribution for the case ergy. Solidanddottedcurves: diffusioncoefficientforT =4.2 B. Solid, dashed, and dotted curves correspond to t =0 ns, K and T =20 K respectively. Dashed line: drift coefficient 1.6 ns, and 3.2 ns respectively. Vertical arrow indicates the which is independentof T. position of ε21. for two different temperatures T = 4.2 K and T = 20 K3.9.1Thu1s0,9thmeeVno2r/msaalnizda9ti.o5n5 co1e0ffi9cmieenVts2/Ds~fωoor≡T =Dξ4=.20 =K 1.0 × × andT =20K,respectively. Anothercoefficient,whichis in1d0e9pemnedVen/ts.oNnotthee, tthematpDer/aVture,TisfoVr~ωTo=≡2V0ξ=K0w=hi5l.e2a2 xc A e × ≃ n visibledistinctionofenergydependenciesappearsatT = / 4 K. We use below the energy-independent coefficients nt0.5 V =3.5 109meV/s,andD =2.2 109meV2/sand6.2 × × × 109 meV2/s for T = 4.2 K and T = 20 K, respectively. B This approximation is valid within an accuracy better than 20% for the numerical parameters in the interval (~ω ε ,~ω ) considered below. Another important o 21 o param−eter is the interlevel relaxation time, ν−1, which 0.0 ε was assumed to be shorter than the time scales of Figs. 0 2 4 6 3,4. Actually, for the above used parameters, ν−1 varies t (ns) ε FIG. 4: Temporal evolution of the concentration for the dis- from 0.21 ns to 0.19 ns over the passive region. tributionsAandB,at T =4.2Kand20K(solid anddashed Equation (8) may be considered with the zero bound- curves,respectively). ary conditions at ξ if ~ω ε > ε . The o 21 21 → ±∞ − initial distribution is located in the region ξ < 0, and electrons do not reach the bottom of the second level, In orderto describe the temporalnegative absorption, i.e., ξ > ε . For the case A we use the initial con- we need the concentration over the negative absorption 21 dtoittiaolne:xfcξit−t=ed0 =connceexnδt∆ra(ξtio−n,ξeξxe)x/2≃ρß−2D9,mwehVereisntehxeisextchie- rTehgeiotnem(~pωoora−leεv21o,lu~tωioo)ndoeffinnte,dobatsaninte=dw2ρitßh2DthRe−0sε2o1ludtξifoξnts. tation energy, and the half-width of the peak is equal to ofEq.(8)forthecasesAandB,andtheQWparameters 2.5 meV. Thus, the solution for Eq.(8) may be written detailed above, is shown in Fig.4. as the moving peak The temporal evolution of the resonant absorption of thenon-equilibriumelectronsisdescribedbytherealpart n f = ex δ (ξ+Vt ξ ) (9) of the conductivity (see8 and Appendix): ξt 2ρ ∆t − ex ß2D e2 v 2n Γ γ ß⊥ t with the time-dependent half-width ∆t = √∆2+4Dt. Reσδε,t = ε| /~| 2 (cid:18)δε2+Γ2 − δε2+γ2(cid:19). (10) 21 For the case B, the broadening of the stepped distribution is equal to 2.5 meV. The solution is given by the Here δε = ~ω ε is the detuning energy, v is the 21 ß⊥ − integralofthe initial distribution multiplied by a similar intersubband velocity matrix element, Γ and γ are the to (2) form-factor. This solution is plotted in Fig.3 for broadeningenergyvaluesduetotheopticalphononemis- different times. sion and the elastic scattering, which correspond to the 4 absorption and stimulated emission processes. Using Eq.(A11) and the parameters given above we make an estimateofΓ 1.3(1.5)meV[forξ = 4( 9)meV,re- 1.5 ∼ ex − − spectively],sothatΓ γandoneobtainsReσ <0. δε=0,t ≫ The contribution to Reσ from electrons distributed δε,t A odvueerttohtehreesgaiomne−brεo21ad>enξing>oεf2t1h−e a~bωsooripstieoqnuaalndtoemzeirso- 3 10) 1.0 x sionprocesses. We havealsosupposed thatelectronsare ( absent from the region ξ < ε ~ω , where only the t absorption due to transitions 121 −2 isopossible. n ( 0.5 B → l III. TRANSIENT AMPLIFICATION 0.0 0 2 4 6 We then consider the amplification of a probe THz t (ns) mode in the ideal waveguide of width L due to the tem- FIG.5: Transientamplification η forthesame conditionsas t poral negative absorption discussed above. The trans- in Fig.4. verse electric field in the resonator Eß⊥exp( iωt+ikx) zt − is ruled by the wave equation: meV under the photoexcited concentration n = 1010 ex ∂2 κ2 Eß⊥+iǫ2ω∂Ezßt⊥ =0 (11) cm−2. We have also used ǫ = 12.9 and L = 19 µm, cor- (cid:18)∂z2 − (cid:19) zt c2 ∂t responding to the ground mode propagation along the THzwaveguide. Onecanseethesaturationoflnη when with κ2 = k2 ǫω2/c2, where ǫ is the dielectric per- n /n approachestozero(cf. Fig.4). Themaximatlvalue − t ex mittivity supposed to be uniform across the structure. oflnη is around103, i.e., a huge temporalamplification t The boundary conditions at z = L/2 takes the form of the probe signal has been obtained. Thus, the stimu- ± Ezß=⊥±L/2,t = 0. At the center of the resonator (z = 0), latedregimeoftheemissionshouldberealizedintheTHz where the MQW is placed, one has to use: waveguide with a weak damping (such waveguides have been studied recently, see Ref. 9 andreferences therein). Eß⊥ 0 =0, ∂Ezßt⊥ 0 iN4πωσ Eß⊥ , (12) zt −0 ∂z (cid:12) ≃− c2 δε,t z=0t (cid:12) (cid:12)−0 (cid:12) (cid:12) IV. CONCLUSIONS where N is the number(cid:12)of wells in the structure. The initial condition for the ground mode propagating along the resonator is: Eß⊥ =Ecos(πz/L). In summary, we have considered a new non-coherent zt=0 transient mechanism of the stimulated emission in the Taking into account the slowness of the temporal evo- lutionundertheconditionα=Nπe2 v 2Ln /γc2 1 THz spectral region. The mechanism neither requires ß⊥ ex | | ≪ coherent response nor inverted distribution but it rather and restricting ourselves to the resonant case, δε = 0, we obtain the solution of Eqs. (11, 12) as Eß⊥ = appears due to the different broadening of emission and zt absorption contributions in Eq. (10). cos(πz/L) where the time-dependent field is governed t E The consideration performed here is based on several by: assumptions. Rather than using a microscopic calcu- ∂ n 1 4αc2 lation of the photogeneration, we have used two mod- t t E = , = , (13) ∂t τ∗n Et τ∗ ǫω L2 els for initial conditions (A and B) to obtain the tem- ex 21 poral evolution of the distribution. We have also ne- with the initial condition: Et=0 =E. glected the energy dependencies of Dξ and Vξ in Eq. Afteraveragingovertheresonator,thePoyntingvector (8). In Eq. (10), which describes the intersubband along OX-direction is introduced as response, we have neglected the Coulomb renormaliza- tion effect and taken into account that Γ γ as it is kc2 ≫ S = dz 2 (14) demonstrated in the Appendix. We also neglected non- t zt 2πω Z |E | parabolicity of the electron dispersion law and the effect of the spatial confinement on phonons and electron- andthetemporalamplificationisdeterminedastheratio phonon interaction10. These approximations are gen- of (14) to the Poynting vector at t=0: erally accepted for the GaAs/AlGaAs heterostructures St t dt′ nt′ with the parameters used here. In addition, instead of η = =exp 2 . (15) t S (cid:18) Z τ∗ n (cid:19) a stimulated emission effect due to the transient nega- t=0 0 ex tive absorption, we have calculated the amplification of InFig. 5weplottedthelogarithmoftheamplificationco- aprobesignal,andtheroleofthewaveguidelossesisnot efficient versus time for a five-layers MQW with γ =0.2 considered here. 5 To conclude, we have found a huge amplification due Here Cß(po) is the matrix element of the Fröhlich inter- ßQ to the temporal negative absorption under intersubband action with the bulk phonon mode characterized by the transitions of photoexcited electrons in a wide MQW wave vector Q=(q,q ). ß⊥ structureplacedatthecenterofaTHzresonator. Weex- For the case of the resonant intersubband excitation, pectthatthepresentedanalysismotivatesanexperimen- ~ω ε ε , only the component (δ ) δF 21 21 ε 21,p εp tal treatment of the transient stimulated emission both a|ppe−ars t|o≪be essential. Using the non-dFiagonal≡matrix for the system considered and for another heterostruc- element δh = (ie/ω)E v we transcform Eqs. (A2) 21 ß⊥ ß⊥ tures with closely-spaced levels (stepped QW, tunnel- and (A3) into coupled structures, etc.). ~ω−ε21+iγp δFεp− wpp′δFε−~ωop′ (cid:0) (cid:1) Xp Acknowledgments ie = 2ωEß⊥vß⊥(cid:20)fε−~ω/2(cid:16)Gßε−A~ω/2−Gßε−R~ω/2(cid:17)11,p ThisworkhasbeensupportedinpartbytheConsejer´ıa ddee ECdanuacaricaiós.n, Cultura y Deportes. Gobierno Autónomo −fε+~ω/2(cid:16)Gßε+A~ω/2−Gßε+R~ω/2(cid:17)22,p(cid:21), (A4) where the broadening energy is given by APPENDIX A: BROADENING OF INTERSUBBAND TRANSITIONS γp = i(cid:20)(cid:16)Σˆßε+R~ω/2(cid:17)22,p−(cid:16)Σˆßε+A~ω/2(cid:17)11,p(cid:21) 2peiQ·r jp′ 2 Below we present the evaluation for the resonant in- i Cß(po) 2 |h | | i| ≃ | ßQ | (cid:20)ε+~ω/2 ~ω +iλ tersubband conductivity givenby Eq. (10). We consider jXQp′ − o the interaction of electrons with optical phonons at zero 1peiQ·r jp′ 2 temperature and demonstrate that only the broadening −ε |h~ω|/2 ~|ω i+| iλ(cid:21) (A5) of the 1 2 transition, which is responsible for the ab- − − o → sorption,takesplaceinthe passiveregionwhilethe 2 1 → and wpp′ is determined by a similar expression which is transitionremainsnarrow(the broadeningisdetermined notessentialbelow. ThelowerEq. (A5)iswritteninthe by the weak elastic scattering). Born approximation and, from the formal point of view, Within the framework of nonequilibrium diagram λ +0. If we take into accountthe elastic scattering, λ technique3,11,onedescribesthetransientresponseonthe sh→ould be replaced by the elastic broadening energy. resonant probe field E exp( iωt) based on the linear ß⊥ The Fourier component of current density due − combination of Green’s functions: to the intersubband transitions under consideration is expressed through δF according to j = dε εp ω Fˆt1t2+Gˆßt1At2−Gˆtß1Rt2 ≃Z 2π~e~iε(t1−t2)Fˆεt1+2t2 +δFt1t2. −ievß⊥ dp dεδFεp/(2πc~)2. Herewehavereplacedthe c(A1) perturbRationRofthcedensitymatrixδρt usingtherelation: Here ˆ =2f (GˆßA GˆßR)describesthe temporalevo- δρ =( i~/2)lim δ . Performingthesubstitution Fεt εt ε − ε t − t1,2 Ft1t2 b lution of energy distribution f governed by the qua- δF =i2eE v ϕ /ω we obtain the conductivity : siclassic kinetic equation (1), εGtˆßA,R is the advanced b εp ß⊥ ß⊥ εpc (A) or retarded (R) equilibrium Green’s function, and e2 v 2 dp ß⊥ σ = | | dεϕ . (A6) δFt1t2 describes the linear response on the perturbation ω π Z (2π~)2 Z εp δhexp( iωt). Usingtheε,t-representationweobtainthe c − The function ϕ is governed by the integral equation linearized equation for δ exp( iωt) in the following εp c Fε − form: c ~ω−ε21+iγp ϕεp− wpp′ϕε−~ωop ~ωδ ε [hˆ,δ ε] δhˆε−~ω/2,t ˆε+~ω/2,tδh (cid:0) (cid:1) Xp = δΩεcGFˆßε−A−~ω/2−cFGˆßε+R−~(cid:16)ω/c2δFΩε −F c(cid:17) = fε−~ω/2(cid:16)Gßε−A~ω/2−Gßε−R~ω/2(cid:17)11,p +cΣˆßε+R~ω/2δFε−δFεΣˆßε+Ac~ω/2, (A2) −fε+~ω/2(cid:16)GAε+~ω/2−GRε+~ω/2(cid:17)22,p, (A7) c c whereΣˆßR,A istheR-orA-self-energiesandtheintegral which is obtained from Eq. (A4). ε contribution to this equation is given by Next we write the resonant conductivity (A6) for the case of the δ-like electron distribution f = ε δΩε ≃XQ |CßßQ(po)|2eiQ·rδFε−~ωoe−iQ·r. (A3) c(naleixz/e2dρißn2Dth)eδ(iεnetxer−vaεl)(w~ωitoh thεeexe,x~cωitoa).tiDonueenteortghyeεwexealko-- c c − 6 ness of the integral term in Eq.(A7) we obtain the con- first term. The corresponding broadening energy is de- ductivity in the form: termined as Γ=Reγp|εp→εex,ε→εex+~ω/2, or σ = e2|vß⊥|2n dε  (cid:16)Gßε−A~ω/2−Gßε−R~ω/2(cid:17)11,p Γ ≃ 2π2ǫe∗2V~ωo p|h2|pei′q/⊥~z2|1+i|2q2 ω 2π exZ p~ω−ε21+iγp |ε→εex+~ω/2 ×δ(εex+εqX2⊥1p−′ |~ω−o−|ε1p′)|εp→⊥εex, (A10) GßA GßR (cid:16) ε−~ω/2− ε−~ω/2(cid:17)22,p . (A8) whereV isthenormalizationvolumeandtheeffectivedi- −~ω−ε21+iγp |ε→εex−~ω/2 electric permittivity, ǫ∗ =(ǫo−ǫ∞)/ǫoǫ∞, is introduced through the static and high-frequency dielectric permit-  tivities, ǫ and ǫ . Performing the integration over the The spectral densities in the numerators, o ∞ energy one obtains: i2λ (cid:16)GAε−~ω/2−GRε−~ω/2(cid:17)jj,p ≃ (εex−εjp)2+λ2, (A9) Γ≃ e24~ǫω∗oρß2DZ dq⊥Z 2πdϕ|h2q|2eiq+⊥zQ|12i|2, (A11) 0 ⊥ ϕ are written through the δ-functions if λ +0. As a re- sult, Eq.(A8) is transformed into Eq.(10→) with the elas- where the denominator contains the factor Q2ϕ = (p2ex+ tic broadening energy in the second contribution while p˜2ex−2pexp˜excosϕ)/~2 with the characteristic momenta the optical phonon induced broadening appears in the p =√2mε and p˜ = 2m[ε (~ω ε )]. ex ex ex ex o 21 − − p ∗ Electronic address: [email protected] (1987). † Electronic address: [email protected] 7 F.T. Vasko,JETP Lett., 79, 431 (2004). 1 J. Shah, Ultrafast Spectroscopy of Semiconductors and 8 A.Hernandez-Cabrera,P.Aceituno,andF.T.Vasko,Phys. Semiconductor Nanostructures(Springer,Berlin,1999);F. Rev.B67,045304(2003); S.Khan-ngernandI.A.Larkin, Rossi and T. Kuhn,Rev.Mod. Phys. 74, 895 (2002). Phys. Rev.B 64, 115313 (2001). 2 E. Gornick and R. Kerstling, Semicond. Semimet. 67 389 9 B.S. Williams., S. Kumar, H. Callebaut, Q. Hu, and J.L. (2001); V.N. Bondar, A.T. Dalakyan, L.E. Vorobev, D.A. Reno, Appl. Phys. Letters 83, 5142 (2003); H.N. 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