Calculation of the nonlinear free-carrier absorption of terahertz radiation in semiconductor heterostructures

Transcription

1 Unversty of Wollongong Research Onlne Faculty of Engneerng - Papers (Archve) Faculty of Engneerng and Informaton Scences 2004 Calculaton of the nonlnear free-carrer absorpton of terahertz radaton n semconductor heterostructures C. Zhang Unversty of Wollongong, czhang@uow.edu.au J. C. Cao Chnese Academy of Scences, Shangha, Chna Publcaton Detals Ths artcle was orgnally publshed as: Zhang, C & Cao, JC, Calculaton of the nonlnear free-carrer absorpton of terahertz radaton n semconductor heterostructures, Physcal Revew B, 2004, 70, 933. Copyrght 2004 Amercan Physcal Socety. The orgnal journal can be found here. Research Onlne s the open access nsttutonal repostory for the Unversty of Wollongong. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au

2 PHYSICAL REVIEW B 70, 933 (2004) Calculaton of the nonlnear free-carrer absorpton of terahertz radaton n semconductor heterostructures C. Zhang and J. C. Cao 2 School of Engneerng Physcs, Unversty of Wollongong, New South Wales 2522, Australa 2 State Key Laboratory of Functonal Materals for Informatcs, Shangha Insttute of Mcrosystem and Informaton Technology, Chnese Academy of Scences, Shangha, Chna (Receved 3 February 2004; revsed manuscrpt receved 23 September 2004; publshed 23 November 2004) Nonlnear absorpton of terahertz waves by electrons n a semconductor heterostructure s calculated. We solve the quantum transport equaton for electrons strongly coupled to terahertz photons. The electrcal feld of the laser radaton s ncluded exactly, and the electron-mpurty nteracton s ncluded up to the second order. It s found that Joule heatng of the electronc system due to mpurty scatterng decreases rapdly due to the strong electron-photon nteracton. Our result s the dynamc equvalence of electron localzaton n a strong feld. In the lmt of weak radaton feld, the current s lnear n the feld strength. DOI: 0.03/PhysRevB PACS number(s): Fs, 7.0.Ca, 7.45.Gm Electromagnetc radaton n the frequency range of terahertz (THz) plays an mportant role n probng and studyng the optcal and transport propertes of low-dmensonal semconductor systems. Because of the nearly resonant match between the photon energy (of order of THz) and all characterstc energes of the electronc system such as Ferm energy, cyclotron energy, subband energy, etc. (all of order of mev or a few THz), varous resonant phenomena and nonlnear propertes n the electronc systems can be nvestgated. THz lasers have been appled to expermental nvestgaton of nonlnear transport and optcal propertes n electron gases such as those n semconductor quantum wells and heterostructures. 8 The dscovery of a new magnetoresstance oscllaton and zero-resstance state n hgh-moblty electronc systems 9 has stmulated new theoretcal nterest n the transport propertes of electrons under electromagnetc radaton. 2 5 A satsfactory understandng of the observed dc magnetoresstance oscllaton has been acheved. 2,4,5 There are three dfferent theoretcal methods of calculatng the electrcal transport under a radaton. The frst s the balance equaton approach 2 whch calculates the resstvty drectly. The second approach 4,5 employs the Green s functon formalsm and Kubo formula to calculate the conductvty or resstvty. We have developed a quantum transport equaton method 6 to study the frequencydependent phenomena. All exstng theoretcal work deal wth the electrcal current response of a strongly coupled electron-photon system to a weak ac or dc force. In these works, the terahertz feld E s strongly coupled to the electrons and the coupled system s respondng n a lnear fashon to another weak potental (or probng feld E). In a recent paper, 6 we calculated the lnear response of a coupled electron-photon system to a weak ac probng feld, j, =, E, where s the radaton frequency and s the probng frequency. It s found that the resonant absorpton of the probng feld can be acheved when the probng frequency s n the vcnty of the frequency of E. In Ref. 6, we used the terahertz feld E to tune the absorpton of E or j,. However, we dd not calculate the drect absorpton of the terahertz feld E by the electrons. The absorpton of E s much stronger than the absorpton of E. The absorpton of E goes to zero as E goes to zero. However, n the absence of the probng feld E, there s stll a strong absorpton of E. To calculate the absorpton of the THz radaton, electrcal current n the absence of any probng feld must be calculated. Such a current cannot be obtaned from the prevous theory 2 6 because the current obtaned there vanshes when the probng feld s removed. Furthermore, n the lmt of zero radaton feld, all exstng theores predcted a quadratc dependence of the current on the radaton feld whle the experment clearly shows a lnear dependence. In ths paper, we study the nonlnear absorpton of an ntense THz electrcal feld by electrons. We calculate the electrcal current drven drectly by the THz radaton feld (.e., wthout any probng feld) to the second order of the electron-mpurty nteracton. It s found that photon absorpton due to mpurty scatterng decreases very rapdly at hgh THz feld, ndcatng a suppresson of electron-mpurty scatterng by ntense photon radaton. Our result of the electrcal current s lnear n the radaton feld as E approaches zero. Our model system s a two-dmensonal electron gas under an ntense laser radaton. We choose the laser feld to be along the x drecton, E t =E 0 cos t e x, where E 0 and are the ampltude and frequency of the laser feld. For the notatonal convenence, both and the speed of lght c have been set to unty. Let us choose the vector potental for the laser feld to be n the form A = E 0 / sn t e x. The tme-dependent Schrödnger equaton for a sngle electron s gven as t r,t = H r,t = p ea 2 r,t. 2 2m* The tme-dependent wave functon can be wrtten as /2004/70(9)/933(4)/$ The Amercan Physcal Socety

3 k r,t = exp 2 t exp 0 k x cos t exp sn 2 t exp k t exp k r, 3 n q,t = F p,p + k. p 0 where 0 = ee 0 /m* 2 and = ee 0 2 / 8m* 3. The effect of electrcal feld s ncluded n ths wave functon exactly. Electrcal current of the system drven by the terahertz laser due to electron random-mpurty scatterng can now be calculated. The Hamltonan of the system can be wrtten as H = H 0 + H ee + H ei, where H 0 s the Hamltonan of a nonnteractng manyelectron system, where H 0 = 2m* p p + ea 2 b p t b p t b k t = b k exp 2 t exp 0 k x cos t exp sn 2 t and b k b k s the creaton (annhlaton) operator for an electron wth momentum k n the absence of THz photon radaton. The term H ee s the electron-electron nteracton H ee = 2 p,p,q V q b p+q t b p q t b p t b p t, where V q =2 e 2 /q s the Fourer transform of the electronelectron nteracton n two dmensons. H ei s the nteracton between the electrons and random mpurtes: H ei = p,q V q b p+q t b p t e q R, where R s the poston of the th mpurty whch s assumed to be sngly charged. Now the total average two-dmensonal (2D) current densty of the system s defned as j = H A = e m* p p + ea b p t b p t. The current gven n Eq. (8) s drectly drven by the radaton feld;.e., t s not a lnear response to any probng feld. The method developed n Ref. 6 can be used to obtan nonlnear current for the present system. There are two basc equatons to descrbe the quantum transport of the system. The frst equaton s the equaton of moton for the sngleelectron densty matrx F p,p+k = b p t b p+k t, t F p,p + k = p+k p + k x 0 sn t F p,p + k + q V q n q,t F p + q,p + k e q R F p,p + k q Here p = p 2 /2m s the knetc energy of an electron havng momentum p and 9 The second equaton descrbes tme-dependent current densty j t : dj t dt = e m* q V q qn q,t e q R. The key dfference between ths work and Ref. 6 s that the present work does not seek a lnear current response to a probng feld. Instead, we solve the exact current densty drven by the radaton feld. The soluton of Eqs. (9) and (), up to the second order n electron-mpurty scatterng, can be wrtten as j = ne m * q where qv q m 0 m m D q,m = V q Q q,m J m q x 0 e q x V qq q,m 0, D q,m 2 3 s the delectrc functon n the random-phase approxmaton and Q q, = dp f p+q f p 2 2 p+q p 4 s the polarzablty for free electrons. Here, f p s the Ferm- Drac dstrbuton and J m x s the Bessel functon of the frst knd. The terms wth dfferent m represent varous photon emsson and absorpton processes. Here t s necessary to pont out that the appearance of the equlbrum dstrbuton f p n the fnal result does not mean that the orgnal nonequlbrum densty matrx F p,p+k = b p t b p+k t has been treated n an equlbrum fashon. Our theory nvolves decomposng the nonequlbrum densty matrx nto successve densty matrces for photon sde bands. Such a densty matrx for the mth photon sdeband s wrtten n the form of the F 0 m R m t where F 0 m s the densty matrx n the absence of radaton feld. 6 Because of the dfferent tme dependences of R m for dfferent photon sdebands, F 0 m R m t s stll a nonequlbrum quantty and has to be determned by the equaton of moton ncludng the electron-mpurty scatterng. Such an equaton of moton s then solved to the lowest order n electron-mpurty scatterng. In the equaton of moton, for those F 0 terms multpled by a term proportonal to mpurty potental, only a zerothorder soluton s requred. Our choce of a zeroth-order soluton s the Ferm-Drac dstrbuton functon. For sotropc systems, the electrc current s along the drecton of polarzaton of the laser feld. The real part of the electrc current s gven as 933-2

4 FIG.. Plot of R= 2 Re j /r versus the reduced electrc feld r for several dfferent frequences: the sold lne s for = Thz, the dotted lne s for =2 Thz, and the dashed lne s for =5 Thz. Re j x = ne m * q J q x V q m q x 0 m 0 m Im D q,m sn q x 0 m /2 Re D q,m cos q x 0 m /2. 5 We now make use of the followng facts: (a) The delectrc functon s only dependent on the magntude of q; (b) the Bessel functons are symmetrc for even m and antsymmetrc for odd m wth respect to the argument q x. Therefore the ntegraton over the drecton of q wll be zero for the second term n the curly brackets. The real part of the electrc current s now wrtten as Re j x = ne m * q J q x V q m q x 0 m 0 m Im D q,m sn q x 0 m /2. 6 The parameter that controls the electrcal feld dependence of the current s 0. We consder the two lmtng cases. () At small 0 (low feld and hgh frequency), we only retan the lnear and cubc terms n E 0. The real part of the current s gven as Re j x + ne 2 m * 2 3 q ne 4 m * 4 5 q q x 2 V q Im D q, E 0 q x 4 V q Im D q, D q,2 E The current s lnear n E 0 as E 0 0. Ths s n contrast to the E 2 0 dependence predcted by other theores. It s consstent FIG. 2. Plot of R= 2 Re j /r versus the laser frequency. The sold lne s for r=0., the dotted lne s for r=0.5, and the dashed lne s for r=.0. wth the lnear dependence found n the experment. Snce the magnetc feld s absent n the current calculaton, we cannot make a drect comparson wth the experment. However, we do not expect that applcaton of a constant magnetc feld wll alter such a lnear dependence. It s also mportant to note that terms of hgh order n E 0 also represent hgh-order multphoton processes. () At large 0, J m q x 0 2/ qx 0 cos q x 0 m /2 + /4. Therefore the dsspatve part of the electrcal current vanshes n the lmt of strong electrcal feld. Ths s the dynamc equvalence of electron localzaton n a strong statc electrc feld. When electrons are strongly coupled to the photons, electron-mpurty scatterng approaches zero. We have numercally calculated the real part of the current. The parameters used n our calculaton are those of GaAs, m* =0.067m 0, and r s =m*e 2 / 0 k F =.0. All energes and frequences are n unts of E F, and all wave numbers are n unts of k F. Here =k F ee/ m* 2 s the dmensonless electron-photon couplng parameter. Fgure shows electrc feld dependence of the real part of the electrc current dvded by the reduced electrc feld, r=k F ee/ m* 0 2 (where 0 = THz). In the plot, we multply Re j /r by the frequency squared so the values R = 2 Re j /r at three dfferent frequences are comparable. The quantty R s a drect measure of the absorpton of THz photons by the electronc system due to electron-mpurty scatterng. At weak electrc felds or hgh frequences, the current s almost lnear n the electrc feld. The electronphoton couplng s nversely proportonal to 2. Therefore the absorpton ncreases as frequency decreases. At fxed frequency, as the laser ntensty ncreases, the current starts to devate from the lnear dependence and the absorpton coeffcent j/e starts to decrease wth the feld ntensty. Ths behavor s a drect consequence of the electron-mpurty scatterng tme beng affected by electron-photon couplng. At low feld ntensty or weak electron-photon couplng, the scatterng tme s ndependent of the electrc feld. As the feld ntensty ncreases, the electron-photon couplng becomes stronger and, as a consequence, the electron-mpurty 933-3

5 scatterng becomes less effectve. Ths reduced electronmpurty scatterng s the orgn of the reducton of the dsspatve current at hgh felds. In Fg. 2, we plot the quantty R= 2 Re j /r versus the laser frequency. At low frequency, the absorpton ncreases wth frequency due to the ncreasng phase space for electron-mpurty scatterng. The energy loss of the electromagnetc waves s weaker but ncreases more rapdly at strong feld. At hgh frequency the real part of the current decreases wth frequency due to rapdly decreasng electronmpurty scatterng at the hgh-frequency regon. At hgh frequency, electron-photon couplng s weak and therefore the conductvty under the strong feld s very close to that under the weak feld. In concluson, we have presented a study of the nonlnear electrcal transport n low-dmensonal systems under ntense THz laser radaton. Due to the strong electron-photon couplng, the electron-mpurty scatterng and ts effect on the energy loss of the laser feld are sgnfcantly reduced. Ths work s supported by the Australa Research Councl and the Natonal Natural Scence Foundaton of Chna. J. C. Cao, Phys. Rev. Lett. 9, (2003). 2 N. G. Asmar, A. G. Markelz, E. G. Gwnn, J. Cerne, M. S. Sherwn, K. L. Campman, P. E. Hopkns, and A. C. Gossard, Phys. Rev. B 5, 8 04 (995). 3 W. Xu and C. Zhang, Appl. Phys. Lett. 68, 3305 (996). 4 W. Xu and C. Zhang, Phys. Rev. B 55, 5259 (997). 5 N. G. Asmar, J. Cerne, A. G. Markelz, E. G. Gwnn, M. S. Sherwn, K. L. Campman, and A. C. Gossard, Appl. Phys. Lett. 68, 829 (996). 6 A. G. Markelz, N. G. Asmar, B. Brar, and E. G. Gwn, Appl. Phys. Lett. 69, 3975 (996). 7 B. N. Murdn, W. Hess, C. J. G. M. Langerak, S.-C. Lee, I. Galbrath, G. Strasser, E. Gornk, M. Helm, and C. R. Pdgeon, Phys. Rev. B 55, 57 (997). 8 C. Zhang, Appl. Phys. Lett. 78, 487 (200). 9 R. G. Man, J. H. Smet, K. von Kltzng, V. Narayanamur, W. B. Johnson, and V. Umansky, Nature (London) 420, 646 (2002). 0 M. A. Zudov, R. R. Du, L. N. Pfeffer, and K. W. West, Phys. Rev. Lett. 90, (2003). P. D. Ye, W. Engel, D. C. Tsu, J. A. Smmons, J. R. Wendt, G. A. Vawter, and J. L. Reno, Appl. Phys. Lett. 79, 293 (200). 2 X. L. Le and S. Y. Lu, Phys. Rev. Lett. 9, (2003). 3 V. Ryzh and V. Vyurkov, Phys. Rev. B 68, (2003). 4 Adam C. Durst, Subr Sachdev, N. Read, and S. M. Grvn, Phys. Rev. Lett. 9, (2003). 5 Junren Sh and X. C. Xe, Phys. Rev. Lett. 9, (2003). 6 C. Zhang, Phys. Rev. B 66, 0805 (2002)