Nonlinear magneto-gyrotropic photogalvanic effect
H. Diehl, V.A. Shalygin, L.E. Golub, S.A. Tarasenko, S.N. Danilov, V.V. Bel'kov, E.G. Novik, H. Buhmann, C. Brüne, E.L. Ivchenko, S.D. Ganichev
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Nonlinear magneto-gyrotropic photogalvanic effect
H. Diehl, V.A. Shalygin, L.E. Golub, S.A. Tarasenko, S.N. Danilov, V.V. Bel’kov, , E.G. Novik, H. Buhmann, C. Br¨une, E.L. Ivchenko, and S.D. Ganichev Terahertz Center, University of Regensburg, 93040 Regensburg, Germany St. Petersburg State Polytechnic University, 195251 St. Petersburg, Russia A.F. Ioffe Physical-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia and Physical Institute (EP3), University of W¨urzburg, 97074 W¨urzburg, Germany
We report on the observation of nonlinear magneto-gyrotropic photogalvanic effect inHgTe/HgCdTe quantum wells. The interband absorption of mid-infrared radiation as well as theintrasubband absorption of terahertz radiation in the heterostructures is shown to cause a dc electriccurrent in the presence of an in-plane magnetic field. A cubic in magnetic field component of thephotocurrent is observed in quantum wells with the inverted band structure only. The experimentaldata are discussed in terms of both the phenomenological theory and microscopic models.
PACS numbers: 73.21.Fg, 72.25.Fe, 78.67.De, 73.63.Hs
I. INTRODUCTION
Much current attention in condensed matter physics isdirected toward understanding the spin dependent phe-nomena, both from the fundamental point of view anddue to increasing interest in spintronics devices that arebased not only on the electron charge, but also on itsspin. Conventional quantum well (QW) structures fab-ricated of III-V and II-VI wide gap materials are in fo-cus of present day investigations. QW structures basedon HgTe appear to be very attractive for the study offundamental spin-orbit effects. Narrow gap HgTe-basedQWs are characterized by an extraordinary large Rashba-type spin-orbit splitting, a parameter crucial for the fieldof spintronics because it allows an electric field controlof spins, determines the spin relaxation rate, and canbe utilized for all-electric spin injection. The lifting ofspin degeneracy is caused by spin-orbit interaction due tostructure and bulk inversion asymmetries which lead toRashba and Dresselhaus spin-orbit terms in the Hamil-tonian, respectively (see Refs. 2,3,4,5,6,7). The Rashbaspin splitting in HgTe-based QWs can reach values ofup to 30 meV, which is several times larger than forany other semiconductor materials, and can be tunedover a wide range.
Last but not least, HgTe-basedQWs are characterized by a highly specific band struc-ture which, depending on the well width and temper-ature, can be either normal or inverted, small effectivemasses about 0 . ÷ . m (Refs. 10,11) and a large g -factor of about 20 (Ref. 12). Despite the enhanced spinfeatures, however, there has been only a low interest inthe HgTe-based QWs. This can be attributed to diffi-culties in the fabrication of HgTe-based devices and itsmoderate mobilities. Recently, a significant progress hasbeen achieved in the growth of HgTe-based QWs. Theseadvances make high mobility samples available. Addi-tionally, lithographical techniques were developed whichmeet the special requirements of HgTe QWs. The appearance of high quality HgTe/HgCdTe QWsresulted in the observation of numerous transport, optical and magneto-optical spin-related effects, like large Zee-man spin splitting, circular photogalvanic effect, enhancement of the subband spin splitting by introducingmagnetic ions in the QW structure and the quantumspin Hall effect. The latter effect is characterized bynondissipative transport of spin-polarized electrons andhas a high potential for spintronics applications.Here, we report on the observation of the magneto-gyrotropic photogalvanic effect (MPGE) in (001)-grownHgTe/HgCdTe QWs. We present the experimental andtheoretical studies of MPGE induced by terahertz as wellas mid-infrared radiation. The effect was detected in awide temperature range from liquid helium to room tem-perature. The MPGE has so far been detected in GaAs,InAs, GaN, and Si QWs for various spectral ranges (fora review see Ref. 22). It has been shown that differ-ent microscopic mechanisms of both paramagnetic (spin dependent) and diamagnetic origins cancontribute to the photocurrent. Recently, we demon-strated that MPGE provides a tool to probe the sym-metry of QWs and gives the necessary feedback to reli-able growth of structures with the controllable strengthand sign of the structure inversion asymmetry. Thus,the observation of MPGE gives new access to the novelmaterial under investigation. According to the previousstudies carried out on III-V-based heterostructures theMPGE current depends linearly on the magnetic fieldstrength B . To our surprise, in HgTe/HgCdTe QWs withinverted band structure we have detected both linear andnonlinear-in- B contributions. By contrast, in QWs withthe normal band ordering the nonlinear-in- B photocur-rent is negligibly small. The paper is organized as fol-lowing. In Sec. II we give a short overview of the experi-mental technique. In Sec. III the experimental results aresummarized. In Sec. IV we present the phenomenologi-cal theory of the MPGE and compare its results with ex-perimental data on polarization dependences. In Sec. Vand VI we show the results of the band structure cal-culations and discuss experimental data in view of themicroscopic background. II. SAMPLES AND EXPERIMENTALTECHNIQUE
The experiments are carried out onHg . Cd . Te/HgTe/Hg . Cd . Te QWs having fourdifferent nominal well widths, L W : 5 nm, 8 nm, 12 nmand 22 nm. Structures are molecular beam epitaxy(MBE) grown on a Cd . Zn . Te substrate with thesurface orientation (001). Samples with the sheet densityof electrons n s from 1 × cm − to 2 × cm − andmobility in the range between 5 × and 2 × cm /Vsat T = 4 . x k [100] and y k [010] crystallographic directions. The (cid:13) FIG. 1: Magnetic field and polarization dependences of thephotocurrent measured in a QW structure with the well width L W = 12 nm at temperature 200 K. Data are presented fornormally incident mid-infrared radiation with ¯ hω = 117 meV( λ = 10 . µ m) and P ≈ . B is ap-plied parallel to the x axis and the photocurrent is measuredin the direction y normal to the vector B . (a) Magnetic fielddependence for two states of polarization with the azimuthangle α equal to 0 ◦ and 90 ◦ . (b) The dependence of thephotocurrent on α measured for two magnetic field strengths.The data on polarization dependence are fitted after Eqs. (1)and (6). The insets show the experimental geometry and theorientation of the light electric field E and the magnetic field B with respect to the sample orientation. FIG. 2: Magnetic field dependence of (a) the polarizationindependent photocurrent J and (b) the polarization depen-dent photocurrent J obtained for the QW structures with L W = 8 nm and 22 nm at temperature 200 K and Data aregiven for normally incident radiation of P ≈ . hω = 117 meV. The data are fitted afterEqs. (3) and (6). For the QW structure with L W = 8 nm thefitting is limited by linear terms. Dashed lines on the rightpanel demonstrate the linear contribution only. Insets showthe experimental geometry and the temperature dependenceof the ratio of polarization independent and dependent pho-tocurrents for QWs with L W = 8 nm, 12 nm, and 22 nm at B = 1 T. photocurrent is measured in unbiased structures viathe voltage drop across a 50 Ω load resistor. Sampleswere mounted in an optical cryostat which allowed us tostudy MPGE in the temperature range from 4.2 K up toroom temperature. An external in-plane magnetic field B up to ± x -direction usinga superconducting magnet.The measurements of magnetic field induced photocur-rents are carried out under excitation of the samples withmid-infrared and terahertz radiation at normal incidence.The geometry of the experiment is sketched in the insetin Fig. 1(a). In (001)-oriented unbiased quantum wellstructures this experimental arrangement excludes othereffects known to cause photocurrents. The source of in-frared radiation is a Q -switched CO laser with operatingwavelengths λ = 9 . ÷ . µ m (corresponding photonenergies ¯ hω = 135 ÷
115 meV). In the investigated nar-row gap QWs the radiation of these photon energies mayinduce inter -band optical transitions or transitions be-tween size-quantized subbands. While the direct opticaltransitions dominate in the radiation absorption, the lessintensive free carrier absorption (Drude-like) may con-tribute substantially to the photocurrent generation. Theradiation power P was varied in the range from 10 W upto 1.2 kW. For the measurements in the terahertz rangewe used molecular laser, optically pumped by a TEACO laser. With NH as active gas, 100 ns pulses oflinearly polarized radiation with peak power ∼ FIG. 3: Magnetic field dependences of the polarization independent photocurrent J obtained for QW structures at differenttemperatures. Data are given for normally incident radiation of P ≈ . hω = 117 meV. Thephotocurrent is measured in the direction perpendicular to B in QWs of three different widths. The data are fitted accordingto Eqs. (3) and (6). The dashed line in the panel (c) is plotted according to the linear law. obtained at wavelengths λ = 90, 148 and 280 µ m (cor-responding photon energies ¯ hω are 13.7 meV, 8.4 meVand 4.4 meV). We also used a CH F as active gas to ob-tain radiation with λ = 496 µ m (¯ hω = 2 . α between the lightpolarization plane and the magnetic field, the plane ofpolarization of the radiation incident on the sample wasrotated. Hereafter the angle α = 0 ◦ is chosen in such away that the incident light polarization is directed alongthe x axis, see inset in Fig. 1(b). In the terahertz range weused λ/ α from 0 ◦ to 180 ◦ covering all possible orientationsof the electric field vector in the QW plane. In the mid-infrared range we applied a Fresnel rhomb converting thelinearly polarized laser radiation into the circularly polar-ized radiation and placed an additional double-Brewster-window polarizer behind the rhomb. Rotation of the po-larizer enabled us to tune the azimuth angle α . III. EXPERIMENTAL RESULTS
First, we discuss the results obtained with the mid-infrared radiation. Irradiating samples at normal inci-dence we observe, for the in-plane magnetic field B k x ,a photocurrent signal in the y direction. The width of the current pulses is about 300 ns which corresponds to theinfrared laser pulse duration. The signal linearly dependson the radiation power up to P ≈ . T = 200 K for twopolarization states of the radiation with the electric field E of the light wave aligned parallel and perpendicularlyto the magnetic field. In the both cases the signal isan odd function of B . Its strength and behavior uponvariation of B depends, however, on the orientation ofthe radiation electric field vector. Figure 1(b) shows thedependence of the photocurrent J y on the orientation ofpolarization plane specified by the angle α . The data canbe well fitted by the equation J y ( α, B x ) = J ( B x )+ J ( B x ) cos 2 α + J ( B x ) sin 2 α. (1)Below we demonstrate that exactly these dependencesfollow from the theory. The measurements in the twofixed polarization directions allow us to extract two in-dividual contributions: the polarization independentbackground and the amplitude of one of the polarizationdependent contributions, namely, J = J y (0 ◦ ) + J y (90 ◦ )2 , J = J y (0 ◦ ) − J y (90 ◦ )2 . (2)Figure 2 shows magnetic field dependence of J and J for samples with the well widths of 8 nm and 22 nm at T = 200 K. The signal behavior is different for thesestructures. We have found that, for the QW with L W =8 nm, the photocurrent depends linearly on the magneticfield. On the other hand, in the QW with L W = 22 nmthe photocurrent can be described by a superposition of FIG. 4: Polarization dependence of the photocurrent J y ex-cited by terahertz radiation in the QW structure with L W =22 nm. The dependence is obtained at T = 200 K, photon en-ergy ¯ hω = 4 . λ = 280 µ m), radiation power P ≈
50 Wand for two magnetic field strengths. The full lines are thefits after Eqs. (1) and (6). linear-in- B and cubic-in- B terms: J y ( B ) = aB + bB . (3)Figure 2 shows that the B -term is more pronounced inthe polarization independent photocurrent J . We fo-cus below particularly on this photocurrent because ourmeasurements reveal that this contribution dominatesthe photocurrent in the almost whole temperature rangeeven at low magnetic fields, where the total photocur-rent is mostly linear in B . While the linear dependenceof the photocurrent on magnetic field is previously re-ported for various structures the observation of the cubicin magnetic field photocurrent is unexpected and has notbeen detected so far. We emphasize that the last term inEq. (3) corresponding to J is strong and overcomes thelinear-in- B contribution at the magnetic field about 6 T.Similar behavior was observed in the structure with L W = 12 nm. Moreover, in this sample the coefficients a and b for polarization independent photocurrent J have opposite signs resulting in a sign inversion observedfor B about 4 T [see Fig. 3(a)]. In the structure with L W = 5 nm the signals were too small to conclude def-initely on the magnetic field dependence (but it is mea-surable at the excitation with THz radiation). The de-crease in temperature drastically affects the experimen-tal data. At intermediate temperature of 120 K we haveobserved that the linear-in- B contribution in QW with L W = 22 nm changes its sign [see Fig. 3(b)]. Now, thesample with L W = 22 nm also shows the sign inversionof the photocurrent J with rising B , in the first sam-ple with L W = 8 nm the data are still well described bythe linear-in- B dependence. Further reduction of tem-perature to the liquid helium temperature results in thesign inversion of the linear-in- B current in sample with L W = 8 nm but also yields to the cubic-in- B component[see Fig. 3(c)]. Now, the magnetic field dependence ofthe photocurrent in all samples is described by the linear- FIG. 5: Magnetic field dependence of the polarization inde-pendent photocurrent J excited by terahertz radiation in theQW structure with L W = 22 nm. (a) The photocurrent ismeasured in the direction perpendicular to B in response tothe radiation of λ = 280 µ m of P ≈
50 W measured at threetemperatures. (b) The photocurrent is measured at liquid he-lium temperature in response to the radiation of two photonenergies. The lines are plotted according to Eqs. (3) and (6). and cubic-in- B terms with pre-factors of opposite signs.The total current tends to the sign inversion, however atsubstantially larger magnetic fields B .Now we turn to the experiments with terahertz radia-tion. We observed magnetic field induced photocurrent inall structures, including sample with L W = 5 nm and atall wavelengths used. Like in the mid-infrared range thesignal depends on the radiation polarization (see Fig. 4)and is well described by Eq. (1). Figure 5(a) shows themagnetic field dependence of the polarization indepen-dent contribution to the photocurrent J obtained in thewide QW with L W = 22 nm in response to the radia-tion of the photon energy ¯ hω = 4 . λ = 280 µ m).Figure 5(a) demonstrates that also in the terahertz rangethe photocurrent in the QW with L W = 22 nm is welldescribed by the Eq. (3) with significant contribution ofthe cubic-in- B term at high magnetic field. At low tem-perature we also detected a peak in the magnetic fielddependence (a dip for absolute value of the signal). Thepeak has minimum at B ≈ µ m wavelength we ob-tained that the magnetic field position of the peak lin-early scales with the photon energy [Fig. 5(b)]. At shorterwavelength, e.g., with the photon energy ¯ hω = 8 . λ = 148 µ m), no peak has been detected at B ≤ J , however the peak in this contri-bution is much less pronounced. Figure 6 demonstratesthat linear-in- B as well as cubic-in- B current contribu-tions J and J drastically increase with increasing ofthe wavelength. We see that at longest wavelength used( λ = 496 µ m) all current contributions are more than twoorders of magnitude larger than that detected in the mid-infrared range. We note that some contributions invertthe sign with wavelength increasing.In contrast to the wide QWs, in the narrowest QWsample ( L W = 5 nm) we observe that the photocurrentdepends only linearly on the magnetic field B . This isdemonstrated in Fig. 7(a) for both, polarization inde-pendent and polarization dependent, photocurrents ob-tained for T = 200 K and excitation with the photonenergy ¯ hω = 4 . λ = 90 µ m (¯ hω = 13 . hω = 4 . L W = 22 nm, but it is much wider and is character-ized by a halfwidth of at least 3 T. Like in the wideQWs, at higher photon energies no peak has been seenfor B ≤ FIG. 6: Wavelength dependences of the absolute values ofcoefficients S − , S − , A − and A − [see Eq. (6)] obtained for theQW structure with L W = 22 nm at T = 200 K. Full symbolscorrespond to negative values of the coefficients. The dashedline is plotted according to the wavelength square law. FIG. 7: Magnetic field dependence of the photocurrent ex-cited by terahertz radiation in the QW structure with L W =5 nm. (a) The polarization independent and polarizationdependent contributions to the photocurrent measured at T = 200 K in the direction perpendicular to B in responseto the radiation with the photon energy ¯ hω = 4 . λ = 280 µ m) and P ≈ hω = 13 . λ = 90 µ m). The full linesare plotted according to Eqs. (3) and (6) with coefficients b and A − , equal to zero. The inset shows the data obtained atliquid helium temperature in response to the radiation with¯ hω = 4 . IV. PHENOMENOLOGY
In order to describe the observed magnetic filedand polarization dependences, we first derive here phe-nomenological equations for the photocurrents in two-dimensional HgTe-based structures. Holding the linearand cubic in the magnetic field strength B terms, MPGEfor unpolarized or linearly polarized radiation at normalincidence is given by j α = X βγδ φ αβγδ B β e γ e ∗ δ + e δ e ∗ γ I (4)+ X βµνγδ Ξ αβµνγδ B β B µ B ν e γ e ∗ δ + e δ e ∗ γ I .
Here φ and Ξ are a fourth- and a sixth-rank pseudo-tensors, respectively, being symmetric in the last twoindices, e γ are components of the unit vector of lightpolarization, and I is the light intensity. We note thatwhile in the theoretical consideration the current density j is used, in the experiments the electric current J ismeasured which is proportional to the current density j .We consider (001)-oriented HgTe-based QWs. De-pending on the equivalence or nonequivalence of the QWinterfaces their symmetry may belong to one of the pointgroups D d or C v , respectively. The present experimentshave been carried out on the asymmetric structures ofC v symmetry and, therefore, here we will focus on theseQWs only. For the C v point group it is convenient to write the components of the magneto-photocurrent in thecoordinate system with x ′ k [1¯10], y ′ k [110], and z k [001]being the growth direction. The advantage of this systemis that the in-plane axes x ′ , y ′ lie in the crystallographicplanes (110) and (1¯10) which are the mirror reflectionplanes containing the twofold axis C k z .In QWs of C v symmetry class the tensors φ and Ξ have, respectively, six and twelve linearly independentcomponents and in the system x ′ , y ′ , z for normal inci-dence of the linearly polarized or unpolarized light andthe in-plane magnetic field Eq. (4) is reduced to: j x ′ = I (cid:2) S B y ′ + S B y ′ ( | e x ′ | − | e y ′ | ) + S B x ′ ( e x ′ e ∗ y ′ + e y ′ e ∗ x ′ ) (cid:3) + I B y ′ [ A B + A ( B x ′ − B y ′ )]+ I B y ′ [ A B + A ( B x ′ − B y ′ )]( | e x ′ | − | e y ′ | )+ I B x ′ [ A B + A ( B x ′ − B y ′ )]( e x ′ e ∗ y ′ + e y ′ e ∗ x ′ ) , (5) j y ′ = I (cid:2) S ′ B x ′ + S ′ B x ′ ( | e x ′ | − | e y ′ | ) + S ′ B y ′ ( e x ′ e ∗ y ′ + e y ′ e ∗ x ′ ) (cid:3) + I B x ′ [ A ′ B + A ′ ( B x ′ − B y ′ )]+ I B x ′ [ A ′ B + A ′ ( B x ′ − B y ′ )]( | e x ′ | − | e y ′ | )+ I B y ′ [ A ′ B + A ′ ( B x ′ − B y ′ )]( e x ′ e ∗ y ′ + e y ′ e ∗ x ′ ) . Here S i and A j are the linearly independent componentsof the tensors φ and Ξ , respectively. The polarizationdependence of the photocurrent is determined by the fac-tors ( | e x ′ | − | e y ′ | ) and ( e x ′ e ∗ y ′ + e y ′ e ∗ x ′ ).In our experiments the magnetic field was orientedalong the cubic axis B k x and the current J y was mea-sured perpendicularly to B . For this experimental geom-etry Eqs. (5) reduce to j y = IB x ( − S − + S − sin 2 α − S − cos 2 α ) (6)+ IB x ( − A − + A − sin 2 α − A − cos 2 α ) , where S − l = ( S l − S ′ l ) / A − l = ( A l − A ′ l ) /
2, and α is anangle between the linear polarization direction and theaxis x k [100], see inset to Fig. 1(b). Thus, for the polar-ization independent and polarization dependent contri-butions to the photocurrent measured in the experimentwe have J ∝ − ( B x S − + B x A − ), J ∝ − ( B x S − + B x A − )and J ∝ ( B x S − + B x A − ).Equation (6) describes well the macroscopic features ofthe photocurrent. In accordance with the experimentaldata it contains both linear- and cubic-in- B contribu-tions and fully describes the observed polarization de-pendence (see Figs. 1(b) and 4]. Figure 2 shows that inthe field B ≤ J and polariza-tion dependent J parts. According to Eq. (6) they aregiven by the coefficients S − and S − , respectively. Thetemperature dependence of the ratio J /J = S − /S − ispresented in the inset to Fig. 2(b) and shows that po- larization independent contribution dominates the totalphotocurrent over almost all the temperature range. Inthe narrowest QW with L W = 5 nm and in QW with L W = 8 nm at high temperature the linear-in- B behav-ior remains up to the highest magnetic fields applied. Inother samples, by contrast, for B >
V. BAND STRUCTURE AND OPTICALTRANSITIONS
Now we calculate the band structure of our samplesand indicate optical transitions responsible for radiationabsorption and the MPGE current generation. HgTe as abulk material is a zero-gap semimetal, whereas a narrowenergy gap opens up in a quantum well. Depending onthe actual well width and temperature, the band struc-ture is either normal or inverted. In the latter case, theordering of the subbands in the QW is reversed comparedto common semiconductors.In Fig. 8 the calculated band structure of 8 nm QWis shown together with possible direct optical transitionscorresponding to the photon energy ¯ hω = 117 meV usedin the experiment with mid-infrared radiation. The bandstructure of (001)-grown HgTe/Hg . Cd . Te QW wascalculated using the eight-band k · p model in envelopefunction approximation. This QW is a type III het-erostructure (see insets to Fig. 8) that causes mixing ofthe electron states and strong coupling between the con-duction and valence bands. In order to take into accountthe coupling and the resulting nonparabolicity of thebands the Kane model with the usual eight-band basis set {| u n i} = | Γ , ± / i , | Γ , ± / i , | Γ , ± / i , | Γ , ± / i was used. Assuming the basis functions u n to be thesame throughout the heterostructure and using the cor-rect operator ordering in the effective-mass Hamiltonianfor the eight-component envelope function vector in ac-cordance with the envelope function theory the bound-ary conditions at material interfaces are automaticallysatisfied. The total eight-band Hamiltonian of the QWsystem is given by H = H + H + H + V H + H BP , where H is the diagonal contribution including the band-edge potentials for the chosen basis set {| u n i} , H and H describe the coupling between the bands of this ba-sis set exactly and their coupling to the remote bands insecond-order perturbation theory, respectively, V H is theself-consistently calculated Hartree-potential, and H BP is the Bir-Pikus Hamiltonian describing the effects ofstrain in the structure. The explicit form of the Hamil-tonian as well as the band structure parameters em-ployed in the calculations are given in Ref. 32. The ob-tained system of eight coupled differential equations ofthe second order for the envelope function componentswas transformed then into a matrix eigenvalue problemby means of the expansion of the envelope function com-ponents in terms of the complete basis set which resultsin the required convergence for type III heterostructures.The subbands in Fig. 8 are labeled as heavy-hole- ( H i ),electron- ( E i ) and light-hole-like ( L i ) in accordance withthe properties of the corresponding wave functions at k k = 0 (see Ref. 34).We emphasize that HgTe-based QW may have a nor-mal or inverted band structure depending on its widthand the temperature. For example, for T = 4 . L W < ∼ T = 4 . L W > ∼ T = 300 K and L W = 8 nm the QW has a normal sequence of the sub-bands. Calculations of the band structure for QWs with L W = 12 nm and 22 nm demonstrate that they have aninverted band structure for all temperatures used in ourexperiments, whereas QW of 5 nm width has noninvertedstructure in the whole temperature range. The analysisof the band structure of investigated samples reveals thatthe nonlinear behavior of the MPGE is detected only insamples having inverted band structure.Our calculations show that mid-infrared radiation with FIG. 8: Calculated band structure for 8 nm QW at (a)300 K and (b) 4.2 K. Arrows show optical transitions in-duced by mid-infrared radiation used in the experiments(¯ hω = 117 meV). Insets sketch the band profile of noninverted(left panel) and inverted (right panel) HgTe-based QWs. the photon energy of the order of 100 meV used in exper-iments causes in all our samples direct interband opticaltransitions (see Fig. 8). The photon energies of appliedterahertz radiation (¯ hω = 3 ÷
14 meV) are much smallerthan the energy gap and intersubband separation, there-fore this radiation causes only indirect (Drude-like) opti-cal transitions. At low temperatures with k B T < ¯ hω ter-ahertz radiation may also cause ionization of impurities,intra-impurity transitions or direct transitions betweenthe Zeeman spin-split subbands. These mechanisms mayhave a resonance-like behavior and be responsible forpeaks observed in the magnetic field dependences of thephotocurrent at liquid helium temperature. A compar-atively large width of these peaks covering several Teslaindicates that they are most probably due to impurityrelated mechanisms. The magnetic field shifts the bandedge as well as the impurity level and tune the bindingenergy to the photon energy making the direct optical ex-citation possible. The mechanism of this additional chan-nel of the radiation absorption and the resulting MPGEare out of scope of this paper. VI. MICROSCOPIC MODELS ANDDISCUSSION
The most surprising result obtained in the experimentis that in samples with L W = 12 nm and L W = 22 nm,as well as in the sample with L W = 8 nm at low tem-perature, the cubic-in- B contribution to J is strong andmay overcome the linear-in- B contribution at the mag-netic field of 4 ÷ FIG. 9: Microscopic models of MPGE (a) due to imbalanceof the spin photocurrents in the in-plane magnetic field and(b) due to the diamagnetic mechanism at direct intersubbandtransitions. fields is not available, we perform here only a qualitativemicroscopic analysis of the effect.First we discuss the terahertz spectral range where ra-diation absorption is dominated by Drude-like processes.In this case, the photocurrent is mainly caused by asym-metry of the electron scattering by phonons and staticdefects in the magnetic field. Such a magnetic-field in-duced scattering asymmetry can be of both spin depen-dent and diamagnetic (spin independent) origins. Thespin dependent mechanism of MPGE comes from theimbalance of the spin photocurrents in the in-plane mag-netic field. Microscopically, it is based on spin depen-dent scattering which accompanies the free-carrier ab-sorption. Figure 9(a) sketches the indirect optical tran-sitions within two spin subbands. Vertical arrows in-dicate optical transitions from the initial state k y = 0while the horizontal arrows describe a scattering eventto a final state with either positive or negative electronwave vector k ′ y . Due to the spin dependence of scatter-ing the transitions to states with positive and negative k ′ y occur with unequal probabilities. This is indicatedby horizontal arrows of different thicknesses. By thatthe free carrier absorption leads to a pure spin currentwhere particles with opposite spin orientations flow inopposite directions. Similarly to the excitation mecha-nism, energy relaxation of electron gas heated by Drudeabsorption, also involving electron scattering, is asym-metric and yields spin separation as well. By applicationof an external magnetic field which polarizes free carriers,the spin photocurrent is converted into an electric cur-rent proportional to the Zeeman splitting for small fields.We note that the mechanism based on asymmetry of thephotexcitation yields polarization dependent photocur-rent while that related to asymmetry of energy relaxationresults in polarization independent signal. In QWs withinverted band structure the ground conduction subband,which is populated in equilibrium, is formed from theΓ -band states [see Fig. 8(b)]. The Zeeman splitting ∆ Z of heavy-hole states in the in-plane magnetic field de-pends strongly nonlinear on B (Ref. 35). Since for thismechanism j ( B ) ∝ ∆ Z ( B ), the photocurrent exhibits a nonlinear behavior in the magnetic field. In the narrowQW with L W = 5 nm the ground conduction subbandis formed from the Γ -band states. Here the Zeemansplitting is linear in B , and a noticeable cubic-in- B con-tribution to the photocurrent is absent as observed in ourexperiments.The diamagnetic mechanism of the MPGE under free-carrier absorption is also related to asymmetry of elec-tron scattering in the magnetic field. However theasymmetry stems from the magnetic field induced mix-ture of states from different quantum subbands, whichis not related to the Zeeman splitting. This mixture ismore efficient for subbands formed from the same Blochstates (i.e., between E E H H
2) and de-termined by the ratio of ¯ heB/ ( m ∗ c ) to the intersubbandenergy separation ∆ ε , where m ∗ is the in-plane effectivemass. In QWs with inverted band structure the ground H E E B .The mechanisms described above may also be respon-sible for the photocurrent caused by mid-infrared radi-ation. Although the contribution from the Drude pro-cesses to the total absorption does not seem to be dom-inant in the spectral range where interband transitionsare possible, it may nevertheless determine the photocur-rent. This scenario is supported by the drastic spec-tral dependence of the photocurrent demonstrated inFig. 6. Indeed, the photocurrent strength increases bymore than an order of magnitude with increasing wave-length, the dependence usually detected for Drude-likeabsorption. Another contribution may come from thedirect intersubband transitions caused by mid-infraredradiation. Such a mechanism of MPGE is proposed inRef. 25 and is based on magnetic field induced shift ofquantum subbands in k -space which is described by thelinear-in- B contribution to the electron energy given by δE ( ν ) k = ( e ¯ h ¯ z ν /cm ∗ )[ B × k ] z , where ¯ z ν is the mean co-ordinate along the growth direction, and ν is the QWsubband index. The energy spectrum of the QW, in-cluding the diamagnetic shift, is sketched in Fig. 9(b).In conventional QW structures with parabolic disper-sion of the valence and conduction subbands, the relativesubband shift leads to a photocurrent at direct opticaltransitions. Indeed, in such systems due to the en-ergy and momentum conservation the points of opticaltransitions are shifted in the k -space resulting in asym-metric distribution of photoexcited carriers with respectto the subbands minima, i.e., to a magnetic field inducedphotocurrent. However, this straightforward mechanismof MPGE gets ineffective in HgTe-based structures understudy where the valence subbands are flat (see Fig. 8).In this particular case, the points of optical transitionsremain symmetric with respect to the conduction sub-band minimum. Therefore, the diamagnetic shift of theconduction subband does lead to a photocurrent only ifthe probability of optical transitions W depends on thewave vector. Such a dependence may come from mixtureof the states at finite in-plane wave vector resulting in W = w + w k . This term together with the diamag-netic shift of the conduction subband give rise to the pho-tocurrent j ( B ) ∝ w B , see Fig. 9(b). Moreover, in QWswith inverted band structure and closely spaced valencesubbands, like in the case of wide HgTe QWs, one canexpect that the parameter w can be large enough andthe magnetic field has a remarkable effect on w . Thisleads to a nonlinear dependence of the photocurrent onmagnetic field. VII. SUMMARY
In conclusion, we have presented experimental dataevincing the first observation of nonlinear magnetic field dependence of MPGE current. One of a probable sce-narios is based on the cubic-in- B Zeeman splitting of theground subband. To prove this statement additional ex-periments like electron spin resonance investigations areneeded. Further access to the origin of the photocur-rent and various mechanisms contributing in its forma-tion should provide an analysis of the temperature andspectral behavior, and, in particular, of the observed signinversions.
ACKNOWLEDGMENTS
We thank M.M. Voronov for helpful discussions. Thefinancial support of the DFG via programs SFB 689,GA 501/6-3 and grant AS327/2-1 as well as support ofRFBR is gratefully acknowledged. Work of L.E.G. isalso supported by ”Dynasty” Foundation — ICFPM andPresident grant for young scientists. H.-A. Engel, E.I. Rashba, and B.I. Halperin,
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