Bulk inversion asymmetry induced magnetogyrotropic reflection from quantum wells
L. V. Kotova, V. N. Kats, A. V. Platonov, V. P. Kochereshko, R. André, L. E. Golub
BBulk inversion asymmetry induced magnetogyrotropic reflection from quantum wells
L. V. Kotova , V. N. Kats , A. V. Platonov , V. P. Kochereshko , R. Andr´e , and L. E. Golub ITMO University, 197101 St. Petersburg, Russia Ioffe Institute, 194021 St. Petersburg, Russia and Universit´e Grenoble Alpes, CNRS, Institut NEEL, F-38000 Grenoble, France
Bulk inversion asymmetry (BIA) of III-V and II-VI semiconductor quantum wells is demonstratedby reflection experiments in magnetic field oriented in the structure plane. The linear in the magneticfield contribution to the reflection coefficients is measured at oblique incidence of s and p polarizedlight in vicinity of exciton resonances. We demonstrate that this contribution to the reflection iscaused by magnetogyrotropy of quantum wells, i.e. by the terms in the optical response whichare linear in both the magnetic field strength and light wavevector. Theory of magnetogyrotropiceffects in light reflection is developed with account for linear in momentum BIA induced termsin the electron and hole effective Hamiltonians. Theoretical estimates agree with the experimentalfindings. We have found the electron BIA splitting constant in both GaAs and CdTe based quantumwells is about three times smaller than that for heavy holes. I. INTRODUCTION
In low-symmetry semiconductor quantum wells(QWs), various remarkable effects are present which areabsent in bulk semiconductors and symmetric structures.The examples are spin-dependent phenomena, e.g. elec-trical spin orientation, conversion of nonequilibrium spininto electric current, electric-field induced spin rotations,as well as nonlinear optical effects like photogalvanics,for a review see Ref. [1]. These effects are symmetry-allowed in gyrotropic systems where some componentsof vectors (e.g. electric current) and pseudovectors(spin, magnetic field) transform identically under pointsymmetry operations [2]. The sources of gyrotropy arethe bulk and the structure inversion asymmetries presentin most of QW structures grown from III-V or II-VIsemiconductors. The Structure Inversion Asymmetry(SIA) is present if the QW has different barrier materialsor an electric field removes symmetry in the growthdirection. Bulk Inversion Asymmetry (BIA) is presenteven in QWs with symmetric heteropotential. BIA iscaused by an absence of inversion center in the bulkmaterial forming the QW layer. The most popular(001) QWs have the point symmetry group D d whichis gyrotropic, and the BIA-induced effects are studiedintensively [3].Gyrotropy manifests itself in optics, and a powerfultool for its study is light reflection experiments. In par-ticular, gyrotropy results in conversion of light polar-ization state at reflection. In QWs, an equivalence ofthe in-plane components of the photon momentum to aneffective magnetic field result in natural optical activ-ity [4]. The new group of phenomena called magneto-gyrotropic effects take place in the presence of an exter-nal magnetic field [5, 6]. Magnetogyrotropy means theterms in the optical response which are linear in boththe magnetic field strength and the photon wavevector.The magnetogyrotropy is absent in centrosymmetric sys-tems, and in QWs it also stems from inversion asym-metry. The SIA induced magnetogyrotropy has been demonstrated recently in reflection experiments [7]. Thecombination of SIA and magnetic field result in interest-ing phenomena in light emission which are greatly en-hanced in semimagnetic QWs with grating [8]. However,for the study of these effects, special asymmetric designof QWs and hybrid plasmonic structures has been usedin Refs. [7, 8]. By contrast, the BIA-induced magneto-gyrotropy does not require special technological efforts.In the present work we use the fact that any QW grownfrom GaAs or CdTe is gyrotropic due to BIA, and demon-strate the BIA-induced magnetogyrotropy effects. We in-vestigate reflection from QWs in vicinity of exciton res-onances where the magnetogyrotropic effects are greatlyenhanced.Symmetry analysis allows us to choose a proper ori-entation of magnetic field and light incidence plane forstudy of the BIA contribution to light reflection. For thispurpose we find the BIA contribution bilinear in both thephoton wavevector q and magnetic field B to the non-local dielectric susceptibility tensor ˆ χ . For (001) QWs(point symmetry group D d ), symmetry analysis and theOnsager principle yield the following magnetogyrotropiccontributions to the susceptibility χ xx ± χ yy = T ± ( q x B x ∓ q y B y ) ,χ xy = χ yx = T ( q y B x − q x B y ) . (1)Here x, y are (cid:104) (cid:105) axes in the QW plane, and T , T + , T − are three linearly-independent functions which will befound below.In experiments, magnetogyrotropy manifests itself asan additional birefringence caused by both the magneticfield B and light wavevector q . These magnetogyrotropiccontributions can be probed in reflection experiments.By contrast, the pure B -linear terms in the reflectionare forbidden by the time-inversion symmetry. It followsfrom Eq. (1) that the reflection coefficients at obliqueincidence acquire the contributions linear in both q and B for s and p polarized light∆ r ∝ q (cid:107) B (cid:107) , (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov where symbol (cid:107) denotes projections onto the QW plane.This relation holds for incidence plane oriented along (cid:104) (cid:105) crystallographic directions. Equation (2) demon-strates that in the B -linear contribution to the reflectioncoefficient is present due to BIA if the magnetic field liesin the light incidence plane. We note that the contribu-tion caused by SIA is absent in this geometry. For itsobservation the magnetic field should be oriented per-pendicular to the incidence plane [7]. This allows us tostudy pure BIA magnetogyrotropic effect by choosing thegeometry B (cid:107) q (cid:107) .The paper is organized as follows. In Sec. II we de-scribe our experiments and deduce the size of the mag-netogyrotropy signal for studied QWs. In Sec. III, the mi-croscopic theory accounting for BIA terms in the electronand hole effective Hamiltonians is developed allowing forfinding the magnetoinduced correction to the reflectioncoefficient. In Sec. IV we compare experimental resultswith theory and estimate the electron and hole BIA spin-splittings. Concluding remarks are given in Sec. V. II. EXPERIMENT
For experimental investigation of BIA-induced magne-togyrotropic effects we studied CdTe- and GaAs basedsamples with single (001) QWs, Fig. 1(a). A trian-gular GaAs/AlGaAs QW was grown by the molecularbeam epitaxy method on a semi-insulating substrate.The structure contains a 200 nm wide Al . Ga . Asbarrier followed by the 8 nm wide QW. Then theother sloping barrier was grown with Al concentra-tion smoothly increasing from 4 % to 28 % on alayer of width 27 nm. The rectangular 8 nm wideCd . Zn . Te/CdTe/Cd . Mg . Te QW with 90 nm widebarriers was grown on a buffer layer to a Cd . Zn . Tesubstrate. The design of both structures is identical tothat of the samples used in Ref. [7].Magnetic field was oriented in the QW plane. Themagnetic field up to 1 T was produced by the electro-magnet with a ferromagnetic core. Experiment was per- (100) X h h P h o t o n e n e r g y ( e V )
Reflectance (arb. units)
FIG. 1. Left: Experimental geometry. The magnetic field B lies the plane of the QW and the light incidence plane con-tains B . Right: Reflectance spectrum measured from CdTeasymmetric heterostructure in the vicinity of the X hh reso-nance. B = 0 . 0 5 T B = 0 . 9 5 T A D P h o t o n e n e r g y ( e V )
FIG. 2. Signal D , Eq. (3), for p polarized incident light re-flected from CdTe sample in the vicinity of the X hh resonanceat magnetic fields B = 0 .
05 T and B = 0 .
95 T. The arrowindicates the signal amplitude A plotted in Fig. 3. formed at temperature T = 3 K in closed cycle heliumcryostat which was located in the core gap. The geometryof electromagnet and cryostat allowed for oblique light in-cidence. We measured polarization of the reflected lightat oblique incidence with the incidence angle θ = 27 ◦ .A halogen lamp was used as a light source for reflec-tion measurements. Lenses and slits formed parallel lightbeam. Glan-Taylor prisms produced linearly polariza-tion. Excitation was linearly polarized in the plane ofincidence ( p polarization) and perpendicular to the in-cidence plane ( s polarization). Spectral dependencies ofthe reflected light were registered by a CCD camera con-joined with a monochromator. Strong exciton resonancesare present in experimental data for both samples. As anexample, the reflection spectrum is shown in Fig. 1(b) forCdTe QW structure. The heavy-hole exciton resonance X hh is clearly seen.We measured the polarization components of the re-flected light in magnetic fields from − s and p polarized light and detected the reflec-tion coefficients r s,p . We analyzed the odd in B contri-bution to reflection: D = r ( B ) − r ( − B ) r ( B ) + r ( − B ) . (3)The spectra D ( ω ) for CdTe QW are plotted in Fig. 2 for p polarization. In order to quantify the effect of magneticfield we determine the amplitude A from each reflectionspectrum. The dependencies of the amplitudes on themagnetic field strength are shown in Fig. 3. The depen-dencies A ( B ) are linear up to B = 0 .
75 T. The amplitudeis larger for p polarization.We performed the same studies on the sample withthe GaAs QW. The spectra D ( ω ) for five values of themagnetic field are shown in Fig. 4. Increase of the signalamplitude is clearly seen. The dependence of the ampli-tude A on the magnetic field is shown in Fig. 5 for both p p o l a r i z a t i o n A (%) s p o l a r i z a t i o n A (%)
M a g n e t i c f i e l d ( T )
FIG. 3. Magnetic field dependencies of the signal amplitudesdefined in Fig. 2 for s and p polarized incident light for theCdTe sample. The lines are guides for eyes. X l h X h h D P h o t o n e n e r g y ( e V ) 1 T 0 . 8 T 0 . 4 T 0 . 2 T 0 . 0 5 T
FIG. 4. The signal for p polarized incident light reflected fromGaAs/AlGaAs asymmetric heterostructure at T = 3 K. Theheavy-hole and light-hole exciton resonances are indicated. s - and p polarization of incident light.To summarize the experimental part, the amplitudesof the B -linear contribution to the reflection coefficientsin the CdTe QWs are equal to 2 . × − B T − and1 . × − B T − for p and s polarization, respectively.For GaAs structure, the amplitudes are 1 . × − B T − and 0 . × − B T − . p p o l a r i z a t i o n A (%) s p o l a r i z a t i o n A (%)
M a g n e t i c f i e l d ( T )
FIG. 5. Magnetic field dependencies of the signal amplitudesfor the GaAs/AlGaAs sample. The amplitudes are defined asin Fig. 2 for s - and p polarized light. III. THEORY
With account for the nondiagonal terms of the Lut-tinger effective Hamiltonian [9], the heavy-hole wavefunc-tions in QWs at B = 0 have the formΨ B =0 hh , ± / = φ hh ( z ) u ± / ∓ γ (cid:126) m (cid:88) n { k z k ± } φ lhn ( z ) E hh − E lhn u ± / . (4)Here k is the hole wavevector, k ± = k x ± ik y , { . . . } de-notes the anticommutator, φ hh and φ lhn are the func-tions of size quantization of corresponding levels of heavyand light holes, E hh and E lhn are their energies, γ isthe Luttinger parameter, and u µ are the Bloch ampli-tudes for the states of the top of the valence band.In magnetic field B ⊥ z , we choose the vector potential A = z ( B y , − B x , k ± → k ± ∓ izB ± e/ ( (cid:126) c ). Therefore we obtain the B (cid:107) -dependent wavefunctionΨ B hh , ± / = φ hh ( z ) u ± / + B ± F ( z ) u ± / ≡ | h, ∓ / (cid:105) , (5)where F ( z ) = γ e (cid:126) m c (cid:88) n { ik z z } φ lhn ( z ) E hh − E lhn . (6)The matrix elements of exciton creation are e · d e,s ; h,m = (cid:104) e, s | e · ˆ d |K ( h, m ) (cid:105) where e is the lightpolarization vector, ˆ d is the dipole momentum operator,and K = iσ y K is the time inversion operator with K being the complex conjugation operation. Here s = ± / m = ± / e · d e,s ; h,m in the basis e, ± / h, ± / e · d = d ⊥ (cid:18) e + + 2 ζB + e z − e − ζB − − e + ζB + e − − ζB + e z (cid:19) . (7)Here d ⊥ = iep cv (cid:104) e | hh (cid:105) / ( m ω √ p cv and ω are theinterband momentum matrix element and the exciton fre-quency, respectively, and the parameter ζ is given by ζ = (cid:104) e | F ( z ) (cid:105)(cid:104) e | hh (cid:105) = γ e (cid:126) m c √ (cid:104) e | hh (cid:105) (cid:88) n (cid:104) e |{ ik z z }| lhn (cid:105) E hh − E lhn . (8)Here summation is performed over the light-hole stateswith even envelopes ( n = 1 , , . . . ).The nonlocal exciton dielectric polarization P in quan-tum wells depends on the growth-direction coordinate z P ( q , z ) = Φ( z ) (cid:88) ν d ∗ ν ( q ) C ν ( q ) . (9)Here summation is performed over four exciton states ν = ( e, s ; h, m ), s, m = ± /
2, and Φ( z ) is the envelopefunction of the exciton size quantization at coincidingcoordinates of electron and hole [9]. The coefficients C ν satisfy the equation[( (cid:126) ω − (cid:126) ω − i Γ) δ νν (cid:48) + H νν (cid:48) ( q )] C ν (cid:48) = (cid:90) dz (cid:48) Φ ∗ ( z (cid:48) ) E ( z (cid:48) ) · d ν ( q ) . (10)Here ω and Γ are the heavy-hole resonant frequency andlinewidth, ˆ H is the contribution to the exciton Hamilto-nian caused by the spin-orbit interaction, and E ( z ) isthe total electric field. The first order in the spin-orbitinteraction correction to the polarization describing themagnetogyrotropy is given by δ P ( q , z ) = − Λ Φ( z )( (cid:126) ω − (cid:126) ω − i Γ) (cid:88) νν (cid:48) d ∗ ν H νν (cid:48) ( d ν (cid:48) · E ) . (11)Here Λ = (cid:82) dz Φ ∗ ( z ) exp ( iq z z ), and we neglect radiativerenormalizations of ω and Γ.We account for BIA via the k -linear spin-orbit split-ting of the conduction- and valence band states. As aresult the total spin-orbit Hamiltonian is a sum of theelectron and hole terms which have the following formsin the basis e ↑ , e ↓ of the electron states and | h, / (cid:105) , | h, − / (cid:105) of the hole states [10, 11]: H = H e + H h , H e,h = β e,h ( σ e,hx k e,hx − σ e,hy k e,hy ) . (12)Here x (cid:107) [100], y (cid:107) [010], k e,h are the electron andhole wavevectors in the QW plane, and β e,h are the two-dimensional Dresselhaus constants. Using Eqs. (7), (11) and (12) we obtain the magnetogyrotropic contributionsto the nonlocal susceptibility ˆ χ defined as δ P ( z ) = (cid:82) dz (cid:48) ˆ χ ( z, z (cid:48) ) E ( z (cid:48) ) in the form of Eqs. (1) with T + = ˜ β e G ( z, z (cid:48) ) , T − = T = ˜ β h G ( z, z (cid:48) ) . (13)Here we used the relation between the wavevectors ofexciton and carriers k e,h = q (cid:107) m e,h / ( m e + m h ) and intro-duced the constants˜ β e,h = β e,h m e,h m e + m h , (14)where m h and m e are the heavy-hole mass in the QWplane and the electron effective mass, respectively. Thefunction G ( z, z (cid:48) ) describes a nonlocality of the QW exci-ton optical response: G ( z, z (cid:48) ) = − ζ Φ( z )Φ ∗ ( z (cid:48) ) | d ⊥ | ( (cid:126) ω − (cid:126) ω − i Γ) . (15)Solving the problem of light reflection from the QW invicinity of the exciton resonance [9], we obtain the mag-netogyrotropic correction to the Jones reflection matrix.For the incidence plane (100) ( q (cid:107) (cid:107) [100]) we get (cid:20) ∆ r s r sp r ps ∆ r p (cid:21) = 4 ζq (cid:107) i Γ cos θ ( (cid:126) ω − (cid:126) ω − i Γ) × (cid:20) ( ˜ β e − ˜ β h ) B x ˜ β h cos θB y ˜ β h cos θB y ( ˜ β e + ˜ β h ) cos θB x (cid:21) . (16)Here q (cid:107) = ( ω/c ) sin θ , θ and θ are the light incidenceangle and the angle of light propagation inside the sam-ple, respectively, and Γ = 2 π (cid:126) | d ⊥ | | Λ | q/ε b is the X hh oscillator strength for normal incidence with ε b being thebackground dielectric constant of the QW material.In the (cid:104) (cid:105) axes, x (cid:48) (cid:107) [1¯10], y (cid:48) (cid:107) [110], the BIA spin-orbit interaction (12) has the form H SO = β e ( σ ex (cid:48) k ey (cid:48) + σ ey (cid:48) k ex (cid:48) ) + β h ( σ hx (cid:48) k hy (cid:48) + σ hy (cid:48) k hx (cid:48) ) . (17)Calculation for the incidence plane (110) when q (cid:107) (cid:107) [110]yields (cid:20) ∆ r s r sp r ps ∆ r p (cid:21) = 4 ζq (cid:107) i Γ cos θ ( (cid:126) ω − (cid:126) ω − i Γ) × (cid:20) ( ˜ β e + ˜ β h ) B y (cid:48) − ˜ β h cos θB x (cid:48) − ˜ β h cos θB x (cid:48) ( ˜ β e − ˜ β h ) cos θB y (cid:48) (cid:21) . (18)In this coordinate system, the reflection plane (110) ex-ists. The corresponding reflection keeps the combination q x (cid:48) B y (cid:48) invariant and, hence, it can be present in ∆ r s and∆ r p . The combination q x (cid:48) B x (cid:48) changes its sign under thereflection in the (110) plane, and this allows for the cor-responding term in r sp = r ps because, at this reflection,the s and p components of the electric field are odd andeven, respectively. IV. DISCUSSION
The derived expressions demonstrate that the geom-etry q (cid:107) (cid:107) B (cid:107) (cid:107) (cid:104) (cid:105) is suitable for observation of∆ r s and ∆ r p caused by BIA, while in the geometry q (cid:107) (cid:107) B (cid:107) (cid:107) (cid:104) (cid:105) the polarization conversion coefficient r sp = r ps induced by BIA can be measured. The exper-imental configuration used in the present work, Fig. 1, q (cid:107) (cid:107) B , allows for maximal corrections to the reflectioncoefficients ∆ r s and ∆ r p . In addition, these correctionshave the resonant behavior ∝ ( ω − ω − i Γ / (cid:126) ) − at the X hh frequency. This is clearly seen in the experimentaldata, Figs. 2 and 4.The derived Eqs. (16), (18) yield the following esti-mation for the magnetogyrotropic corrections: ∆ r s,p ∼ ( βq Γ / Γ )( a/l B ) , where a is the QW width and l B is the magnetic length. For q = 2 . × cm − , β = 140 meV ˚A [11], Γ = 1 meV, Γ = 0 . a = 100 ˚A, we obtain ∆ r s,p ∼ − B T − . This valueagrees in the order of magnitude with the experimentaldata for both GaAs and CdTe based QWs, Figs. 3 and 5.We have also estimated other magnetogyrotropic con-tributions caused by BIA. Account for k -odd terms inthe bulk valence-band Hamiltonian as well as the inter-face inversion asymmetry terms also results in magneto-gyrotropy due to admixture of the ∓ / ± / γ v q/ (∆ E lh l B ) where γ v isthe valence-band cubic in k spin-orbit splitting constantand ∆ E lh ∼
10 meV is the energy splitting between thesize-quantized heavy- and light-hole levels. This value at B = 1 T has an order of 10 − which is two orders of mag-nitude smaller than the contributions (16), (18). There-fore we see that the BIA-induced spin-orbit splitting ofthe electron and hole states in QWs gives the dominantcontribution to the magnetogyrotropic corrections to thereflection coefficients. The ratio of the corrections for p and s polarized inci-dent light according to Eq. (16) is given by (cid:12)(cid:12)(cid:12)(cid:12) ∆ r p ∆ r s (cid:12)(cid:12)(cid:12)(cid:12) = cos θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ β e + ˜ β h ˜ β e − ˜ β h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)In the experiment, the signal for p polarization is abouttwice stronger for both samples, see Figs. 3 and 5. Sincecos θ ≈ m e ≈ m h , the ratio agrees with the exper-iment at coinciding signs of β e and β h and at β h ≈ β e . V. CONCLUSION
From magnetoreflection experiments in vicinity of ex-citon resonances we registered and analysed the magne-togyrotropic terms in the optical response of II-VI andII-V semiconductor QWs. We demonstrate that the q (cid:107) -and B (cid:107) -linear contribution to the reflection has an orderof 0.1 % in both QWs under study. The developed theoryaccounting for BIA spin-orbit splittings of electron andhole states in QWs agrees with the experimental find-ings. Comparison of the theory with experimental dataallowed for determination of the ratio of the electron andheavy-hole BIA spin-splitting constants. ACKNOWLEDGMENTS
We thank I. A. Akimov, A. N. Poddubny and E. L.Ivchenko for fruitful discussions. L. V. K. is supportedby Russian Science Foundation (project 16-12-10503).L. E. G. thanks the Presidium of RAS and the Foun-dation for advancement of theoretical physics and math-ematics “BASIS”. [1] S. D. Ganichev and L. E. Golub, Interplay ofRashba/Dresselhaus spin splittings probed by photogal-vanic spectroscopy - A review, Phys. Status Solidi B ,1801 (2014).[2] Y. Tokura and N. Nagaosa, Nonreciprocal responses fromnon-centrosymmetric quantum materials, Nat. Commun. , 3740 (2018).[3] E. Marcellina, A. R. Hamilton, R. Winkler, and D. Cul-cer, Spin-orbit interactions in inversion-asymmetric two-dimensional hole systems: A variational analysis, Phys.Rev. B , 075305 (2017).[4] L. V. Kotova, A. V. Platonov, V. N. Kats, V. P.Kochereshko, S. V. Sorokin, S. V. Ivanov, and L. E.Golub, Optical activity of quantum wells, Phys. Rev. B , 165309 (2016).[5] V. Kochereshko, V. Kats, A. Platonov, V. Sapega, L.Besombes, D. Wolverson, H. Mariette, Nonreciprocalmagneto-optical effects in quantum wells, Phys. StatusSolidi C , 1610 (2014). [6] S. Hayami, M. Yatsushiro, Y. Yanagi, and H. Kusunose,Classification of atomic-scale multipoles under crystallo-graphic point groups and application to linear responsetensors, Phys. Rev. B , 165110 (2018).[7] L. V. Kotova, V. N. Kats, A. V. Platonov, V. P.Kochereshko, R. Andr´e, and L. E. Golub, Magnetospatialdispersion of semiconductor quantum wells, Phys. Rev.B , 125302 (2018).[8] F. Spitzer, A. N. Poddubny, I. A. Akimov, V. F. Sapega,L. Klompmaker, L. E. Kreilkamp, L. V. Litvin, R.Jede, G. Karczewski, M. Wiater, T. Wojtowicz, D. R.Yakovlev, and M. Bayer, Routing the emission of a near-surface light source by a magnetic field, Nat. Phys. ,1043 (2018).[9] E. L. Ivchenko, Optical Spectroscopy of SemiconductorNanostructures (Alpha Science Int., Harrow, UK, 2005).[10] A. A. High, A. T. Hammack, J. R. Leonard, S. Yang, L.V. Butov, T. Ostatnick´y, M. Vladimirova, A. V. Kavokin,T. C. H. Liew, K. L. Campman, and A. C. Gossard, Spin
Currents in a Coherent Exciton Gas, Phys. Rev. Lett. , 246403 (2013).[11] M. V. Durnev, M. M. Glazov, and E. L. Ivchenko,Spin-orbit splitting of valence subbands in semiconductor nanostructures, Phys. Rev. B89