Alternative method for the quantitative determination of Rashba- and Dresselhaus spin-orbit interaction using the magnetization
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Alternative method for the quantitativedetermination of Rashba- and Dresselhausspin-orbit interaction using the magnetization
M A Wilde and D Grundler
Lehrstuhl f¨ur Physik funktionaler Schichtsysteme, Technische Universit¨at M¨unchen,Physik Department, James-Franck-Str. 1, D-85747 Garching b. M¨unchen, GermanyE-mail: [email protected]
Abstract.
The quantum oscillatory magnetization M of a two-dimensional electronsystem in a magnetic field B is found to provide quantitative information onboth the Rashba- and Dresselhaus spin-orbit interaction (SOI). This is shown byfirst numerically solving the model Hamiltonian including the linear Rashba- andDresselhaus SOI and the Zeeman term in an in particular doubly tilted magneticfield and second evaluating the intrinsically anisotropic magnetization for differentdirections of the in-plane magnetic field component. The amplitude of specific magneticquantum oscillations in M ( B ) is found to be a direct measure of the SOI strengthat fields B where SOI-induced Landau level anticrossings occur. The anisotropic M allows one to quantify the magnitude of both contributions as well as their relative sign.The influence of cubic Dresselhaus SOI on the results is discussed. We use realisticsample parameters and show that recently reported experimental techniques providea sensitivity which allows for the detection of the predicted phenomena.PACS numbers: 73.21.Fg, 75,70.Tj, 85.75.-d Keywords : Spin-Orbit Interaction, Rashba effect, Dresselhaus effect, de Haas-vanAlphen effect, Magnetization
Submitted to:
New J. Phys.
ONTENTS Contents1 Introduction 22 Energy spectrum and magnetization of 2DESs with R-SOI and D-SOIin tilted magnetic fields 4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 k D-SOI . . . . . . . . . . . . . . . . . . 14
The spin-orbit interaction (SOI) in low-dimensional electron systems in semiconductorshas been investigated intensely both theoretically and experimentally due to the rich spinphysics and potential application in the field of spintronics [1, 2, 3]. In a two-dimensionalelectron system (2DES) spin-orbit interaction arises due to either structure inversionasymmetry or bulk inversion asymmetry of the host crystal. These contributions areknown as Rashba (R) and Dresselhaus (D) SOI, respectively. Early spintronics proposals[4] focused on the R-SOI that can be tuned by an external electric field [5]. Here, theenvisaged devices were based on ballistic transport. More recently, proposals emergedthat rely on the interplay of R-SOI and D-SOI [6, 7] and work for diffusive transport.In any case, a prerequisite for the realization of spintronic devices is the unambiguousand quantitative knowledge of R-SOI and D-SOI present in a given electron system.This evaluation is still challenging. A widespread method for the determination of SOIcoupling constants is based on the oscillatory resistivity ρ ( B ) measured in a magneticfield B at low temperature T . The analysis of the relevant beating patterns does notyield unambiguous results however, when both R-SOI and D-SOI play a comparable role.Further on, the weak-antilocalization phenomenon in ρ near B = 0 is used to determineSOI coupling constants. Here, different models have been put forward which help toevaluate experimental data only in specific parameter regimes [8, 9, 10, 11, 12, 13, 14, 15].Spin photocurrents [16] provided the ratio and relative sign, but the absolute values ofthe SOI coupling constants were not accessible. Recently, the R-SOI and D-SOI strengthwas inferred from spin precessional motion as a function of an in-plane electric field ONTENTS g -factor and the spin relaxation rate by spin-quantum beatspectroscopy [18]. In a pioneering theoretical work [19] Fal’ko considered cyclotron andelectric-dipole spin resonance in tilted magnetic fields and provided an approach toextract both SOI coupling constants. The author developed an analytical expression forthe anticrossing energy gap depending on both the R-SOI and D-SOI contributions butleft the exact diagonalization of the problem outside the scope of the paper.Magnetization experiments addressing in particular the ground state of a 2DES haveso far not been thoroughly discussed in the framework of spintronics and the separateextraction of R-SOI and D-SOI coupling constants. This is surprising as magnetizationmeasurements have been used already to gain quantitative insight into 2DES propertiessuch as, e.g., quantum lifetimes [20], self-consistent electron redistribution [21], giantspin splitting [22] and electron-electron interaction [23]. Early on, theoretical workspredicted beating patterns in the quantum oscillations of M due to SOI [24, 25, 26, 27]but corresponding magnetization measurements have been reported only recently[28, 29]. A rigorous theoretical treatment of M ( B ) and its in-plane anisotropy resultingfrom the two SOI contributions in tilted magnetic fields has not yet been presented. Therelevance of M ( B ) for the quantitative determination of both SOI coupling constantshas not been profoundly explored at all. In contrast to other techniques magnetizationexperiments do not address excited states and no knowledge about microscopic detailsof the scattering processes in the sample is needed. A detailed theoretical analysisand description of M ( B ) is now timely as the magnetization is a non-invasive probe ofelectronic band structure properties that does not require contacts to the sample on theone hand and measures the system in equilibrium on the other hand.In this paper we address the oscillatory magnetization M ( B ) of a 2DEStheoretically in detail and show that in particular tilted field experiments can be usedto extract both, R-SOI and D-SOI coupling constants as well as their relative sign.For this we perform numerical and analytical calculations based on the Hamiltonianfirst written out by Das et al. [30]. We focus on the numerical diagonalization of theHamiltonian including linear R-SOI, D-SOI terms and the Zeeman term in an arbitrarilytilted magnetic field. We further follow the analytical approaches of [19, 31]. In magneticfields collinear with the 2DES normal, Landau levels (LLs) with different spin indicesexhibit multiple crossings when SOI is present. The uneven spacing of LLs at the Fermienergy leads to the well-known beating patterns in the quantum oscillations. When thefield direction is tilted away from the 2DES normal an anticrossing of LLs occurs [32]that is caused by an SOI-induced mixing of levels. The anticrossing opens an energy gapin the ground-state energy spectrum when the Zeeman splitting equals the cyclotronenergy. We will show that the gap provokes a characteristic quantum oscillation of M whose amplitude ∆ M is a direct measure of the SOI coupling constant if one SOIterm dominates. For the case where both R-SOI and D-SOI terms play a role at thesame time, the spectrum of ∆ M is shown to be anisotropic with respect to the in-planemagnetic field direction. From the amplitude ∆ M and its in-plane anisotropy, both, the ONTENTS M ( B ) in order to unambiguously identifythe SOI contributions. As the magnetization M is calculated directly from a modelHamiltonian without further assumptions, it is thus ideally suited for a quantitativemodeling of experimental data [33].The paper is organized in the following way: In section 2 we define the modelHamiltonian of the problem and calculate the quantum oscillatory magnetization withemphasis on the anticrossing behavior in the regime of high magnetic fields and hightilt angles. The results are put into relation to the well-known SOI-induced beatingpatterns occurring in quantum oscillations in the low-field regime. In section 3 themodel calculations are applied to specific relevant systems. We discuss the feasibility ofthe proposed experiments and the modeling of experimental data in section 4. Finally,we draw conclusions in section 5.
2. Energy spectrum and magnetization of 2DESs with R-SOI and D-SOI intilted magnetic fields
The magnetization M = − ∂F/∂B | T,n s is a thermodynamic state function that candirectly be linked to a model Hamiltonian H . The function reduces to M = − ∂U/∂B | T =0 ,n s at T = 0 K. Here F ( U ) denotes the free (ground state) energy ofthe system and n s is the sheet electron density. In the following we formulate theproblem for the exact numerical calculation of the quantum oscillatory magnetization M , i.e., the de Haas-van Alphen (dHvA) effect, and give analytical approximations thatallow for detailed simulations of experimental data. In the following we consider zinc blende semiconductor quantum wells grown in the[001] direction. This class of 2DESs is most relevant for conducting experiments. Thecalculations are based on the model Hamiltonian H = H + H SO, taking into account thelinear Rashba (R-SOI) term and the linear Dresselhaus (D-SOI) term. The latter arisesfrom the k bulk spin-orbit interaction and is the relevant term in the limit of narrowquantum wells and small in-plane wave vectors k of the electrons in the conduction band.Additional effects due to the k D-SOI relevant in the limit of wide quantum wells andlarge k will be discussed in section 3.3. The Hamiltonian for the k -linear terms reads: H = H + α R ~ ( σ x π y − σ y π x ) + β D ~ ( σ x π x − σ y π y ) . (1)Here π x,y = p x,y + eA x,y where p x,y = − ı ~ ∂∂ ( x,y ) are the components of the in-plane momentum operator, A x,y are the components of the vector potential, e is ONTENTS α R ( β D) is the R-SOI (D-SOI) spin-orbit coupling strength. H = π m ∗ + g ∗ µ B Bσ , π = ( π x , π y ) includes the Zeeman contribution with the effectiveLand´e factor g ∗ , the Bohr magneton µ B = e ~ / m e and the vector of Pauli spin matrices σ = ( σ x , σ y , σ z ). Here, m ∗ is the effective mass, and m e is the free electron mass. Theeffects of g -factor anisotropy will be discussed in section 4 The magnetic field is chosenas B = B b = B (sin θ cos φ, sin θ sin φ, cos θ ), where θ is the tilt angle between B andthe 2DES normal n and φ is the azimuthal angle with respect to the [100] direction.We consider the lowest subband of the narrow quantum well to be occupied.In the presence of both SOI terms in tilted magnetic fields no analytical solution isknown. In this paper, we focus on the exact numerical diagonalization of H . For this, weuse the matrix elements of H between the eigenstates of H in perpendicular magneticfields, i.e., between the well known Landau states | n, σ i with LL index n = 0 , , , . . . andenergies E σn = ( n + / ) ~ ω c + g ∗ µ B Bσ . Here, ω c = eB ⊥ /m ∗ is the cyclotron frequencyand B ⊥ is the magnetic field component perpendicular to the 2DES. The correspondingmatrix elements have been first written out by Das et al. [30]. ‡ The resulting matrixis diagonalized numerically by truncating the matrix dimensions while including asufficient number of LLs. It is instructive, however, to reexpress the Hamiltonian (1)in a basis where the spin quantization axis is chosen along the magnetic field directionfollowing [19, 31]. While this bears no particular advantage for the exact numericaldiagonalization, it allows us to gain additional analytical insight into the anticrossingas we will discuss below. After the rotation σ x → σ x cos θ cos φ − σ y sin φ + σ z sin θ cos φσ y → σ x cos θ sin φ − σ y cos φ + σ z sin θ sin φ (2) σ z → − σ x sin θ + σ z cos θ (2) yields for the spin-orbit part of (1) [31] H SO = π + σ + ( η + + η − cos θ ) − σ − ( η + − η − cos θ )+ σ z η − sin θ ] + H.c. , (3)where we used σ ± = ( σ x ± ıσ y ) / π ± = π x ± ıπ y and the convenient notation η ± = β D e ıφ ± ıα R e − ıφ introduced in [31]. The corresponding nonvanishing matrixelements can be written as h n, ±| H | n, ±i = ~ ω c( n + 12 ) ± g ∗ µ B B (4) h n + 1 , ±| H SO | n, ∓i = ± ı s n + 12 l B [ η + ( φ ) ± η − ( φ ) cos θ ] (5) h n + 1 , ±| H SO | n, ±i = ± ı s n + 12 l B η − ( φ ) sin θ , (6) ‡ Note that for α R = 0 , β D = 0 , g ∗ = 0 , θ = 0, the energy spectrum is anisotropic with respect tothe direction of the in-plane magnetic field component, and the Zeeman terms in H have to take intoaccount the in-plane field direction. This case has not been discussed in [30]. ONTENTS
10 151216 (a) (b)2, B/B E ( m e V )
1, 10 15012345 ene r g y gap E ( m e V ) B/B
ACc cAC
Figure 1. (a) Anticrossing of levels (1, ↓ ), (2, ↑ ) (solid lines) numerically calculatedfor a 10 nm wide InGaAs/InP quantum well. Sample parameters are g ∗ = − . m ∗ = 0 . m e, n s = 2 . × m − , α R = 9 . × − eVm and β D = 0. Forvanishing SOI the levels coincide (dashed lines) at an angle θ c. (b) Energy gap ∆ E between levels in (a) as a function of B/B ⊥ = 1 / cos θ . Note that the gap ∆AC at θ cis solely due to SOI. and their Hermitian conjugates. Here, we have introduced the magnetic length l B =( ~ /eB ⊥ ) / .In a system without SOI the interplay of Landau quantization energy ( n + / ) ~ eB ⊥ /m ∗ which depends on B ⊥ and Zeeman energy splitting g ∗ µ B B which dependson the total B = B ⊥ / cos θ leads to artificial degeneracies between spin-split LLswhenever l cos θ c = ( g ∗ / m ∗ /m e) ( l = 1 , , . . . ). Two levels with different Landauand spin indices ( ↑ , ↓ ) thus cross as a function of B/B ⊥ = 1 / cos θ . A crossing betweene.g. ( n = 1, ↓ ) and ( n = 2, ↑ ) LLs is sketched as dashed lines in figure 1 (a) assumingInGaAs/InP quantum well (QW) parameters as reported by Guzenko et al. [34] whofound that D-SOI is negligible in their samples. The condition for crossing [35, 36] -commonly named coincidence condition - has been used extensively in experiments tomeasure | g ∗ | m ∗ [36, 37]. When SOI is present, an anticrossing gap ∆AC is opened at thecoincidence angle θ c due to the resonant level mixing (solid lines). The gap ∆ E betweenthe levels (1, ↓ ) and (2, ↑ ) at integer filling factor ν = n s / ( eB/h ) = 4 shown in figure 1has a finite value ∆AC at θ c [figure 1 (b)] (whereas ∆ E = 0 would be expected withoutSOI). At this angle θ c, the SOI-induced mixing of levels is strong. To first order, thetwo levels are coupled by the matrix elements as illustrated in figure 2. Following this,the anticrossing gap ∆AC is approximately given by 2 |h n + 1 , ±| H SO | n, ∓i| at θ c, i.e.,we have ∆AC ≈ p n + 1) l B × r β D(1 − g ∗ | g ∗ | b z ) + 4 α R β D b x b y + α R(1 + g ∗ | g ∗ | b z ) . (7)This is consistent with the results of [19, 31]. ∆AC is anisotropic with respect to thedirection of the in-plane magnetic field component. ONTENTS (cid:1328)(cid:2033) (cid:3030)(cid:2869)(cid:2870) + (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:3) -i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879)(cid:1499) sin (cid:2016) (cid:3) i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878)(cid:1499) (cid:3398) (cid:2015) (cid:2879)(cid:1499) cos (cid:2016) )
0 0 0 (cid:1328)(cid:2033) (cid:3030)(cid:2869)(cid:2870) (cid:3398) (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:3) -i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878)(cid:1499) + (cid:2015) (cid:2879)(cid:1499) cos (cid:2016) ) (cid:3) i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879)(cid:1499) sin (cid:2016)
0 0 (cid:3) i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879) sin (cid:2016) i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878) + (cid:2015) (cid:2879) cos (cid:2016) ) (cid:1328)(cid:2033) (cid:3030)(cid:2871)(cid:2870) + (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:3) -i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879)(cid:1499) sin (cid:2016) (cid:3) i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878)(cid:1499) (cid:3398) (cid:2015) (cid:2879)(cid:1499) cos (cid:2016) ) -i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878) (cid:3398) (cid:2015) (cid:2879) cos (cid:2016) ) (cid:3) -i (cid:3117)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879) sin (cid:2016) (cid:1328)(cid:2033) (cid:3030)(cid:2871)(cid:2870) (cid:3398) (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:3) -i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878)(cid:1499) + (cid:2015) (cid:2879)(cid:1499) cos (cid:2016) ) (cid:3) i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879)(cid:1499) sin (cid:2016)
0 0 (cid:3) i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879) sin (cid:2016) i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878) + (cid:2015) (cid:2879) cos (cid:2016) ) (cid:1328)(cid:2033) (cid:3030)(cid:2873)(cid:2870) + (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) ( (cid:2015) (cid:2878) (cid:3398) (cid:2015) (cid:2879) cos (cid:2016) ) (cid:3) -i (cid:3118)(cid:3118)(cid:3287)(cid:3251)(cid:3118) (cid:2015) (cid:2879) sin (cid:2016) (cid:1328)(cid:2033) (cid:3030)(cid:2873)(cid:2870) (cid:3398) (cid:3034) (cid:1499) (cid:2870) (cid:2020) (cid:3003) (cid:1828) (cid:1709) … Figure 2.
Matrix of the problem. For a negative g -factor the coincidence conditionin the absence of SOI is met when the diagonal matrix elements inside the shaded2 × g -factor the corresponding diagonal terms arecoupled by the matrix elements highlighted by dotted lines. In the following we relate ∆AC to the magnetization M as the experimentally accessiblequantity. To calculate M , first the field-dependent Fermi energy E F is determinedfrom the condition of constant density n s using the energy eigenvalues ǫ n . At T = 0 K the ground state energy U is calculated by summing up the energies of alloccupied states. The magnetization perpendicular to the 2DES is obtained through M = − ∂U/∂B ⊥ | n s ,T =0 [figure 3 (a)] § . The magnetization is found to oscillate witha sawtooth-like waveform that is characteristic of the dHvA effect in a 2DES [38].To show how the anticrossing gap is reflected in the magnetization [figure 3 (b)] weconsider an InGaAs/InP sample with parameters as in figure 1 at the coincidenceangle θ c ( α R = 9 . × − eVm). M jumps discontinuously in figure 3 (a) at integerfilling factors ν , whenever the Fermi energy E F jumps to the next lower lying LL withincreasing magnetic field. The peak-to-peak amplitude ∆ M , normalized to the electronnumber N = n s · A , where A is the area of the 2DES, is directly related to the jump∆ E in E F (i.e., the energy gap) via ∆ M = ∆ E/B ⊥ [39, 40, 41]. Thus, the anticrossingmanifests itself as a sharp jump in M , denoted by the arrow in figure 3 (a). ∆ M exhibitsthe minimum amplitude ∆ M AC at θ = θ c [figure 3(b)] and is directly related to the § The jump ∆ M corresponding to an anticrossing gap ∆AC occurs in the magnetization componentcollinear with the 2DES normal. ONTENTS c E F ( m e V ) B (T) c (c) 2, 80 82 84 86024 M ACAC cAC ene r g y gap E ( m e V ) angle (deg) cAC = M AC BM AC (d)80 82 84 8601 M pe r e l e c t r on ( m e V / T ) angle (deg) (b)0 1 2 3-101 M pe r e l e c t r on ( m e V / T ) B (T)(a)
Figure 3. (a) Magnetization M numerically calculated at the coincidence angle θ c.Parameters as in figure 1 (a) for α R = 9 . × − eVm and β D = 0 (solid line). Thepeak-to-peak magnetization amplitude at ν = 4, ∆ M ν =4 ( θ c) = ∆ M AC, is markedby an arrow. (b) ∆ M ν =4 as a function of tilt angle θ (solid line). For vanishing SOI(dashed line) the amplitude ∆ M reaches zero at the coincidence angle. For α R = 0the minimal amplitude at θ c (arrow) directly reflects the corresponding anticrossinggap ∆AC = ∆ M AC B ⊥ highlighted in the oscillatory Fermi energy E F shown in (c).Dotted lines in (c) indicate the Landau levels (2 , ↑ ) and (1 , ↓ ). (d) ∆ E ν =4 as a functionof tilt angle. Dashed lines are always for α R = β D = 0 for comparison. anticrossing gap ∆AC [figure 3 (c) and (d)] via∆ M AC = ∆AC /B ⊥ . (8)In the case of finite SOI, the critical angle θ c is found experimentally by looking for theminimum magnetization amplitude ∆ M = ∆ M AC in large B providing the minimumgap size. Dashed lines in figure 3 show the behaviour for α R = β D = 0. For θ = θ c,oscillations occur only at odd integer ν [figure 3 (a) and (c)] since the coincidencecondition is met for even ν and an anticrossing gap does not form. B Beating patterns in M at small B are a consequence of the nonlinear LL dispersioninduced by the SOI, leading to small ∆ M (nodes) when adjacent levels at E F areequally spaced, i.e., where the total spin splitting at the Fermi level - denoted by δ inthe following - is δ = ( p + / ) ~ ω c, with p = 0 , , , . . . [42]. The beating pattern isillustrated in figure 4 (a) considering the sample parameters of figure 1. Most earlierinvestigations have focused on the low field regime, and evaluated such beating patternsoccurring in magneto-oscillations of ρ [43, 30, 5, 44, 45]. However, as noted above, their ONTENTS M pe r e l e c t r on ( m e V / T ) B (T)(a) 0.0 0.5 1.0 M pe r e l e c t r on ( m e V / T ) B (T)
Figure 4. (a) Beating pattern in M for the InGaAs/InP QW containing an ideal2DES considered above (light curve). The angle θ = 15 ◦ is fixed and B ⊥ is varied.The beating pattern is more clearly seen if a real 2DES is considered where electronsundergo scattering. To account for the scattering and LL broadening, a Gaussianlevel broadening of Γ = 0 . M for a 2DES with α R = β D = 5 × − eVm. All other parameters are chosen like in (a). Here, the LLspectrum is linear and a beating pattern in M does not occur although SOI is present. analysis is sometimes ambiguous, especially when both α R and β D play a role. Toillustrate the ambiguity, we address the relevant case α R = β D (cf. Schliemann et al. [6] and Cartoix‘a et al. [7]) in figure 4 (b). The resulting LL spectrum is linear in B as inthe case α R = β D = 0, and a beating pattern is not formed. This way, no informationabout SOI can be gained in small fields B . Our theoretical approach and analysisoutlined above predict however that for large B , ∆ M still experiences the anticrossingof LLs. By measuring the amplitude ∆ M AC( φ ) one still obtains information about SOI,even if the well-known beating is absent.
3. Application to specific electron systems
Of great interest in the field of spintronics are in particular two cases where (i) a strong(tunable) R-SOI dominates and D-SOI can be neglected [4] and where (ii) R-SOI andD-SOI are of comparable or even identical strength [6, 7]. We will discuss the behaviourof ∆ M for the two relevant scenarios in section 3.1 and section 3.2, respectively. We first focus on the case where D-SOI can be neglected and R-SOI dominates. Theresult from the exact diagonalization is shown as open symbols in figure 5. Thedependence deviates from a linear slope for ∆R ≥ M AC is relatedto the energy gap ∆AC at the Fermi energy E F we can reexpress (7) in terms of thezero-field spin splitting energy at E F given by ∆R = 2 α R k F. Here, k F = √ πn s is theFermi wave vector of the 2DES. Using (8) the analytical approximation to the amplitude ONTENTS M A C ( m e V / T ) R (meV) Figure 5.
Oscillation amplitude ∆ M AC at ν = 4 and θ = θ c vs Rashba splittingparameter ∆R. Open symbols are the results of the exact diagonalization. Thedashed line marks the first-order analytical approximation of equation (10). Thedeviation from the exact result increases with increasing ∆R. Equation (10) is agood approximation for ∆ M AC B ⊥ / ~ ω c ≪ of the magnetic oscillation simplifies to∆ M AC ≈ q ∆ R(1 + g ∗ | g ∗ | b z ) B ⊥ . (9)In this approximation ∆ M AC is a linear measure of the SOI coupling constant thatis independent of the direction of the in-plane magnetic field component. For theInGaAs/InP quantum well considered in figure 1 g ∗ is negative. k In this case, weget ∆ M AC ≈ ∆R sin θ c B ⊥ . (10)This approximate analytical result for ∆ M AC is shown as a function of ∆R = 2 α R k F(dashed line) in figure 5. We evaluate the gap ∆AC at the small filling factor ν = 4 here,since it occurs at sufficiently high B ⊥ for the total spin splitting δ to approach ~ ω c forthe anticipated coincidence condition on the one hand and is still within experimentalreach [cf. section. 4] on the other hand. By comparing the analytical with the exactresult we find that (10) is valid for ∆ M AC B ⊥ / ~ ω c ≪
1. Higher order couplings becomesignificant in the limit of large SOI coupling constants and small cyclotron energy ~ ω c.These are not included in (10) but relevant for the exact calculation. Equation 10overestimates the real gap size. In the analysis of experimental data, (9) can be usedas a starting point for modeling of the magnetization. Analysis and modeling of thespecific purely SOI-induced magnetic oscillation at, e.g., ν = 4 allows an independentdetermination of the Rashba constant that is complementary to the analysis of beatingpatterns in small magnetic fields. The analysis is applicable even in situations wherebeating patterns are not resolved. For ∆R = 0 and ∆D = 2 β D k F = 0 one needs toreplace ∆R by ∆D in this section and apply a minus sign before g ∗ | g ∗ | in (9). k A negative g ∗ can be assumed for systems with large SOI, see [25]. ONTENTS °°° ° °°°° Figure 6.
Polar plot of ∆ M AC ,ν =4 vs azimuthal angle φ for an InSb QW for whichwe assume different R-SOI values (symbols interconnected by a solid line). InSb QWparameters are m ∗ = 0 . m e, g ∗ = − n s = 2 . × m − and ∆D = 8 .
07 meV.∆R takes the values 0 . , , ≃ . × ∆D (symbols are defined in the top left inset). Notethat for ∆R / ∆D = 1 beating patterns do not exist in a perpendicular B . Still, ∆ M AC(squares) exhibits a characteristic anisotropy. Both, the size and relative sign of ∆Rand ∆D determine the anticrossing gap value and anisotropy. For ∆R / ∆D = − . M AC ,ν =14 vs φ is rotated by π/ / ∆D = +2 . M AC is isotropicsince ∆D = 0. The dashed lines in all plots correspond to the approximate solutions(11). For large SOI splitting energies (∆R / ∆D = ≃ .
1) the approximate solutionoverestimates the magnetization amplitudes.
We now turn to the case where R-SOI and D-SOI are of the same order, but considerdifferent relative strengths. For zero in-plane magnetic field, the magnetization for thiscase has been treated in [25]. Here, we focus on the general case of an arbitrarily tiltedmagnetic field applied under angles θ and φ defined above.Exact numerical results for the oscillation amplitude ∆ M AC corresponding to ananticrossing at ν = 4 versus azimuthal angle φ are shown as symbols interconnected bysolid lines in figure 6. We consider a 7 nm wide InSb QW with n s = 2 . × m − . Thecalculations are performed for m ∗ = 0 . m e , g ∗ = −
51 and β D = 3 . × − eVmtaken from [46]. α R = ∆R / k F is varied as shown in the legend. These parameterslead to θ c = 69 . ◦ . We find that the presence of, both, R-SOI and D-SOI provokes apronounced anisotropy of ∆ M AC with respect to the direction of the in-plane magneticfield component. The anisotropic ∆ M AC contains information on the absolute valuesof α R and β D as well as their relative sign as will be detailed later. Dashed lines infigure 6 denote the approximate analytical results. Again, the analytical approximationis valid in the regime ∆ M AC B ⊥ ≪ ~ ω c. From the first order approximation, one gets ONTENTS ¶ ∆ M AC( φ ) ≈ B ⊥ × r ∆ R sin θ c2 + ∆ D cos θ c2 + ∆R∆D2 sin θ c sin 2 φ . (11)For the discussion we have chosen filling factor ν = 4, which occurs at B ⊥ ≃ .
59 Tleading to B ≃ .
25 T at the coincidence angle. This value is within experimental reach[47]. With the R-SOI strength approaching the D-SOI value from below, a pronouncedanisotropy of the oscillation amplitude develops. The strength of the anisotropy dependson α R /β D = ∆R / ∆D. The minimum amplitude occurs at φ = − ◦ and the maximumamplitude at φ = 45 ◦ . The orientation of the double-loop figure depends on the relativesign of α R and β D (bottom right inset of figure 6): For α R /β D = − .
1, the positionof minima and maxima occur at φ = 45 ◦ and φ = − ◦ , respectively. We note that thedefinition of the sign of β D varies in the literature [30, 19, 6, 16, 48, 31]. The isotropicoscillation amplitude expected for the InGaAs/InP structure with ∆D = 0 is shown inthe top right inset for comparison. The case | ∆R | = | ∆D | = ∆ is special due to itsrelevance for spintronics [6] and because beating patterns do not exist in perpendicularmagnetic fields. Here, (11) reduces to ∆ M AC ≈ ∆ B ⊥ q sin θ c(sin 2 φ ∓
1) and thevalue of ∆ and the relative sign of the contributions can be extracted. The exactcalculation and the analytical approximation match well as shown in figure 6 as solidinterconnected squares and dashed line, respectively.Starting values ∆R and ∆D for the exact modeling can be extracted from themaximal and minimal oscillation amplitudes in the experimental data occurring at φ = ± π/ M ± AC( φ = ± π/ ≈ (cid:12)(cid:12)(cid:12) ∆R sin θ c ± ∆D cos θ c (cid:12)(cid:12)(cid:12) B ⊥ . (12)Here, we assume ∆R / ∆D > g ∗ <
0. For ∆R / ∆D < φ = ∓ π/ φ = − π/ M AC = 0 for α R /β D = 1 / tan θ c /
2, i.e., theanticrossing gap vanishes to first order for this specific in-plane field direction. For theInSb quantum well this condition is matched for α R /β D ≃ .
1. Equation (12) yieldsin general four solutions for ∆R , ∆D. Acknowledging that we can only determine therelative sign of ∆R and ∆D but not the absolute sign of both we end up with twosolutions ∆R ≈ (∆ M + ± ∆ M − ) B ⊥ θ c , ∆D ≈ (∆ M + ∓ ∆ M − ) B ⊥ θ c . (13)These reduce to one for ∆R / ∆D = 1 / tan ( θ c / M − = 0. We pointout that for the general case of two nondegenerate solutions, the correct solution isobtained by comparing the numerically calculated magnetization with the experiment:the two solutions are distinguished by the behavior of M in the low-field regime where ¶ This equation holds for negative sign of g ∗ . For positive g ∗ , ∆ R and ∆ D need to be exchanged.
ONTENTS M A C ( m e V / T ) M pe r e l e c t r on ( m e V / T ) B (T)2 4 -2024 B (T) M pe r e - ( m e V / T ) Figure 7.
Beating pattern in M for InSb QW parameters at the coincidence angle θ c = 69 . ◦ and φ = 45 ◦ . The curves for ∆R = ∆D = 8 .
07 meV (dark) and∆R = 17 .
03 meV, ∆D = 3 .
82 meV (light) differ in the low field regime, where beatingpatterns are present. The curves are offset for clarity. In large B ⊥ the situation isdifferent (upper left inset). Here, the magnetization traces are almost identical, sinceinside an anticrossing region (circle) the amplitudes are the same to first order asgiven by (13). Outside the anticrossing regions the spin splitting is governed by the(identical) Zeeman energy. Upper right inset: ∆ M AC( φ ) for ν = 4 for both parameterpairs. In this exact calculation, the degeneracy implied by the analytical approximationis lifted. the beating patterns occur. To substantiate this, we show numerically calculatedmagnetization traces in figure 7 for the case ∆R = ∆D = 8 .
07 meV and thecorresponding second solution of (13) given by ∆R = 17 .
03 meV and ∆D = 3 .
82 meV.While the behavior of M at large B ⊥ (left inset of figure 7) is similar for both cases,the beating patterns at low fields differ (main graph). Thus, the exact diagonalizationallows us to distinguish the two cases corresponding to the two solutions of (13) andidentify α R, β D and their relative sign. As can be seen in the right inset at φ = 45 ◦ (arrow), the full numerical treatment in general lifts the degeneracy of the two solutionsfor ∆ M AC( φ ) present in the approximation of (13). We note here, that the sameprocedure applies when only one term of ∆R an ∆D is finite, and it is not known apriori which on it is (unlike the situation considered in section 3.1, where we consideredR-SOI). In such a case ∆ M AC( φ ) = ∆ M is isotropic, and the two possible solutionsare ∆R ≈ ∆ M B ⊥ / sin θ c and ∆D ≈ ∆ M B ⊥ / cos θ c . Again, the applicable solution ONTENTS M , since the beating patterns for the twosolutions differ.The analysis of the anisotropy of ∆ M AC based on (13) can be used as a startingpoint for the exact diagonalization. The full numerical treatment then provides thedefinite values of α R and β D by one-to-one comparison with the experimental data overa broad field regime. k D-SOI
We have focussed on the limit of narrow quantum wells and low n s up to now. Thisallowed us to neglect the Dresselhaus terms that are cubic in the wave vector k . Suchconditions are met for many realistic systems and we showed that in this limit itis in particular possible to use the analytical approximation (11) for the analysis ofexperimental data. With the numerical model at hand it is however instructive toconsider the impact of the k D-SOI in the limit of wide quantum wells. In the followingwe thus consider the Hamiltonian [30, 49] H = H + α R ~ ( σ x π y − σ y π x )+ β D ~ ( σ x π x − σ y π y )+ γ D ~ (cid:0) σ y π x π y − σ x π x π y (cid:1) .(14)For lack of experimental values in the literature, we consider values α R = 5 . × − eVm, β D = 3 . × − eVm and γ D = 4 . × − eVm calculated in theextensive work of Gilbertson et al. [49] for a 30 nm wide In . Al . Sb/InSb/In . Al . Sbasymmetric quantum well with n s = 2 . × m − . We show the results in figure 8.Solid squares interconnected by lines represent the values including the full Hamiltonian(14), i.e., including all k D-SOI terms. For comparison, we show the result where k D-SOI is neglected as open circles interconnected by lines. The difference in the absolutevalues shows that the k D-SOI is important in such wide quantum wells, in contrastto the systems considered in the previous sections. However, the angular dependenceis qualitatively the same. This raises the question, whether one can determine the k D-SOI contribution from experimental data on ∆ M AC and if analytical approximationssimilar to (11) can be found. To answer this, it is more instructive to consider the matrixelements of the problem as given in the appendix of [30], i.e. without rotation of thespin quantization axis. This minimizes the number of entries in the matrix that arerelevant for the discussion. In brief, the last term in (14) leads to two non-vanishingmatrix elements. One couples different spin levels that differ in the Landau index by3, i.e., it couples levels that are far apart in energy in high fields. A calculation of theanticrossing gap neglecting this matrix element [solid red line in figure 8] is virtuallyidentical to the calculation for the full Hamiltonian [solid squares].The other matrix element couples exactly the same levels as the only non-vanishing k -linear Dresselhaus matrix element, i.e., different spin levels of neighboring Landaulevels. It has opposite sign compared to the k -linear element, leading to a decreasedvalue of this entry. Fitting (11) to experimental data thus yields useful informationalso in the limit of wide quantum wells, when ∆ ′ D = 2 k F( β D − γ D k F /
4) is substituted
ONTENTS (a) (b) Figure 8. (a) Polar plot of ∆ M AC ,ν =4 vs azimuthal angle φ . Calculationsare performed for α R = 5 . × − eVm, β D = 3 . × − eVm and γ D =4 . × − eVm given in [49] for a 30 nm wide In . Al . Sb/InSb/In . Al . Sbasymmetric quantum well. Solid squares interconnected by lines represent the valuesincluding the full Hamiltonian (14), i.e., including all k D-SOI terms. Open circlesinterconnected by lines denote results neglecting k D-SOI. The curves differ in theabsolute values, highlighting the importance of k D-SOI in wide quantum wells. Thesolid red line shows the result of a calculation where only the k D-SOI matrix elementis included that couples neighboring Landau levels as discussed in the main text. (b)Low field beating patterns in M for θ = θ c and φ = 45 ◦ including k D-SOI (uppercurve) and neglecting k D-SOI (lower curve). The curves are shifted in verticaldirection for clarity. .for ∆D. Whether the k D-SOI can be neglected in the analysis of experimental data∆ M AC can again be determined experimentally from the different beating patterns inlow magnetic fields. This is demonstrated in figure 8 (b), where we show the low fieldbeating patterns at θ c predicted for the full Hamiltonian including the k D-SOI (uppercurve) and neglecting the k D-SOI (lower curve). Details of the anomalous beatingsthat occur in M due to (14) in perpendicular magnetic fields have been discussed in [25].Information about the relative strength of β D and γ D could be obtained by variationof k F = √ πn s via the carrier density.
4. Discussion
In the following we discuss the observability of the predictions using state-of-the-artexperimental techniques. The magnetization of single-layered 2DESs in semiconductorheterostructures is weak and has been resolved after optimizing custom designedmagnetometers. These include torsion wire magnetometers [50, 39, 51, 52], acustom designed superconducting quantum interference device (SQUID) [53], andmicromechanical cantilever magnetometers [54, 20, 47]. A field modulation technique
ONTENTS M per electron better than 0 .
001 meV/T has been demonstrated for conditions [33] thatwould be required for the proposed investigations. Cantilever magnetometers are thusparticularly suited for experiments addressing the anisotropic M .In general, the magnetization experiment needs to be performed as a function ofboth the out-of-plane and in-plane field angles θ and φ , respectively. For the case where itis known a priori that either R-SOI or D-SOI is absent, the experiment requires rotationabout θ only. To be specific, for the InGaAs/InP quantum well parameters the Zeeman-induced artificial level degeneracy (without SOI) is expected to occur at θ c ≃ . ◦ .The purely SOI-induced dHvA oscillation at ν = 4 highlighted in figure 3 occurs at aperpendicular magnetic field B ⊥ ≃ .
26 T. This corresponds to a total magnetic fieldof B ≃ . B = 33 T has beendemonstrated in [56]. After measuring ∆ M AC( θ ) and determining its minimum value,the Rashba parameter α R is extracted from modeling of the data.When both R-SOI and D-SOI play a role, the experimental procedure is as follows:First, the coincidence angle θ c is determined from the measured minimum of ∆ M AC( θ )for a given φ . Second, θ = θ c is fixed and the field is rotated in the plane about φ . Thisprocedure automatically accounts for a g -factor anisotropy between out-of-plane and in-plane directions [58] in that the experimentally determined critical angle θ c correspondsto the g -factor for that specific field direction. In case of an in-plane anisotropy ofthe g -factor [59], the coincidence angle θ c becomes a function of φ and should thusbe determined for each φ . Modeling the measured anisotropic behaviour of ∆ M ACusing the numerical model introduced above yields the absolute values of α R and β Dand their relative sign. For the InSb based quantum well, the coincidence conditionamounts to θ c ≃ . ◦ . At this angle, the filling factor ν = 4 discussed above occursat B ≃ . φ = ± π/ T for the sakeof clarity. Considering a finite temperature T and disorder broadening of the LLs inthe calculations is straightforward as demonstrated in figure 4. This way a one-to-onecorrespondence between experimental and theoretical data becomes possible for a real ONTENTS
5. Conclusions
Considering recent advances in magnetometry on 2DESs we have presented a detailednumerical analysis of the quantum oscillatory magnetization when Rashba andDresselhaus SOI are relevant. The formalism predicts that R-SOI and D-SOI constants α R and β D, respectively, can be extracted from the de Haas-van Alphen effect ifmeasured in tilted magnetic fields. For the evaluation, the anticrossing of energy levelsis found to be of particular importance. In the case of one dominant SOI term, themagnetization oscillation amplitude at the anticrossing is a direct measure of the SOIstrength. In the general case of α R = 0 and β D = 0, the oscillation amplitude exhibitsa pronounced anisotropy with respect to the direction of the in-plane magnetic fieldcomponent. Using realistic sample parameters we argue that state-of-the-art torquemagnetometry techniques allow one to study experimentally the predicted anisotropyof M . Via numerical modeling α R and β D as well as their relative sign are obtained.Experiments addressing M lift ambiguities and circumvent model assumptions that existwith the analysis of beating patterns in magnetotransport data and the investigationof weak-antilocalization features, respectively. Magnetization measurements thus allowone to further promote the understanding of SOI in low-dimensional electron systems. Acknowledgments
We thank E. I. Rashba for valuable discussions and gratefully acknowledge financialsupport by the DFG via SPP1285, grant no. GR1640/3.
References [1] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Moln´ar S, Roukes M L,Chtchelkanova A Y and Treger D M 2001
Science
Physics World Rev. Mod. Phys. Appl. Phys. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Applied Physics Letters Progress of Theoretical Physics Journal of the Physical Society of Japan Pisma Zh. Eksp. Teor. Fiz. [JETP Lett.] Phys.Rev. B Phys. Rev. Lett. Phys. Rev. B ONTENTS [14] Kallaher R L, Heremans J J, Goel N, Chung S J and Santos M B 2010 Phys. Rev. B Phys. Rev. B Phys. Rev. Lett. Nature Physics Phys. Rev. B Phys. Rev. B Phys. Rev. B Phys. Rev. B Phys. Rev. Lett. Physica E:Low-dimensional Systems and Nanostructures
191 ISSN 1386-9477 Proceedings of the 16thInternational Conference on Electronic Properties of Two-Dimensional Systems (EP2DS-16)[24] Bychkov Y A and Rashba E I 1984
J. Phys. C Phys. Rev. B Physica E Phys. Rev. B Phys. Rev. B (3) 035307[29] Wilde M A, Reuter D, Heyn C, Wieck A D and Grundler D 2009 Phys Rev. B Phys. Rev. B Phys. Rev. B Sov. Phys. JETP Magnetization of Interacting Electrons in Low-Dimensional Systems (Springer Nanoscience and Technology) chap 10, p 245[34] Guzenko V A, Sch¨apers T and Hardtdegen H 2007
Phys. Rev. B Phys. Rev.
Phys. Rev. B Phys. Rev. B R12743[38] Wilde M A, Schwarz M P, Heyn C, Heitmann D, Grundler D, Reuter D and Wieck A D 2006
Phys. Rev. B Phys. Rev. Lett. Phys. Rev. B Phys. Rev. B Semicond. Sci. Technol. R1[43] Luo J, Munekata H, Fang F F and Stiles P J 1990
Phys. Rev. B Phys. Rev. B R1958[45] Grundler D 2000
Phys. Rev. Lett. Phys. Rev. B Phys. Status Solidi B
Phys. Rev. B Phys.Rev. B Phys.Rev. Lett. ONTENTS [51] Zhu M, Usher A, Matthews A J, Potts A, Elliott M, Herrenden-Harker W G, Ritchie D A andSimmons M Y 2003 Phys. Rev. B Appl. Phys. Lett. Appl. Phys.Lett. Appl. Phys. Lett. Phys. Rev. Lett. Phys. Rev. B Phys. Rev. B Pisma Zh. Eksp. Teor. Fiz. [1992
JETP Lett. , 253] Pisma Zh. Eksp. Teor. Fiz. [1993
JETP Lett. , 571]57