Spin-Orbit Interaction in Symmetric Wells with Two Subbands

(1)

Spin-Orbit Interaction in Symmetric Wells with Two Subbands

Esmerindo Bernardes,¹John Schliemann,^2,3Minchul Lee,³J. Carlos Egues,^1,3,4and Daniel Loss^3,4

1Instituto de Fı´sica de Saõ Carlos, Universidade de Saõ Paulo, 13560-970 Saõ Carlos, Saõ Paulo, Brazil

2Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany

3Department of Physics and Astronomy, University of Basel, CH-4056 Basel, Switzerland

4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA (Received 11 July 2006; published 17 August 2007)

We investigate the spin-orbit (SO) interaction in two-dimensional electron gases in quantum wells with two subbands. From the 88 Kane model, we derive a new intersubband-induced SO term which resembles the functional form of the Rashba SO but is nonzero even insymmetricstructures. This follows from the distinct parity of the confined states (even or odd) which obliterates the need for asymmetric potentials. We self-consistently calculate the new SO coupling strength for realistic wells and find it comparable to the usual Rashba constant. Our new SO term gives rise to a nonzero ballistic spin-Hall conductivity, which changes sign as a function of the Fermi energy ("_F) and can induce an unusual Zitterbewegungwith cycloidal trajectorieswithoutmagnetic fields.

DOI:10.1103/PhysRevLett.99.076603 PACS numbers: 72.25.Dc, 71.70.Ej, 73.21.Fg, 85.75.d

The rapidly developing field of spintronics has generated a great deal of interest in spin-orbit (SO) coupling in semiconductor nanostructures [1]. For an n-doped zinc- blende semiconductor quantum well with only the lowest subband occupied, i.e., in a strictly 2D situation, there are two main contributions to the interaction of the spin and orbital degrees of freedom of electrons. One contribution is the Dresselhaus term, which results from the lack of inversion symmetry of the underlying zinc-blende lattice [2]

and is to lowest order linear in the crystal momentum [3].

This linearity is shared by the other contribution known as the Rashba term [4], which is due to structural inversion asymmetry and can be tuned by an electric gate across the well [5]. These two contributions can lead to an interesting interplay in spintronic systems [6].

In this Letter we consider yet another type of electronic SO coupling which, as we show, occurs in III-V (or II-VI) zinc-blende semiconductor quantum wells with more than one subband. We derive a new intersubband-induced SO interaction which resembles that of the ordinary Rashba model; however, in contrast to the latter, ours is nonzero even insymmetricstructures (Fig.1). We self-consistently determine the strength of this new SO coupling for realistic single and double wells and find it comparable to the Rashba constant [Figs. 2(a) and 2(b)]. We have investigated the spin-Hall effect and the dynamics of spin- polarized electrons due to this new SO term. We find (i) a nonzero ballistic spin-Hall conductivity which changes sign as a function of "_F and (ii) an unusual Zitterbewegung [7] with cycloidal trajectories without magnetic fields (Fig.3). As derived below, for a symmetric well with two subbands our44electron Hamiltonian is

H p~²

2m

11^z1

@^x p_x^yp_y^x; (1)

wherem is the effective mass, "_o "_e=2,"_eand

"_o are quantized energies of the lowest (even) and first excited (odd) subbands (corresponding to eigenstates jei and joi), respectively, measured from the bottom of the quantum well, ^x;y;z denote the Pauli matrices describing the subband (or pseudospin) degree of freedom, and^x;y;z are Pauli matrices referring to the electron spin. The new intersubband-inducedSO couplingis

1

E²_g 1 E_g²

P²

3 hej@_zVzjoi

_v

E²_g E_g²

P²

3 hej@_zhzjoi; (2) where E_g and are the fundamental and split-off band gaps in the well region [8],P is the Kane matrix element [9]. The parameters_vanddenote valence-band offsets between the well and the barrier regions [10],Vzis the Hartree-type contribution to the electron potential, and hzis the structural quantum-well profile [11]. Note that

) (

: z

o ϕo

0 ) ( ) ( ) ( )

( − − − ≠

∝ϕeaϕoa ϕe aϕo a η

o e i a

a _i

i

i∝ϕ()²−ϕ(−)²=0, =, α

δc

εe

εo ) (

: z

e ϕe

−a a z

FIG. 1 (color online). Square well with its ground-state’ez and first excited-state ’_oz wave functions. The new intersubband-induced SO coupling in Eq. (4) is nonzero even in symmetric wells due to the distinct parities of ’_ez (even) and ’_oz (odd), which yield a nonvanishing matrix element for the derivative of the symmetric potential.

PRL99,076603 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending 17 AUGUST 2007

(2)

can be varied via external gates (Fig.2). Next we outline the derivation ofH in Eq. (1).

Kane Hamiltonian. —We start from the usual 88 Kane Hamiltonian describing the s-type conduction and thep-type valence bands around thepoint [12],

H88 H_c H_cv H_vc H_v

; (3)

where H_c is a 22 diagonal matrix with elements

p²=2m₀V_c~r,m₀is the bare electron mass,H_vis a6 6diagonal matrix with elementsp²=2m₀V_v~r E_gfor the heavy- and light-hole bands, and p²=2m₀V~r E_gfor the split-off band, V_i~r (ic; v;) denote arbitrary potentials (see below), andH_cv H_vc^yis

H_cv

^p₂

2 3

q _z ^p₆ 0 ^p₃^z ^p₃ 0 ^p₆

2 3

q_z ^p₂ ^p₃ ^p^z₃ 0

B@

1 CA; (4)

where~ P ~k,k~p=~ @is the electron wave vector,k k_x ik_y, and P i@hSjp_xjXi=m₀ parametrizes the conduction-to-valence-band coupling, jSiandjXiare the usual periodic Bloch functions at thepoint.

Effective electron Hamiltonian: Folding down.—The Kane Hamiltonian (3) acts on an eight-component spinor ^y _c _v^y in which the last six components _v represent valence-band states. By eliminating the hole components from the Schro¨dinger equation H88

", where"is the eigenenergy, we can fold down this8 8 equation into a 22 effective equation for the conduction-band states only: H"~

c H_cH_cv"

H_v¹H_vc~

c, ~

c is a renormalized conduction-electron spinor.

SO in symmetric wells. —Applying the above procedure to a quantum well, defined by the confining potentials [11]

V_i~r !V_iz Vz _ihz,ic; v;, we find H" H_QW P²

3@²p¹₁ ¹₂ ; p_z; (5) H_QW p 1

2mz; "pp_z 1

2mz; "p_zV_cz; (6) where 1=mz; " 2P²=3@²2=₁1=₂ 1=m₀, ₁ " p²=2m₀Vz _vhz E_g and ₂

" p²=2m₀Vz hz E_g. Equation (5) describes an electron in a quantum well (H_QW term) with spin-orbit interaction (last term) [13]. The kinetic-energy operators above are complicated due to the position- and energy-dependent effective mass mz; ". Since E_g and E_gare the largest energy scales in our system, we can simplify (5) and (6) by expanding1=₁ and1=₂ in the form 1=₁ E¹_g f1 "p²=2m₀Vz _vhz=

E_g g and 1=₂ E_g¹f1 "p²=2m₀ Vz hz=E_g g. To zeroth order ₁ E_g, ₂E_g, and H_QWp²_k=2mp²_z=2m V_cz with (a constant effective mass) 1=m 2P²=3@²2=E_g1=E_g 1=m₀ [14]. Since the SO operator ¹₁ ¹₂ ; p_z !@_z1=₁ @_z1=₂, we need to keep the first-order terms in the expansions of ¹₁ and¹₂ which yield the leading nonzero contribution to the SO term in (5). We find¹₁ ¹₂ ; p_z 1=E²_g 1=E_g²@_zVz _v=E²_g =E_g ²@_zhz.

Finally, we project this SO operator into the twolowest (spin-degenerate) eigenstates jii_z jk~kiij_zi, h~rjk~kii expi ~kk~rk’_iz,ie; o, and_z";#, of thesymmetric

-6 -4 -2 0 2 4

-0.2 0.0 0.2 0.4 0.6 0.8

η,αi,βi(meVnm)

Vb(eV)

(a) GaInAs single well _α^η

αeo βe βo Vb=0

-20 -10 0 10 20 30

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

η,αi,βi(meVnm)

V_b(eV) (b) InSb double well

αηe αo βi

Vb=0 12 13

-0.02 0.02

ε±(k)(meV)

k(nm⁻¹) Vb=0

FIG. 2 (color online). Calculated SO coupling strengths as a function of the external gate V_b for realistic wells. (a) For the single GaInAs [16,20] well studied, the intersubband coupling is larger than the Dresselhaus_i and the Rashba_i constants (ie; o). Note that j_ej j_oj and both change sign across V_b0(in contrast to_i and). (b) For the InSb double well considered,shows a ‘‘resonant behavior’’ aboutV_b0[symmetric configuration, lower-left inset in (b)]. This occurs because the subband splitting"_o"_ereaches a minimum atV_b0and the double-well wave functions are very similar (though of distinct parities) forV_b0. This also makes_e _oaround V_b0. Upper-right inset in (b): Energy dispersions " k~ [Eq. (10)] of the symmetric double well.

FIG. 3 (color online). Zitterbewegungdue to the SO coupling for distinct ratios =2"_SO. Note the peculiar trajectories with the forward injected electrons moving backward (I) and even in aclosed path (II). This follows from the SO induced change in the curvature of the bands which renormalizes the effective masses. Here we useSOk0y =10,¹_SOm=@.

076603-2

(3)

well (H_QW) (Fig.1). This directly leads to the H in (1) with the SO coupling (2) [15]. Note that this new SO interaction is nonzero even in symmetric wells asarises from the coupling between the ground state (even) and the first excited state (odd) [Eq. (2)]. We can generalizeH to include the Rashbaand the linearized DresselhausSO couplings. Next we determine the magnitude of(and, ) for realistic quantum wells with two subbands.

Self-consistent calculation of the SO couplings. —We consider modulation-doped quantum wells similar to those experimentally investigated in Ref. [16]. Our wells, however, havetwooccupied subbands. Similarly to Ref. [16], we study cases with constant chemical potentials [17]. By self-consistently solving Poisson and Schro¨dinger’s equations we determine the energy levels "_e; "_o and the confined wave functions’_iz,ie; oof the wells. We then calculate (i)via Eq. (2), (ii)_ifrom equations similar to Eq. (2) for each subband, and (iii)_i_chijk²_zjii,_cis the bulk Dresselhaus SO parameter [18]. The structural symmetry of the wells and their charge densities can be changed via a gate potentialV_b.

Our calculated SO couplings,_i, and_i,ie; o, for anInAlAs=InGaAs=InAlAs single quantum- well (‘‘sam- ple 3’’ in [16]) are all comparable in magnitude [Fig.2(a)].

Note that our _e=_e ratio is consistent with the experimental one in Ref. [19]. In addition,_e vsV_b here agrees well with the experimental data in Fig. 3 (‘‘triangle up’’) of Ref. [16] (see also Fig. 4 in [20]). Our_i[21] and_iare also consistent with those of Ref. [22]. Note that for the single well studied heredoes not vary appreciably with the gateV_b, similarly to_iand as opposed to_i.

For adouble-well structure, on the other hand, we find that has a ‘‘resonant behavior,’’ changing by about an order of magnitude asV_bis swept acrossV_b0[Fig.2(b)]

(this may have a dramatic effect on Shubnikov– de Haas measurements).V_b0corresponds to a fully symmetric double well. In contrast to the single-well case,_eand_o have opposite signs and undergo abrupt changes in magni- tudes nearV_b 0[Fig.2(b), dashed lines]. Similarly to the single-well case,_e and _o are also essentially constant for a double well [Fig. 2(b), dotted lines]. A detailed account of our results will be presented elsewhere.

Having established that the new SO couplingis sizable, in what follows we focus on a fully symmetric well to investigate physical effects arising solely from.

Fully symmetric case: Eigensolutions. —Let us consider a two-subband well (single or double) described by the Hamiltonian H in (1) (we assume a negligible Dresselhaus term [19]). In the basis fjei_";joi_#;joi_";jei_#g H becomes

H~

"²k²

2m"_e ik 0 0

ik ^"_2m²^k²"_o 0 0

0 0 ^"_2m²^k²"_o ik

0 0 ik "²k²

2m "_e 0

BB BB

@

1 CC CC A:

(7)

Both the upper-left (U) and lower-right (L) blocks of H~ have eigenvalues

" k ~ _k @; (8) with _k @²k²=2m, @² ²k²², and ei- genvectors

j ₁i^Usin=2jei_"cos=2eⁱjoi_#; (9) j ₂i^Lcos=2joi"sin=2eⁱjei#; (10) j ₃i^Ucos=2jei_"sin=2eⁱjoi_#; (11) j ₄i^Lsin=2joi_"cos=2eⁱjei_#: (12)

Here, eⁱ k_yik_x=k, cos 1=

1 k=² p , and k~ksin;cos (here we drop the ‘‘k’’ in k~_k).

For k2 we can expand " k~ in (8) and define effective massesm m=1 2"_SO=, where"_SO ²m=2@² is the energy scale of the new SO coupling. For the double well of Fig. 2(b), m is reduced by 5%

compared to the bulk value m. This could be measured via, e.g., cyclotron-resonance experiments [23].

Novel Zitterbewegung. — The dynamics of electron wave packets in wells with SO interaction exhibit an oscillatory motion [7]— theZitterbewegung. For our new SO interaction, a wave packet ji moves according to hj~r_Htji where ~r_Ht U^y~rUis the position operator in the Heisenberg picture [UexpiHt="] with components

x_Ht 11x0 11p_x mt

@t^x^y

2@²

^y^y

@p_y1^z

cos2t 1 2@³

²^x^y

@^z1p_x

@ ₂

p_y^x p_x^yp_y^x

sin2t 2t; (13) and y_Ht, obtained from Eq. (13) via the replacements p_x; ^x哫p_y; ^y, p_y; ^y哫p_x;^x (i.e., a =2 rotation about the z axis). Similar expressions can be derived for the spin components ⁱ_Ht,ix; y; z[24].

For simplicity, we evaluate the expectation value of

~r_Ht for planes waves (‘‘wide wave packets’’). For a spin-up electron injected into the lowest subband along theyaxis with (group) velocityv~_g @k_0y=my, we find^

hx_Hti ²k_0y

2@²1cos2t; (14)

hy_Hti @k_0y

m t²k_0y

2@³ sin2t 2t; (15) assuming x0 y0 0. Equations (14) and (15) show that cycloidal motion is possible in our system. This differs PRL99,076603 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending

17 AUGUST 2007

076603-3

(4)

qualitatively from the Rashba SOZitterbewegungwhich is always perpendicular to the initialv~_g.

Figure 3 shows trajectories for three distinct

~

v_g @k_0y=my—all with^ k_0y>0. We find motion op- posite toand along theyaxis (orbits I and III, respectively) and even a closed path (II). To understand this behavior we note that for k_0y"_o"_e the linear-in-t terms in hy_Hti can be recast into@k_0yt=m) the injected wave moves with the renormalized velocity v_g@k_0y=m. Hence, for orbit I, <1)m<0[25] andv_g<0, for orbit II, 1)m! 1andv_g0, and for orbit III, >1)m>0 and v_g>0. Though remarkable, we stress that the orbits I and II occur forunusualparameters (e.g., "_F< "_SO=10). However, these orbits do show that our SO Hamiltonian has a physical mechanism allowing for cyclotronic motion without magnetic fields.

Spin-Hall conductivity ^z_xy.—The spin-Hall effect is a convenient probe for SO effects in wells [26]. We have calculated^z_xy(‘‘clean limit’’) in the presence of an external magnetic fieldBby following the approach of Rashba [27], which allows us to properly account for both the intrabranch and interbranch contributions in the Kubo formula [28]. Here we focus on the B!0 limit where we find^z_xy0for"_F> "_o(two subbands occupied) and

^z_xy e 8

1 ₁

1 ₃2

₂₁=2 2³₃

(16) for "_e< "_F< "_o (upper subband empty), where ₁ 2"_SO=, ₂ "_F=2, and ₃

²₁=4₁₂1=4

q . Note that^z_xy is nonzero and non- universal in this range, shows a discontinuity at"_F "_o, and changes sign as a function of "_F. Details of our calculation of ^z_xyand a thorough discussion will be presented elsewhere [29]. Here we just note that measurements of^z_xy(versus"_F) in symmetric two-subband wells offer a possibility to probe our new SO interaction [26].

Note that the Rashba and the linearized Dresselhaus spin- Hall conductivities are identically zero in the dc limit [27,30].

We have introduced an intersubband-induced SO interaction in quantum wells with two subbands. The corresponding SO coupling (whose magnitude is similar to Rashba coupling) is nonzero even in symmetric wells. This new SO interaction gives rise to a nonzero spin-Hall conductivity, renormalizes the bulk mass by5%(measurable via cyclotron resonance [23]) in double wells, and can induce a cycloidalZitterbewegung. Weak antilocalization [20,31] should offer another possibility to measure.

We thank S. Erlingsson, D. S. Saraga, D. Bulaev, J. Lehmann, M. Duckheim, L. Viveiros, G. J. Ferreira, R.

Calsaverini, and E. Rashba for useful discussions. This work was supported by the Swiss NSF, the NCCR Nanoscience, DARPA, ONR, CNPq, FAPESP, DFG via SFB 689, and U.S.A. NSF PHY99-07949.

[1] Semiconductor Spintronics and Quantum Computation, edited by D. D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002); I. Zutic et al., Rev. Mod. Phys.

76, 323 (2004).

[2] G. Dresselhaus, Phys. Rev.100, 580 (1955).

[3] M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys.

Semicond. 20, 110 (1986); G. Bastard and R. Ferreira, Surf. Sci.267, 335 (1992).

[4] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).

[5] G. Engelset al., Phys. Rev. B55, R1958 (1997); J. Nitta et al., Phys. Rev. Lett.78, 1335 (1997).

[6] J. Schliemannet al., Phys. Rev. Lett.90, 146801 (2003);

M. Trushin and J. Schliemann, Phys. Rev. B 75, 155323 (2007).

[7] J. Schliemann et al., Phys. Rev. Lett. 94, 206801 (2005).

[8] I. Vurgaftmanet al., J. Appl. Phys.89, 5815 (2001).

[9] E. O. Kane, J. Phys. Chem. Solids1, 249 (1957).

[10] Note that _vE^b_gE_g_c and _v_b, whereE^b_g and_b are, respectively, the fundamental and split-off band gaps in the barriers andcthe conduction- band potential offset [see, e.g., P. Pfeffer and W.

Zawadzki, Phys. Rev. B68, 035315 (2003)].

[11] R. Lassnig, Phys. Rev. B31, 8076 (1985).

[12] T. Darnhofer and U. Ro¨ssler, Phys. Rev. B 47, 16020 (1993).

[13] In (5) we neglect an effective Darwin term as it arises in relativistic quantum mechanics. This term leads to a rigid intrabandshift and can be absorbed into"_e; "_o.

[14] We treat m as a parameter in our model (the Kane effective mass neglects corrections from higher-lying bands).

[15] In Eq. (1) we assume thathej@_zVjoiand (in turn) are real. If jjeⁱ, we have to replace ^xbycos^x sin^y in (1). The freedom in fixinghas no physical consequences since a change incan be compensated by applying the phase factor expi^z to the basis states:

cos^xsin^yexpi^z ^x. Thus,H enjoys a U(1) gauge symmetry corresponding to a rotation about thezdirection in the~space.

[16] T. Kogaet al., Phys. Rev. Lett.89, 046801 (2002).

[17] Wells with constant areal densities yield similar results in the parameter range studied.

[18] J.-M. Jancuet al., Phys. Rev. B72, 193201 (2005).

[19] S. Giglbergeret al., Phys. Rev. B75, 035327 (2007).

[20] T. Kogaet al., Phys. Rev. B74, 041302(R) (2006).

[21] C. M. Huet al.[Phys. Rev. B60, 7736 (1999)] find a larger Rashba constant in the second subband.

[22] E. Shafiret al., Phys. Rev. B70, 241302(R) (2004).

[23] H. K. Nget al., Appl. Phys. Lett.75, 3662 (1999).

[24] E. Bernardeset al., Phys. Status Solidi C3, 4330 (2006).

[25] L. Senaet al., Phys. Rev. B72, 235309 (2005).

[26] V. Sihet al., Nature Phys.1, 31 (2005).

[27] E. I. Rashba, Phys. Rev. B70, 201309(R) (2004).

[28] J. Sinovaet al., Phys. Rev. Lett.92, 126603 (2004).

[29] M. Leeet al.(to be published).

[30] J. Sinova et al., Solid State Commun. 138, 214 (2006).

[31] V. A. Guzenko et al., Phys. Status Solidi C 12, 4227 (2007).

076603-4