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Quantum Spin Hall Effect SLAC-PUB-13912 Quantum Spin Hall Effect B. Andrei Bernevig and Shou-Cheng Zhang Department of Physics, Stanford University, Stanford, CA 94305 The quantum Hall liquid is a novel state of matter with profound emergent properties such as fractional charge and statistics. Existence of the quantum Hall effect requires breaking of the time reversal symmetry caused by an external magnetic field. In this work, we predict a quantized spin Hall effect in the absence of any magnetic field, where the intrinsic spin Hall conductance is e quantized in units of 2 4π .The degenerate quantum Landau levels are created by the spin-orbit coupling in conventional semiconductors in the presence of a strain gradient. This new state of matter has many profound correlated properties described by a topological field theory. Recently, the intrinsic spin Hall effect has been the- field E~ is not constant but is proportional to the radial oretically predicted for semiconductors with spin-orbit coordinate ~r, as it would be, for example, in the inte- coupled band structures[1, 2]. The spin Hall current is rior of a uniformly charged cylinder E~ ∼ E(x, y, 0), then induced by the external electric field according to the the spin-orbit coupling term in the Hamiltonian takes the equation form Eσz(xpy − ypx). We see that this system behaves in such a way as if particles with opposite spins experi- ji = σ ǫ E (1) j s ijk k ence the opposite “effective” orbital magnetic fields, and i a Landau level structure should appear for each spin ori- where jj is the spin current of the i-th component of the spin along the direction j, Ek is the electric field and ǫijk entations. is the totally antisymmetric tensor in three dimensions. However, such an electric field configuration is not easy The spin Hall effect has recently been detected in two to realize. Fortunately, the scenario previously described different experiments[3, 4], and there is strong indication is realizable in zinc-blende semiconductors such as GaAs, that at least one of them is in the intrinsic regime[5]. where the shear strain gradients can play a similar role. Because both the electric field and the spin current are Zinc-blende semiconductors have the point-group sym- even under time reversal, the spin current could be dis- metry Td which is half of the cubic-symmetry group Oh, sipationless, and the value of σs could be independent and does not contain inversion as one of its symmetries. of the scattering rates. This is in sharp contrast with Under the Td point group, the cubic harmonics xyz trans- the extrinsic spin Hall effect, where the effect arises only form like the identity, and off-diagonal symmetric tensors from the Mott scattering from the impurity atoms[6]. (xy + yx, etc.) transform in the same way as vectors on The independence of the intrinsic spin Hall conduc- the other direction (z, etc), and represent basis func- tance σs on the impurity scattering rate naturally raises tions for the T1 representation of the group. Specifically, the question whether it can be quantized under certain strain is a symmetric tensor ǫij = ǫji, and its off-diagonal conditions, similar to the quantized charge Hall effect. (shear) components are, for the purpose of writing down We start off our analysis with a question: Can we have a spin-orbit coupling Hamiltonian, equivalent to an elec- Landau Level (LL) -like behavior in the absence of a mag- tric field in the remaining direction: netic field [7]? The quantum Landau levels arise physi- cally from a velocity dependent force, namely the Lorentz ǫxy ↔ Ez; ǫxz ↔ Ey; ǫyz ↔ Ex (2) force, which contributes a term proportional to A~ · ~p in the Hamiltonian. Here ~p is the particle momentum and The Hamiltonian for the conduction band of bulk zinc- A~ is the vector potential, which in the symmetric gauge blende semiconductors under strain is hence the anal- ~ is given by A~ = B (y, −x, 0). In this case, the velocity ogous to the spin-orbit coupling term (~v × E) · ~σ. In 2 2 dependent term in the Hamiltonian is proportional to addition, we have the usual kinetic p term and a trace of the strain term trǫ = ǫ + ǫ + ǫ , both of which B(xpy − ypx). xx yy zz In condensed matter systems, the only other ubiqui- transform as the identity under Td: tous velocity dependent force besides the Lorentz force 2 p 1 C3 is the spin-orbit coupling force, which contributes a term H = 2m + Btrǫ + 2 ~ [(ǫxypy − ǫxzpz)σx + ~ ~ proportional to (~p × E) · ~σ in the Hamiltonian. Here E is +(ǫzypz − ǫxypx)σy + (ǫzxpx − ǫyzpy)σz] (3) the electric field, and ~σ is the Pauli spin matrix. Unlike C3 5 the magnetic field, the presence of an electric field does For GaAs, the constant ~ = 8×10 m/s [8]. This Hamil- not break the time reversal symmetry. If we consider the tonian is not new and was previously written down in Ref. particle momentum confined in a two dimensional geom- [9, 10, 11, 12] but the analogy with the electric field and etry, say the xy plane, and the electric field direction con- its derivation from a Lorentz force is suggestive enough to fined in the xy plane as well, only the z component of the warrant repetition. There is yet another term allowed by spin enters the Hamiltonian. Furthermore, if the electric group theory in the Hamiltonian [13], but this is higher SIMES, SLAC National Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94309 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 order in perturbation theory and hence not of primary 1 D R R 2 2m (1 ∓ 2 )m + (1 ± 2 )n +1 . We shall focus on importance. theq case of R = 2 where there is no additional static Let us now presume a strain configuration in which potential within the Landau level. ǫ = 0 but ǫ has a constant gradient along the y di- xy xz For the spin up electron, the vicinity of R ≈ 2 is charac- rection while ǫyz has a constant gradient along the x di- 1 C3 terized by the Hamiltonian H = ~ g(2aa†+1) with the n 2 ⋆ rection. This case then mimics the situation of the elec- ↑z zz LLL wave function φ↑ (z)= exp( − ). a is the op- tric field inside a uniformly charged cylinder discussed n √πn! 2 erator moving between different Landau levels, while b is above, as ǫxz(↔ Ey)= gy and ǫyz(↔ Ex)= gx, g being the magnitude of the strain gradient. With this strain the operator moving between different degenerate angu- configuration and in a symmetric quantum well in the lar momentum states within the same LL: Lz = bb† −aa†, xy plane, which we approximate as being parabolic, the Lzφn↑ (z)= nφn↑ (z). The wave function, besides the con- above Hamiltonian becomes: fining factor, is holomorphic in z, as expected. These up spin electrons are the chiral, and their charge conduc- p2 + p2 1 C tance is quantized in units of e2/h. H = x y + 3 g(yp − xp )σ + D(x2 + y2) (4) 2m 2 ~ x y z For the spin down electron, the situation is exactly the opposite. The vicinity of R ≈ 2 is characterized We first solve this Hamiltonian and come back to the 1 C3 by the Hamiltonian H = 2 ~ g(2bb† +1) with the LLL experimental realization of the strain architecture in ↓ ⋆ m ⋆ (z ) zz wave function φ (z) = exp( − ). b is the oper- the later stages of the paper. We make the coordi- m↓ √πm! 2 1/4 1/4 nate change x → (2mD)− x, y → (2mD)− y and ator moving between different Landau levels, while a is 2 1 C3 2m 2mgC3 the operator between different degenerate angular mo- R = 2 ~ D g. R = 2 or D = D0 ≡ 16~2 is q mentum states within the same LL: Lz = bb† − aa†, a special point, where the Hamiltonian can be writ- L φ (z) = −mφ (z). The wave function, besides the 1 ~ 2 z m↓ m↓ ten as a complete square, namely H = 2m (~p − eAσz) confining factor, is anti-holomorphic in z. These down ~ mC3g with A = 2~e (y, −x, 0). At this point, our Hamilto- spin electrons are anti-chiral, and their charge conduc- nian is exactly equivalent to the usual Hamiltonian of tance is quantized in units of −e2/h. a charged particle in an uniform magnetic field, where the two different spin directions experience the oppo- site directions of the “effective” magnetic field. Any generic confining potential V (x, y) can be written as 2 2 D0(x + y ) + ∆V (x, y), where the first term completes the square for the Hamiltonian, and the second term 2 2 ∆V (x, y) = V (x, y) − D0(x + y ) describes the addi- tional static potential within the Landau levels. Since [H, σz] = 0 we can use the spin on the z direction to char- acterize the states. In the new coordinates, the Hamilto- nian takes the form: H 0 H = ↑ 0 H ↓ D 2 2 2 2 H , = 2m [px + py + x + y ± R(xpy − ypx)] (5) ↓ ↑ q The H , is the Hamiltonian for the down and up spin σz respectively.↓ ↑ Working in complex-coordinate formalism and choosing z = x+iy we obtain two sets of raising and FIG. 1: Spin up and down electrons have opposite chiral- lowering operators: ity as they feel the opposite spin-orbit coupling force. Total charge conductance vanishes but spin conductance is quan- ⋆ ⋆ z z tized.
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