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Home , Quantum Hall effect

SLAC-PUB-13912

Quantum Hall Effect

B. Andrei Bernevig and Shou-Cheng Zhang Department of Physics, Stanford University, Stanford, CA 94305 The quantum Hall is a novel state of 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 in the presence of a strain gradient. This new 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 -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 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 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 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. The inset shows the lattice displacement leading to the a = ∂z + 2 , a† = −∂z + 2 ⋆ strain configuration. z ⋆ z b = ∂z + 2 , b† = −∂z + 2 (6) in terms of which the Hamiltonian decouples: The picture that now emerges is the following: our system is equivalent to a bilayer system; in one of the D R R layers we have spin down electrons in the presence of a H , =2 (1 ∓ )aa† + (1 ± )bb† +1 (7) ↓ ↑ r2m  2 2  down-magnetic field whereas in the other layer we have spin up electrons in the presence of an up-magnetic field. The eigenstates of this system are harmonic oscil- These two layers are placed together. The spin up elec- m n , lators |m,ni = (a†) (b†) |0, 0i of energy Em,n↓ ↑ = trons have positive charge Hall conductance while the 3 spin down electrons have negative charge Hall conduc- present case, we have re-created the so-called symmet- tance. As such, the charge Hall conductance of the whole ric gauge in magnetic-field language, but, with different system vanishes. The time reversal symmetry reverses strain architectures, one can create the Landau gauge the directions of the “effective” orbital magnetic fields, Hamiltonian and indeed many other gauges. The Landau but interchanges the layers at the same time. However, gauge Hamiltonian is maybe the easiest to realize in an the spin Hall conductance remains finite, as the chiral experimental situation, by growing the quantum well in states are spin-up while the anti-chiral states are spin- the [110] direction. This situation creates an off-diagonal 1 down, as shown in Figure[1]. The spin Hall conductance strain ǫxy = S44T , and ǫxz = ǫyz = 0 where T is the 2 4 e ~ e lattice mismatch (or impurity concentration), s44 is a ma- is hence quantized in units of 2 h 2e = 2 4π . Since an electron with charge e also carries spin ~/2, a factor of terial constant and x,y,z are the cubic axes. The spin- ~ C3 orbit part of the Hamiltonian is now ǫxy(pxσy −pyσx). 2e is used to convert charge conductance into the spin ~ conductance. However, since the growth direction of the well is [110] While it is hard to experimentally measure the spin we must make a coordinate transformation to the x′,y′,z′ Hall effect, and supposedly even harder to measure the coordinates of the quantum well (x′,y′ are the new coor- quantum version of it, the measurement of the charge dinates in the plane of the well, whereas z′ is the growth quantum Hall effect has become relatively common. In coordinate, perpendicular to the well and identical to the our system, however, the charge Hall conductance σxy [110] direction in cubic axes). The coordinate transfor- mation reads: x = 1 (x − y), y = −z, z = 1 (x + y), vanishes by symmetry. However, we can use our physi- ′ √2 ′ ′ √2 cal analogy of the two layer system placed together. In and the momentum along z′ is quantized. We now vary each of the layers we have a charge Quantum Hall effect the impurity concentration T (or vary the speed at which at the same time (since the filling is equal in both lay- we deposit the layers) linearly on the y′ direction of the ers), but with opposite Hall conductance. However, when quantum well so that ǫxy = gy′ where g is strain gradient, on the plateau, the longitudinal conductance σxx also linearly proportional to the gradient in T . In the new co- vanishes (σxx = 0) separately for the spin-up and spin- ordinates and for this strain geometry, the Hamiltonian down electrons, and hence vanishes for the whole system. reads: Of course, between plateaus it will have non-zero spikes 2 p C3 2 (narrow regions). This is the easiest detectable feature ′ ′ H = + ~ gy′px σz + Dy (8) of the new state, as the measurement is entirely electric. 2m Other experiments on the new state could involve the in- where we have added a confining potential. At the suit- jection of spin-polarized edge states, which would acquire able match between D and g, this is the Landau-gauge different chirality depending on the initial spin direction. Hamiltonian. One can also replace the soft-wall condition We now discuss the realization of a strain gradient of (the Harmonic potential) by hard-wall boundary condi- the specific form proposed in this paper. The strain tion. tensor is related to the displacement of lattice atoms We now estimate the Landau Level gap and the strain from their equilibrium position ui in the familiar way gradient needed for such an effect, as well as the strength ǫij = (∂ui/∂xj + ∂uj/∂xi)/2. Our strain configuration of the confining potential. In the case R ≈ 2 the en- is ǫxx = ǫyy = ǫzz = ǫxy = 0 as well as the strain gra- ergy difference between Landau levels is ∆ELandau = ~ 1 C3 dients ǫzx = gy and ǫyx = gx. Having the diagonal 2 × 2 ~ g = C3g. For a gap of 1mK, we hence need strain components non-zero will not change the physics a strain gradient or 1% over 60µm. Such a strain gra- as they add only a term to the Hamil- dient is easily realizable experimentally, but one would tonian. The above strain configuration corresponds to a probably want to increase the gap to 10mK or more, displacement of atoms from their equilibrium positions for which a strain gradient of 1% over 6µm or larger is of the form ~u = (0, 0, 2gxy). This can be possibly re- desirable. Such strain gradients have been realized ex- alized by pulverizing GaAs on a substrate in MBE at a perimentally, however, not exactly in the configuration rate which is a function of the position of the pulveriz- proposed here [14, 15]. The strength of the confining po- 15 ing beam on the substrate. The GaAs pulverization rate tential is in this case D = 10− N/m which corresponds should vary as xy ∼ r2 sin(2φ), where r is the distance roughly to an electric field of 1V/m for a sample of 60µm. from one of the corners of the sample where the GaAs In systems with higher spin-orbit coupling, C3 would be depositing was started. Conversely, we can keep the pul- larger, and the strain gradient field would create a larger verizing beam fixed at some r and rotate the sample with gap between the Landau levels. an angle-dependent angular velocity of the form sin(2φ). We now turn to the question of the many-body wave We then move to the next incremental distance r, in- function in the presence of interactions. For our system crease the beam rate as r2 and again start rotating the this is very suggestive, as the wave function incorporates substrate as the depositing procedure is underway. both holomorphic and anti-holomorphic coordinates, by The strain architecture we have proposed to realize the contrast to the pure holomorphic Laughlin states. Let Quantum Spin Hall effect is by no means unique. In the the up-spin coordinates be zi while the down-spin coordi- 4 nates be the wi. zi enter in holomorphic form in the wave and the ground state wave function is Laughlin-type. In function whereas wi enter anti-holomorphically. While if contrast to our case, the theory is still T breaking, the the spin-up and spin-down electrons would lie in sepa- magnetic field is replaced by a rotation axial vector, and rate bi-layers the many-body wave function would be just the lowest Landau level is chiral. 1 ⋆ ⋆ m ⋆ ⋆ m ( zizi + wkwk) (zi − zj) (w − w ) e− 2 i k , i