## Abstract

Although microscopic laws of physics are invariant under the reversal of the arrow of time, the transport of energy and information in most devices is an irreversible process. It is this irreversibility that leads to intrinsic dissipations in electronic devices and limits the possibility of quantum computation. We theoretically predict that the electric field can induce a substantial amount of dissipationless quantum spin current at room temperature, in hole-doped semiconductors such as Si, Ge, and GaAs. On the basis of a generalization of the quantum Hall effect, the predicted effect leads to efficient spin injection without the need for metallic ferromagnets. Principles found here could enable quantum spintronic devices with integrated information processing and storage units, operating with low power consumption and performing reversible quantum computation.

Our work is driven by the confluence of the important technological goals of quantum spintronics (*1, 2*) with the quest of generalizing the quantum Hall effect (QHE) to higher dimensions. The QHE is a manifestation of quantum mechanics observable at macroscopic scales. In contrast to most transport coefficients in solid-state systems, which are determined by the elastic and the inelastic scattering rates, the Hall conductance σ_{H} in QHE is quantized and completely independent of any scattering rates in the system, where the transport equation is given by *j*_{α} = σ_{H}ϵ_{αβ} *E*_{β} [*j*_{α} and *E*_{β} (α,β = 1,2) are the charge current and the electric fields respectively, ϵ_{αβ} is the fully antisymmetric tensor in two dimensions]. Though dissipative transport coefficients are expressed in terms of states in the vicinity of the Fermi level, the nondissipative quantum Hall conductance is expressed in terms of equilibrium response of all states below the Fermi level. The topological origin of the QHE is revealed through the fact that the Hall conductance can also be expressed as the first Chern number of a *U*(1) gauge connection defined in momentum space (*3*). Recently, the QHE has been generalized to four spatial dimensions (*4*). In that case, an electric field *E*_{ν} induces an *SU*(2) spin current through the nondissipative transport equation , where is the t'Hooft tensor, explicitly given by and σ is a dissipationless transport coefficient. The quantum Hall response in that system is physically realized through the spin-orbit coupling in a time-reversal symmetric system. At the boundary of this four-dimensional quantum liquid, when both the electric field and the spin current are restricted to the three-dimensional sub-space, the dissipationless response is given by (1) This fundamental response equation shows that it is possible to induce a purely topological and dissipationless spin current by an electric field in the physical, three-dimensional space.

We consider a realization of this electric field–induced topological spin current in conventional hole-doped semiconductors. In a large class of semiconductors, including Si, Ge, GaAs, and InSb, the valence bands are fourfold degenerate at the Γ point (Fig. 1). The effective Luttinger Hamiltonian (*5*) for holes is given by (2) where *S _{i}* is the spin-3/2 matrix. We take the hole picture, and reverse the sign of the energy. Good quantum numbers for this Hamiltonian are the helicity and the total angular momentum . This kinetic Hamiltonian is diagonalized in the basis where the helicity operator λ is diagonal and the eigenvalue is given by For a given wave vector

**k**, the Hamiltonian (Eq. 2) has two eigenvalues, and forming Kramers doublets. They are referred to as the light-hole (LH) and heavy-hole (HH) bands. In semiconductors with zincblende structure, such as GaAs, inversion symmetry breaking causes an additional tiny splitting in the LH and HH bands. We can neglect the inversion symmetry breaking when the temperature is much higher than this splitting. The band structure of semiconductors deviates from the spherical to the cubic symmetry. We also neglect this effect for simplicity, because physics described below are not so much affected by it.

We shall consider the effect of a uniform electric field **E**. Our full Hamiltonian is thus given by *H* = *H*_{0} + *V*(**x**), where *V*(**x**) = *e***E**·**x**, and –*e* is the charge of an electron. We assume that the split-off band is totally occupied. We first define a 4 by 4 unitary matrix *U*(**k**) which diagonalizes the kinetic Hamiltonian *H*_{0}. *U*(**k**) is defined by *U*(**k**)(**k**·**S**) *U*^{†}(**k**) = *k S*_{z}. In the spherical coordinates where **k** = *k*(sinθcosϕ, sinθsinϕ, cosθ), *U*(**k**) can be expressed as *U*(**k**) = exp(*i*θ*S _{y}*) exp(

*i*ϕ

*S*). Under this unitary transformation, the new Hamiltonian H̃ ≡

_{z}*U*(

**k**)

*HU*

^{†}(

**k**) becomes Eigenvalues of

*S*physically describe the helicity in the original basis. The kinetic part

_{z}*H*

_{0}now becomes diagonal, in the representation where

*S*is diagonal. Because

_{z}**x**=

*i*δ

**, the potential term becomes**

_{k}*V*(

**D̃**), where the covariant derivative

**D̃**is defined by

**D̃**=

*i*δ

**–**

_{k}**Ã**and

**Ã**= –

*i U*(

**k**)δ

_{k}*U*

^{†}(

**k**). Because

**Ã**is a pure gauge potential, there is no curvature associated with it. Up to this point, the transformation is exact. We now consider adiabatic transport and make a corresponding approximation. As is usually assumed in the transport theory, we neglect the interband transitions, i.e., the off–block-diagonal matrix elements of

**Ã**connecting the LH and HH bands. Then we arrive at a nontrivial adiabatic gauge connection

**A**[supporting online material (SOM) text], which takes a block-diagonal form in the LH and HH subspace. Because each band is twofold degenerate, the gauge connection is, in general, non-Abelian. However,

**A**has no matrix elements connecting the λ = 3/2 and λ = –3/2 states in the HH band, because the gauge field

**Ã only**connects states with helicity difference Δλ = 0, ±1. Therefore, the non-Abelian structure is only present in the LH band. For simplicity of presentation, we shall first make an additional, Abelian approximation (AA), in which only the diagonal components in

**A**are retained. Afterward, we shall give our final results, including fully the non-Abelian corrections.

Within the AA, **A** is a diagonal 4 by 4 matrix in the helicity basis. Because a band-touching point acts as a Dirac magnetic monopole in momentum space (*6*), each diagonal component of **A**(**k**) is given by that of a Dirac monopole at **k** = 0, with the monopole strength *eg* given by λ. The associated field strength is given by (4) The effective Hamiltonian takes the form (5) Henceforth, *x _{i}* denotes a covariant derivative in momentum space:

*x*=

_{i}*D*=

_{i}*i*δ/δ

*k*–

_{i}*A*(

_{i}**k**). The definition of

*x*has changed by projecting the original Hamiltonian

_{i}*H*onto the HH or LH band. Whereas

*H*

^{eff}seems to be trivial, its nontrivial dynamics are revealed through the nontrivial commutation relations (6) Such a situation also happens in the Gutzwiller projection of the SO(5) model (

*7*). It also resembles the nontrivial commutation relation between the position operators of a two-dimensional electron gas projected onto the lowest Landau level (

*8*), where

*F*=

_{ij}*B*ϵ

*, and*

_{ij}*B*is the external magnetic field. This general algebraic structure, called “noncommutative geometry,” also underlies the four-dimensional QHE model (

*4*). In our present context, the non-commutativity between the three-dimensional coordinates arises from the magnetic monopole in momentum space, and it is a natural generalization of the QHE to three dimensions.

The equation of motion for holes can be derived easily from Eqs. 5 and 6 as (7) The last term, proportional to *F _{ij}*, is a topological term, describing the effect of the magnetic monopole on the orbital motion. It represents a “Lorentz force” in momentum space, making the hole velocity noncollinear with its momentum, in contrast to the usual situations. In fact, if we interchange the roles of

*x*and

*k*in this term, this term becomes the Lorentz force for a charged particle moving in the presence of a magnetic monopole in real space. This set of equations can be integrated analytically (SOM text), and the resulting trajectory is shown in Fig. 2. The hole motion in real space obtains a shift perpendicular to

**S**. This shift is analogous to the deflection of a charged particle by a magnetic monopole in a direction perpendicular to the plane spanned by its position and velocity vectors (

*9*). It causes a spin current perpendicular to both

**E**and

**S**. For example, for

**E**parallel to the +

*z*direction, the spin current for each band at zero temperature, with spin parallel to the

*x*axis, flowing to the

*y*direction is given by (8) which is obtained by summing contributions from all the filled states. Here, we assumed that the equilibrium momentum distribution is attained by the random impurity scattering that causes the charge relaxation. Eq. 7 describes only the ballistic motion, and scattering by random impurities would lead to additional contributions to the spin current. As one can see from the detailed discussions in the SOM text, these extrinsic effects are not only small, but also scale with a higher power of

*k*∼

_{F}*n*

^{1/3}, where

*n*is the hole density. Therefore, by plotting σ

*/*

_{s}*n*

^{1/3}against

*n*and extrapolating to the limit of

*n*→ 0, the constant intercept would uniquely determine our predicted dissipationless spin conductivity.

It is worth noting that this AA becomes exact in zero-gap semiconductors, e.g., α-Sn. In this class of materials, the bottom of the conduction band and the top of the valence band correspond to the LH and HH bands in other semiconductors like GaAs. These two bands touch at **k** = 0. In this case, p-doping introduces holes only into the HH band, and the AA becomes exact.

The spin current induced by the electric field can also be understood in terms of the conservation of the total angular momentum . As remarked earlier, **J** commutes with *H*_{0}. When **E** is parallel to the *z* direction, *J _{z}* also commutes with the potential. Therefore, substituting , we obtain
(9)
The second term, representing the torque, vanishes in our case because

**k̇**points along the

*z*direction. The first term vanishes in usual problems; however, it does not in our case, due to the noncollinearity of the velocity and the momentum. Furthermore, the first term, describing the time derivative of the orbital angular momentum , is proportional to the spin current. The third term , describing the time derivative of the spin angular momentum

**S**, can be easily evaluated from the acceleration equation in Eq. 7. Therefore, we see that the conservation of the total angular momentum in Eq. 9 directly implies the spin current in Eq. 8. The spin current flows in such a way that the change of

**L**exactly cancels the change of

**S**.

We now discuss the correction due to the non-Abelian nature of the gauge connection of the LH band. Remarkably, even though the gauge connection is non-Abelian, the associated field strength is Abelian and gives a correction factor of –3, compared with the AA (*10*–*12*). The equation of motion modified accordingly agrees with that obtained by generalizing the wave packet formalism (*13*) to the non-Abelian case. This non-Abelian correction gives the following result for the spin current: (10) Here, we defined σ* _{s}* to have the same dimension as the electrical conductivity, to facilitate comparison. The spin current equation is rotationally invariant, with the covariant form given in Eq. 1, and is the central result of our paper. In contrast with similar effects (

*14, 15*), this spin current has a topological character; the spin conductivity σ

_{s}in Eq. 1 is independent of the mean free path and relaxational rates, and all states below the Fermi energy contribute to the spin current, where each contribution is determined purely by the gauge curvature in momentum space, similar to the QHE (

*3*). Assuming the hole density

*n*= 10

^{19}cm

^{–}

^{3}, the mobility of the holes at room temperature in GaAs is μ = 50 cm

^{2}/V·s (

*16*), and the conductivity is σ =

*e n*μ = 80 Ω

^{–}

^{1}cm

^{–}

^{1}. On the contrary, the spin Hall conductivity σ

_{s}in Eq. 10 is estimated as σ

_{s}∼ 80 Ω

^{–}

^{1}cm

^{–}

^{1}, being of the same order with σ. For lower carrier concentration, σ

_{s}becomes larger than σ; for

*n*= 10

^{16}cm

^{–}

^{3}, we have σ = 0.6 Ω

^{–}

^{1}cm

^{–}

^{1}and σ

_{s}= 7 Ω

^{–}

^{1}cm

^{–}

^{1}. At finite temperature, Eq. 10 is modified only through the Fermi distribution function

*n*

^{λ}(

**k**). Because the typical energy difference between the LH and HH bands at the same wavenumber is about 0.1 eV, which largely exceeds the energy scale of the room temperature ∼0.025 eV, our predicted effect remains of the same order even at room temperature.

The nondissipative spin transport equation Eq. 1 does not violate the time-reversal symmetry *T*. Our microscopic Hamiltonian *H*, the electric field **E**, and the spin current are all *T* invariant. Therefore, the electric field and the spin current can be related by a *T* symmetric, dissipationless transport coefficient σ_{s}. This situation is to be contrasted with Ohm's law. As the charge current is odd under *T*, while the electric field is even, they can only be related by a *T* antisymmetric, dissipative transport coefficient, namely, the charge conductivity. One of the main objectives in quantum computing is to achieve reversible computation (*17, 18*). From the above analysis, we see that there is a fundamental difference between the ordinary irreversible electronics computation based on Ohm's law and the reversible spintronics computation based on Eq. 1. The time reversal symmetry property encoded in Eq. 1 could provide a fundamental principle for the reversible quantum computation.

This spin current is also useful for spin injection into semiconductors. Although effective spin injection is necessary for spintronic devices, it has been an elusive issue (*2*). Usage of ferromagnetic metals is not practical because most of the spin polarizations are lost at the interface due to conductivity mismatch between metal and semiconductor (*19, 20*). Spin injection from ferromagnetic semiconductors such as Ga_{1}_{–}* _{x}*Mn

*As has been successful (*

_{x}*21*–

*23*). Nevertheless,

*T*

_{c}is at most 110 K for Ga

_{1}

_{–}

*Mn*

_{x}*As, still too low for practical use at room temperature. Thus, it is desirable to find an effective method for spin injection. The electric field–induced spin current serves as a spin injector, because it creates a spin current inside the semiconductor. One might worry that the short relaxation time τ*

_{x}_{s}∼ 100 fs (

*24*) of the hole spins. This shortness of τ

_{s}is because the strong spin-orbit interaction in the valence band combines the relaxation of momenta and spins (

*25*). Most of the efforts on the spintronics in GaAs have been focused on electrons in the conduction bands, which have much longer spin-relaxation time (∼100 ps) (

*26*). Nevertheless, our spin current is free from such rapid relaxation of spins of holes, because it is a purely quantum mechamical effect with equilibrium spin-momentum distribution. Only when the spin-momentum distribution deviates from equilibrium, e.g., in spin accumulation at boundaries of the sample, does the rapid relaxation of hole spins become effective.

One can consider following experimental setups for detection of the constant spin supply from p-GaAs. When the electric field is applied along the *z* direction and the electric current *J _{z}* is induced, the

*s*spin current

^{x}*j*will flow along the

_{y}^{x}*y*direction. One possibility is to see the spin-dependent electric transport through a ferromagnetic electrode with the magnetization

**M**along ±

*x*direction attached to the positive

*y*side of the sample. With a lead connecting this electrode and the other (negative

*y*) side of p-GaAs as shown in Fig. 3A, one should see a change of the electric current

*I*depending on the direction of

**M**. The ratio of

*I*when

**M**is along ±

*x*-direction,

*I*(+

*x*)/

*I*(–

*x*), is expected to be well larger than unity. For the ferromagnetic electrode, ferromagnetic metals are not efficient because of the conductance mismatch (

*20*). Instead, ferromagnetic semiconductors will be suitable. Another possibility is to measure circular polarization of light emitted via recombination with electrons. This can be achieved by a similar experimental setup in (

*21*), replacing (Ga,Mn)As by p-GaAs, where the quantum well structure of (In,Ga)As is sandwiched by p-GaAs and n-GaAs (Fig. 3B). The spin current injected along the

*y*direction will be recombined with the electrons supplied from the n-GaAs in the (In,Ga)As quantum well.

When the system is not connected to the leads along the *y* direction, spins accumulate near the edges of the sample. This spin polarization can, in principle, be measured by the Kerr rotation. The spin distribution is determined by a balance between the spin current supply and the spin relaxation. At room temperature, because τ_{s} = 100 fs (*24*) is rather short, the area density of spin accumulation at the sample surface, *j ^{x}_{y}*τ

_{s}, is too small to be observed. However, there are several ways to make τ

_{s}longer. One is to lower the temperature. Another is to inject the spin into n-GaAs through the p-n junction, as demonstrated recently (

*27, 28*). The spin lifetime of electrons in GaAs is 100 ps, 10

^{3}times longer than that of holes. Thereby, the spin current can be detected by the Kerr rotation through surface reflection.