Spin-Hall effect in semiconductor heterostructures with cubic Rashba spin-orbit interaction

O. Bleibaum; S. Wachsmuth

REVIEW

Spin-Hall effect in semiconductor heterostructures with cubic Rashba spin-orbit interaction

O. Bleibaum, S. Wachsmuth

Disordered Systems and Neural Networks · Submitted 22 Feb 2006 · 46 references

Pith's verdict: A serious analytical derivation for the cubic Rashba model whose bulk spin physics is plausible, but the headline edge-accumulation formula depends on an unproven field-distortion term and an acknowledged boundary ambiguity.

T0 · AI review

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arXiv IDcond-mat/0602517 (version 1)

Pith ID6WLEDEV2, Pith's permanent record of this paper and its review

CitationO. Bleibaum, S. Wachsmuth. "Spin-Hall effect in semiconductor heterostructures with cubic Rashba spin-orbit interaction." arXiv:cond-mat/0602517 (2006). https://pith.science/paper/6WLEDEV2

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Claim paper

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Editor's note

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Read this one if you care about how spin-current tensors actually couple to diffusion equations in hole gases. The paper derives the diffusive transport theory for the cubic Rashba model and makes two genuinely new moves: it shows that in zero field the spin and charge diffusion equations decouple completely (Eqs. 7-8), and it writes down an explicit non-homogeneous spin-current tensor (Eqs. 17-18) that reduces to the Schliemann-Loss bulk spin-Hall current in the homogeneous limit. That last check is important and they pass it. The Hall-like current in Eq. (43) is also new, and it is a concrete experimental target.

The calculations use the standard ladder / hydrodynamic expansion, and the comparison with the linear Rashba model is instructive. The authors are unusually honest about the ambiguity in reconstructing the current from a divergence: Section III.B and footnote 45 concede that curl-type terms cannot be retrieved, and that only in the presence of an electric field do some of them become visible.

Now the soft spots, in proportion. The big one is the boundary condition behind the edge accumulation formula Eq. (42). They justify J_yz=0 by identifying the spin-current tensor with the tensor of transport currents. That identification hinges on Eq. (31), the field-induced correction to the Green's function, which is stated but not derived. The stress-test note is right that even if Eq. (31) is correct, terms of the form (e_z × ∇)_i(ω(µ_r) S_γ) are explicitly left unretrievable, and those can contribute a normal component at a hard wall. So Eq. (42) is a conditional prediction, not an established one. This is not a fatal flaw in the bulk physics, but it is load-bearing for the headline result.

Two smaller gaps: the Kubo-formalism check of zero dc spin accumulation is mentioned but not shown, and the high-momentum regularization of the cubic dispersion is described in one sentence with no details. Both are addressable. The self-citation pattern is heavy, but the cited works do contain the formalism they rely on, and the new results are distinct.

If I were refereeing this, I would send it out with a request to derive Eq. (31) and to close or explicitly bound the boundary ambiguity. It deserves serious review, not desk rejection. The bulk results are likely correct and useful; the edge accumulation needs a fix.

Referee report

Significance

If the identification of J_ik with the transport current is correct, this is a valuable analytical result: it provides the first explicit connection between the spin-Hall current and the edge spin accumulation in the diffusive cubic Rashba model, and it sharply contrasts with the linear Rashba model, where the operator-defined spin current is known not to be a transport quantity. The paper also makes concrete, falsifiable predictions: the penetration depth of the edge accumulation is independent of disorder, while the magnitude decreases with disorder, and the edge accumulation grows with Fermi energy if the stated scaling of Omega is used. The derivation is systematic within the standard ladder/hydrodynamic framework, and the homogeneous spin-Hall current is checked against the independent result of Schliemann and Loss (Ref. [35]). The main weaknesses are that the key field-distortion expression (31) is asserted rather than derived, the curl-term ambiguity is not fully resolved, and there is an internal contradiction in the energy dependence of Omega between Eq. (9) and the discussion in Section IV.B.

Major objections (3)

[Sec. IV.A, Eq. (31)] The field-induced correction to the Green's function, Eq. (31), is stated without derivation. This expression is load-bearing: it is the stated reason why the spin-charge coupling term in Eq. (18) 'arises naturally' in the presence of an electric field, which underpins the claim that J_ik is the transport current in Eq. (32). The reference to Ref. [26] for a linear-Rashba analogue is not sufficient, because the cubic expression differs in spin structure (sigma_z rather than sigma x k), in the power of the momentum factors (p_+^3 p_-^2), and in the presence of (g_+ g_-)^2. An error in its prefactor or angular structure would propagate directly into the boundary condition (41) and the central prediction (42). Please provide the derivation or a complete reference to it.
[Sec. III.B, Eq. (23) and footnote 45] The identification of J_ik with the tensor of transport currents is not rigorously established. The authors correctly note that the diffusion equation only fixes the divergence of the current, so terms proportional to a curl are ambiguous, and footnote 45 explicitly concedes that the equation could contain a (curl T)_k term. They further state that terms of the form (e_z x nabla)_i (omega(mu_r) S_gamma) cannot be retrieved even after switching on a field, but they do not prove that such terms are absent. Since the boundary condition (41) and the edge spin accumulation (42) rely on J_yz being exactly the transport current, this is a load-bearing gap. In the specific half-plane, translation invariance may suppress some such terms, but the paper does not make or demonstrate that argument.
[Eq. (9) and Sec. IV.B] There is a direct contradiction between Eq. (9) and the scaling statement in Section IV.B. Eq. (9) gives Omega = 16 N^2 m^3 / (hbar^2 E^3 tau), which implies Omega(epsilon_F) scales as epsilon_F^{-3}. However, the text after Eq. (42) states that 'the magnitude of the edge spin-accumulation increases with increasing epsilon_F since Omega(epsilon_F) proportional to epsilon_F^3', which is the opposite scaling. This contradiction affects the physical interpretation of the central result (42) and should be resolved either by correcting Eq. (9) or by revising the scaling statement after a re-derivation.

Minor objections (6)

[Throughout] There are several typographical errors: 'expection' should be 'expectation' in the Introduction and Section III.B; 'cu bic' in the abstract; and Eq. (33) is missing an equals sign ('div j(r|E,t),' should read 'div j(r|E,t) = 0').
[Eq. (42)] The typesetting of the square root in Eq. (42) is corrupted ('/radicaltp/radicalvertex/radicalvertex√'); it should be written as a conventional square root, e.g., (3/2) sqrt(Omega/(2D)) ... .
[References] Reference [27] is cited as '35322', which likely should be '035322' in the standard Phys. Rev. B format; please check the reference list for similar formatting errors.
[Introduction] The sentence 'a paper has recently been published, in which the spin-Hall effect has been investigated numerically in the cubic Rashba model 9' appears twice in the Introduction, nearly verbatim; one occurrence should be removed.
[Sec. III.A] The claim that the zero bulk spin-charge coupling was 'checked by calculating the spin accumulation in the Kubo-formalism' is not substantiated, since the calculation is not shown. If this check is important, it should be presented or described; otherwise the sentence should be removed or softened.
[Sec. IV.B] In the comparison with nearly clean systems, the phrase 'There both the magnitude of the edge spin-accumulation and the penetration depth decrease with increasing particle number 9' is unclear because 'There' refers to the nearly clean case but the preceding discussion is about the authors' own regime; please rephrase for clarity.

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The paper

The paper claims

In a cubic Rashba hole gas, the spin-current tensor is the transport current that sets boundary conditions.

The reading

This paper asks whether the standard symmetrized spin-current tensor used in spin-Hall theory is the quantity that actually enters the spin diffusion equation and fixes boundary conditions in a two-dimensional heavy-hole gas with cubic Rashba spin-orbit interaction. In the diffusive regime with weak spin-orbit coupling, the authors derive particle and spin diffusion equations from Green's functions and show that the spin-current tensor coincides with the transport current once an electric field is present. On that basis they obtain an explicit edge spin accumulation and a new Hall-like contribution to the electric current. If the identification is right, the much-debated spin-current tensor is a physical transport quantity in this model, not an artifact.

What carries the argument

The load-bearing object is the spin-current tensor $J_{ik} = \langle \frac{1}{2}\{\sigma_i, \hat v_k\} \rangle$, evaluated from retarded and advanced Green's functions in the ladder approximation and reduced by the hydrodynamic expansion (small wave vector, small frequency, weak spin-orbit coupling). Its identification with the tensor of transport currents in the diffusion equation is what turns a bulk current into the boundary condition $J_{yz}=0$ at a hard wall. The mechanism that makes this identification work is the field-distorted Green's function of Eq. (31), proportional to $\sigma_z N^2 (g_+ g_-)^2 (p_-^3 p_+^2 F_+ - p_+^3 p_-^2 F_-)$, which produces exactly the spin-charge coupling terms in the diffusion equations that cannot be recovered from the equilibrium curl-free expression.

Extended reading notes

Core claim

On the paper's own terms, the central claim is that, in a two-dimensional heavy-hole gas with weak cubic Rashba spin-orbit interaction in the diffusive regime, the symmetrized spin-current tensor $J_{ik}$ with operator $\hat j_{ik} = \frac{1}{2}\{\sigma_i, \hat v_k\}$ is the transport current appearing in the spin diffusion equation $\partial_t S_k + \Omega_{ki} S_i + \nabla_i J_{ik} = 0$, and therefore it sets the boundary conditions. In equilibrium the diffusion equations contain no spin-charge coupling, because angular averages of the cubic interaction vanish, and the identity $\nabla_i J_{ik} = -D\Delta S_k$ lets the spin diffusion equation be written with this tensor. An applied electric field distorts the Green's function (Eq. (31)) and brings spin-charge coupling terms explicitly into the diffusion equation, which the authors take as resolving the ambiguity of curl-like terms. The consequences are the edge spin accumulation $S_z(y) = -\frac{3}{2}\sqrt{\Omega/(2D)}\,\frac{\hbar}{m}\tau n_0 F_x e^{-\sqrt{2\Omega/D}\,y}$ for a half-plane with a hard wall at $y=0$, and a Hall-like current $\delta j = -\frac{3}{2}\frac{\hbar\tau}{m} e_z \times F\, S_z\, d\Omega/d\epsilon_F$ induced by an optically oriented perpendicular magnetization.

Load-bearing premise

The entire edge-accumulation formula rests on the claim that the symmetrized spin-current tensor is exactly the transport current that must vanish at a hard wall; the paper grants that curl-like terms cannot be recovered from the diffusion equation alone, and the field argument used to remove that ambiguity depends on a perturbative Green's function (Eq. (31)) whose derivation is not shown.

Editorial extensions

If this is right

In a half-plane sample with the field along the edge, the spin accumulation decays into the bulk as $e^{-\sqrt{2\Omega/D}\,y}$, so its penetration depth depends only on the ratio $\Omega/D$ and is independent of disorder strength.
The magnitude of the predicted edge accumulation grows with Fermi energy (since $\Omega \propto \epsilon_F^3$) and decreases with disorder through the prefactor $\tau$, the opposite of the behavior reported for nearly clean strong-coupling systems.
A homogeneous out-of-plane spin polarization created by optical orientation generates a Hall-like electric current $\delta j = -\frac{3}{2}\frac{\hbar\tau}{m} e_z \times F S_z\, d\Omega/d\epsilon_F$, a signal that is quadratic in the Rashba coupling strength $N$.
Without an applied field, the particle and spin diffusion equations decouple in the cubic Rashba model: there is no spin-galvanic current and no field-free spin accumulation, in contrast to the linear Rashba model.
Because the spin-current tensor is the transport current, spin-charge coupling enters through boundary conditions at the sample edge even though the bulk diffusion equations look decoupled.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A numerical test on a finite strip, comparing Kubo-formula spin densities with the diffusion solution imposing $J_{yz}=0$ at the boundary, would settle whether the transport-current identification is exact or only an approximation in the weak-coupling regime.
The Hall-like current of Eq. (43) suggests an experiment: prepare a known out-of-plane spin polarization by optical orientation in a heavy-hole quantum well and measure the transverse voltage as a function of Fermi energy; its size tracking $d\Omega/d\epsilon_F$ would confirm the mechanism.
The contrast with the linear Rashba model hints that the linear model's pure-gauge character is what cancels physical spin currents, and a diffusion-equation treatment of intermediate or quartic momentum dispersion could reveal whether spin-charge coupling reappears continuously or only for certain powers of momentum.

Original abstract

We study the spin-Hall effect in systems with weak cubic Rashba spin-orbit interaction. To this end we derive particle and spin diffusion equations and explicit expressions for the spin-current tensor in the diffusive regime. We discuss the impact of electric fields on the Green's functions and the diffusion equations and establish a relationship between the spin-current tensor and the spin diffusion equations for the model under scrutiny. We use our results to calculate the edge spin-accumulation in a spin-Hall experiment and show that there is also a new Hall-like contribution to the electric current in such systems.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,593 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the spin-Hall effect in systems with weak cubic Rashba spin-orbit interaction. To this end we derive particle and spin diffusion equations and explicit expressions for the spin-current tensor in the diffusive regime. We discuss the impact of electric fields on the Green's functions and the diffusion equations and establish a relationship between the spin-current tensor and the spin diffusion equations for the model under scrutiny. We use our results to calculate the edge spin-accumulation in a spin-Hall experiment and show that there is also a new Hall-like contribution to the electric current in such systems.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the spin-Hall effect in systems with weak cubic Rashba spin-orbit interaction.	contribution 0.87	0.89	0.05	asserted 0.62
s1	To this end we derive particle and spin diffusion equations and explicit expressions for the spin-current tensor in the diffusive regime.	method 0.93	0.95	0.05	asserted 0.95
s2	We discuss the impact of electric fields on the Green's functions and the diffusion equations and establish a relationship between the spin-current tensor and the spin diffusion equations for the model under scrutiny.	method 0.72	0.94	0.07	asserted 1.00
s3	We use our results to calculate the edge spin-accumulation in a spin-Hall experiment and show that there is also a new Hall-like contribution to the electric current in such systems.	result 0.99	0.95	0.69	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (46)

[35]

J. Schliemann and D. Loss, Phys. Rev. B 71 , 85308 (2005)

View on Pith 2005
[26]

O. Bleibaum, Phys. Rev. B 71 , 195329 (2005)

View on Pith 2005
[1]

J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92 , 126603 (2004)

View on Pith 2004
[2]

J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Phys. Rev. Lett. 94 , 047204 (2005)

View on Pith 2005
[3]

B. K. Nikoli \'c , L. P. Z \^a bo, and S. Souma, Phys. Rev. B 72 , 075361 (2005), Phys. Rev. B 73 , 075303 (2006)

View on Pith 2005
[4]

B. K. Nikoli \'c and S. Souma, Phys. Rev. B 71 , 195328 (2005)

View on Pith 2005
[5]

J. Schliemann and D. Loss, Phys. Rev. B 69 , 165315 (2004)

View on Pith 2004
[6]

L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 94 , 016602 (2005)

View on Pith 2005
[7]

S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301 , 1348 (2003) (see also the supporting online material)

View on Pith 2003
[8]

B. A. Bernevig, and S. C. Zhang , Phys. Rev. Lett. 95 , 016801 (2005)

View on Pith 2005
[9]

K. Nomura, J. Wunderlich, J. Sinova, B. Kaestner, A. H. MacDonald, and T. Jungwirth, Phys. Rev. B 72 , 245330 (2005)

View on Pith 2005
[10]

K. Nomura, J. Sinova, N. A. Sinitsyn, and A. H. MacDonald, Phys. Rev. B 72 , 165316 (2005)

View on Pith 2005
[11]

M. W. Wu and J. Zhou, Phys. Rev. B 72 , 115333 (2005)

View on Pith 2005
[12]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 93 , 176601 (2004)

View on Pith 2004
[13]

A. Yu. Silov, P. A. Blajov, J. H. Woller, R. Hey, K. H. Ploog, and N. S. Averkiev, Appl. Phys. Lett. 85 , 5929 (2004)

View on Pith 2004
[14]

S. D. Ganichev, S. N. Danilov, P. Schneider, V. V. Bel'kov, L. E. Golub, W. Wegschneider, D. Weiss, and W. Prettl, cond-mat/0403641 (unpublished)

View on Pith Pith review arXiv
[15]

V. M. Edelstein, Solid State Commun. 73 , 233 (1990)

View on Pith 1990
[16]

J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 67 , 033104 (2003)

View on Pith 2003
[17]

D. Culcer, Y. Yao, A. H. MacDonald, and Q. Niu, Phys. Rev. B 72 , 045215 (2005)

View on Pith 2005
[18]

P. Kleinert, V. V. Bryksin and O. Bleibaum, Phys. Rev. B 72 , 195311 (2005)

View on Pith 2005
[19]

R. Raimondi, P. Schwab, Phys. Rev. B 71 , 33311 (2005)

View on Pith 2005
[20]

O. Chalaev and D. Loss, Phys. Rev. B 71 , 245318 (2005)

View on Pith 2005
[21]

S. Murakami, Phys. Rev. B 69 , 241202(R) (2004)

View on Pith 2004
[22]

E. I. Rashba, Phys. Rev. B 68 , 241315(R) (2003), J. Supercond. 18 , 137 (2005)

View on Pith 2003
[23]

S. Zhang and Z. Yang, Phys. Rev. Lett. 94 , 066602 (2005)

View on Pith 2005
[24]

A. A. Burkov, A. S. Nunez and A. H. MacDonald, Phys. Rev. B 70 , 155308 (2004)

View on Pith 2004
[25]

E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93 , 226602 (2004)

View on Pith 2004
[27]

O. Bleibaum, Phys. Rev. B 73 , 35322 (2006)

View on Pith 2006
[28]

I. Adagideli, G. E. W. Bauer, Phys. Rev. Lett. 95 , 256602 (2005)

View on Pith 2005
[29]

O. Bleibaum, Phys. Rev. B 72 , 75366 (2005)

View on Pith 2005
[30]

T. Damker, H. B\"ottger and V. V. Bryksin, Phys. Rev. B 69 205327 (2004)

View on Pith 2004
[31]

O. Bleibaum, cond-mat/0503471 (unpublished) (2005)

View on Pith Pith review arXiv 2005
[32]

P. Zhang, J. Shi, Si Xiao, and Q. Niu, cond-mat/0503505v4 (2005)

View on Pith Pith review arXiv 2005
[33]

N. Sugimoto, S. Onoda, S. Murakami and N. Nagaosa, cond-mat/0503475v3 (2006)

View on Pith Pith review arXiv 2006
[34]

P. L. Krotkov, and S. Das Sarma, cond-mat/0510114v1 (2005)

View on Pith Pith review arXiv 2005
[36]

M. G. Pala, M. Governale, J. K\"onig, U. Z\"ulicke, G. Iannaccone, Phys. Rev. B 69 , 045304 (2004)

View on Pith 2004
[37]

M. G. Pala, M. Governale, J. K\"onig, U. Z\"ulicke, Europhys. Lett. 65 , 850 (2004)

View on Pith 2004
[38]

R. Winkler, Phys. Rev. B 62 , R4245 (2000)

View on Pith 2000
[39]

R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B 65 , 155303 (2002)

View on Pith 2002
[40]

O. Bleibaum, Phys. Rev. B 69 , 205202 (2004)

View on Pith 2004
[41]

M. Q. Weng, M. W. Wu, and L. Jiang, Phys. Rev. B 69 , 245320 (2004)

View on Pith 2004
[42]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 87 , 22503 (2005)

View on Pith 2005
[43]

P. Kleinert and V. V. Bryksin, Phys. Rev. B (in print)

View on Pith
[44]

L. D. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik VI Hydrodynamik (Akademie-Verlag, Berlin 1991)

View on Pith 1991
[45]

Note that, since S is a vector we could as well expect that the diffusion equation can be written in the form \[ _tS_k+ _ ki S_i+ _i J _ ik +( curl T)_k =0,\] where T is a vector field, which depends on S_ . The choice of the structure ( SC9 ) is finally only based on the observation, that hydrodynamic equations can usually be written in the form ( SC9 ) ...

View on Pith
[46]

P. Zhang and Q. Niu, cond-mat/0406436v1 (2004)

View on Pith Pith review arXiv 2004

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [35]

J. Schliemann and D. Loss, Phys. Rev. B 71 , 85308 (2005)

View on Pith 2005

[2] [26]

O. Bleibaum, Phys. Rev. B 71 , 195329 (2005)

View on Pith 2005

[3] [1]

J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92 , 126603 (2004)

View on Pith 2004

[4] [2]

J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Phys. Rev. Lett. 94 , 047204 (2005)

View on Pith 2005

[5] [3]

B. K. Nikoli \'c , L. P. Z \^a bo, and S. Souma, Phys. Rev. B 72 , 075361 (2005), Phys. Rev. B 73 , 075303 (2006)

View on Pith 2005

[6] [4]

B. K. Nikoli \'c and S. Souma, Phys. Rev. B 71 , 195328 (2005)

View on Pith 2005

[7] [5]

J. Schliemann and D. Loss, Phys. Rev. B 69 , 165315 (2004)

View on Pith 2004

[8] [6]

L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 94 , 016602 (2005)

View on Pith 2005

[9] [7]

S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301 , 1348 (2003) (see also the supporting online material)

View on Pith 2003

[10] [8]

B. A. Bernevig, and S. C. Zhang , Phys. Rev. Lett. 95 , 016801 (2005)

View on Pith 2005

[11] [9]

K. Nomura, J. Wunderlich, J. Sinova, B. Kaestner, A. H. MacDonald, and T. Jungwirth, Phys. Rev. B 72 , 245330 (2005)

View on Pith 2005

[12] [10]

K. Nomura, J. Sinova, N. A. Sinitsyn, and A. H. MacDonald, Phys. Rev. B 72 , 165316 (2005)

View on Pith 2005

[13] [11]

M. W. Wu and J. Zhou, Phys. Rev. B 72 , 115333 (2005)

View on Pith 2005

[14] [12]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 93 , 176601 (2004)

View on Pith 2004

[15] [13]

A. Yu. Silov, P. A. Blajov, J. H. Woller, R. Hey, K. H. Ploog, and N. S. Averkiev, Appl. Phys. Lett. 85 , 5929 (2004)

View on Pith 2004

[16] [14]

S. D. Ganichev, S. N. Danilov, P. Schneider, V. V. Bel'kov, L. E. Golub, W. Wegschneider, D. Weiss, and W. Prettl, cond-mat/0403641 (unpublished)

View on Pith Pith review arXiv

[17] [15]

V. M. Edelstein, Solid State Commun. 73 , 233 (1990)

View on Pith 1990

[18] [16]

J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 67 , 033104 (2003)

View on Pith 2003

[19] [17]

D. Culcer, Y. Yao, A. H. MacDonald, and Q. Niu, Phys. Rev. B 72 , 045215 (2005)

View on Pith 2005

[20] [18]

P. Kleinert, V. V. Bryksin and O. Bleibaum, Phys. Rev. B 72 , 195311 (2005)

View on Pith 2005

[21] [19]

R. Raimondi, P. Schwab, Phys. Rev. B 71 , 33311 (2005)

View on Pith 2005

[22] [20]

O. Chalaev and D. Loss, Phys. Rev. B 71 , 245318 (2005)

View on Pith 2005

[23] [21]

S. Murakami, Phys. Rev. B 69 , 241202(R) (2004)

View on Pith 2004

[24] [22]

E. I. Rashba, Phys. Rev. B 68 , 241315(R) (2003), J. Supercond. 18 , 137 (2005)

View on Pith 2003

[25] [23]

S. Zhang and Z. Yang, Phys. Rev. Lett. 94 , 066602 (2005)

View on Pith 2005

[26] [24]

A. A. Burkov, A. S. Nunez and A. H. MacDonald, Phys. Rev. B 70 , 155308 (2004)

View on Pith 2004

[27] [25]

E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93 , 226602 (2004)

View on Pith 2004

[28] [27]

O. Bleibaum, Phys. Rev. B 73 , 35322 (2006)

View on Pith 2006

[29] [28]

I. Adagideli, G. E. W. Bauer, Phys. Rev. Lett. 95 , 256602 (2005)

View on Pith 2005

[30] [29]

O. Bleibaum, Phys. Rev. B 72 , 75366 (2005)

View on Pith 2005

[31] [30]

T. Damker, H. B\"ottger and V. V. Bryksin, Phys. Rev. B 69 205327 (2004)

View on Pith 2004

[32] [31]

O. Bleibaum, cond-mat/0503471 (unpublished) (2005)

View on Pith Pith review arXiv 2005

[33] [32]

P. Zhang, J. Shi, Si Xiao, and Q. Niu, cond-mat/0503505v4 (2005)

View on Pith Pith review arXiv 2005

[34] [33]

N. Sugimoto, S. Onoda, S. Murakami and N. Nagaosa, cond-mat/0503475v3 (2006)

View on Pith Pith review arXiv 2006

[35] [34]

P. L. Krotkov, and S. Das Sarma, cond-mat/0510114v1 (2005)

View on Pith Pith review arXiv 2005

[36] [36]

M. G. Pala, M. Governale, J. K\"onig, U. Z\"ulicke, G. Iannaccone, Phys. Rev. B 69 , 045304 (2004)

View on Pith 2004

[37] [37]

M. G. Pala, M. Governale, J. K\"onig, U. Z\"ulicke, Europhys. Lett. 65 , 850 (2004)

View on Pith 2004

[38] [38]

R. Winkler, Phys. Rev. B 62 , R4245 (2000)

View on Pith 2000

[39] [39]

R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B 65 , 155303 (2002)

View on Pith 2002

[40] [40]

O. Bleibaum, Phys. Rev. B 69 , 205202 (2004)

View on Pith 2004

[41] [41]

M. Q. Weng, M. W. Wu, and L. Jiang, Phys. Rev. B 69 , 245320 (2004)

View on Pith 2004

[42] [42]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Appl. Phys. Lett. 87 , 22503 (2005)

View on Pith 2005

[43] [43]

P. Kleinert and V. V. Bryksin, Phys. Rev. B (in print)

View on Pith

[44] [44]

L. D. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik VI Hydrodynamik (Akademie-Verlag, Berlin 1991)

View on Pith 1991

[45] [45]

Note that, since S is a vector we could as well expect that the diffusion equation can be written in the form \[ _tS_k+ _ ki S_i+ _i J _ ik +( curl T)_k =0,\] where T is a vector field, which depends on S_ . The choice of the structure ( SC9 ) is finally only based on the observation, that hydrodynamic equations can usually be written in the form ( SC9 ) ...

View on Pith

[46] [46]

P. Zhang and Q. Niu, cond-mat/0406436v1 (2004)

View on Pith Pith review arXiv 2004