Topological spin Hall and spin Nernst effects in a bilayer graphene

# Topological spin Hall and spin Nernst effects in a bilayer graphene

A. Dyrdał, J. Barnaś Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznań
Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland
July 19, 2019
###### Abstract

We consider intrinsic contributions to the spin Hall and spin Nernst effects in a bilayer graphene. The relevant electronic spectrum is obtained from the tight binding Hamiltonian, which also includes the intrinsic spin-orbit interaction. The corresponding spin Hall and spin Nernst conductivities are compared with those obtained from effective Hamiltonians appropriate for states in the vicinity of the Fermi level of a neutral bilayer graphene. Both conductivities are determined within the linear response theory and Green function formalism. The influence of an external voltage between the two atomic sheets is also included. We found transition from the topological spin Hall insulator phase at low voltages to conventional insulator phase at larger voltages.

###### pacs:
73.43.-f, 72.25.Hg, 73.61.Wp

## I Introduction

Four decades ago Dyakonov and Perel showed that a system with strong spin-orbit interaction should reveal transverse spin current and spin accumulation in the presence of external longitudinal electric field dyakonov (); dyakonovlett () – even if the system is nonmagnetic. This effect, known now as the spin Hall effect (SHE) hirsch (), was studied extensively in the last few years hirsch (); murakami03 (); sinova (); kato (); kimura (); brune (); engel (), and is still of current interest – mainly because it offers a new possibility of spin manipulation with electric field only. The possibility of pure electrical manipulation of spin degrees of freedom is interesting not only from fundamental reasons, but also from the point of view of possible applications in future spintronics devices and information processing technologies wolf (); zutic (); sih (); awschalom2009 ().

The crucial interaction responsible for SHE, i.e. the spin-orbit coupling, may be either of intrinsic (internal) or extrinsic origin. The corresponding extrinsic SHE is associated with mechanisms of spin-orbit scattering on impurities and other defects (skew scattering and/or side jump), while the intrinsic SHE is a consequence of a nontrivial trajectory of charge carriers in the momentum space due to the spin-orbit contribution of a perfect crystal lattice to the corresponding band structure. The intrinsic SHE may be described in terms of the Berry phase formalism berry (); sundaram () and therefore it is also referred to as the topological SHE.

It is well known that various spin effects, like for instance spin current and spin accumulation, may be generated not only by external electric field, but also due to a temperature gradient. Indeed, there is a great interest currently in spin related thermoelectric effects. One of such phenomena is the spin Seebeck effect, where longitudinal spin current and spin voltage are generated by a temperature gradient uchida (). Of particular interest, however, are spin thermoelectric effects in systems with spin-orbit interaction, where a temperature gradient gives rise to transverse spin accumulation and/or spin currents. Thus, the temperature gradient in such systems may lead to anomalous (in case of ferromagnetic systems) and spin Nernst effects xiao2007 (); czhang2009 (); jaworski (), which correspond to the anomalous and spin Hall effects induced by longitudinal electric field. Similarity of the spin Hall and spin Nernst effects in nonmagnetic systems with spin-orbit interaction is presented in Fig.1, which clearly shows that the SHE is generated by external electric field, while the spin Nernst effect (SNE) is a similar effect generated by a temperature gradient instead of electric field (gradient of electrostatic potential).

In this paper we consider the topological contribution to the spin Hall and spin Nernst effects in a bilayer graphene. Graphene is a two-dimensional crystal of carbon atoms. A monolayer graphene in a free standing form was obtained few years ago and owing to its unusual and peculiar properties quickly became one of the most extensively studied materials Geim2007 (); katsnelson (); castro (). After the pioneering paper by Kane and Mele kane (), spin Hall effect in a monolayer graphene was studied in many papers and in various physical situations. Recently, bilayer graphene is extensively studied as a more appropriate for applications than a single-layer one. Moreover, it has been shown that spin-orbit coupling in a bilayer can be enhanced in comparison to that in a single-layer graphene guinea (). This motivated us to consider topological contributions to the SHE and SNE in a bilayer graphene.

Both SHE and SNE in (nonmagnetic) graphene are generated by spin-orbit interaction. In general, one can distinguish two different forms of the spin-orbit interaction having the crystal lattice periodicity and contributing to the relevant band structure – internal and Rashba spin-orbit interactions. The latter interaction is due to a substrate and can be controlled by an external gate voltage. In this paper we consider the contributions to SHE and SNE due to the intrinsic spin-orbit interaction only. It is known that this interaction opens an energy gap at the Dirac points. It was also shown that the energy gap can be tuned externally by applying a voltage bias between the layers castro2007 (); zhangtangwang (). The gate voltage dependence of the spin Hall conductivity, leading to phase transition between the spin Hall insulator and classical insulator will also be considered in this paper.

The description of transport properties of graphene is usually based on some effective Hamiltonian, which properly describes electronic spectrum near the Fermi level of a neutral system. However, it is well known that the topological contribution to the spin Hall effect includes contributions from electronic states far below the Fermi level, and therefore a more accurate electronic spectrum is required to describe the effect properly. Accordingly, in this paper we calculate the spin Hall and spin Nernst conductivities from a more realistic electronic spectrum based on a tight binding Hamiltonian, and compare them with the corresponding conductivities obtained on the basis of effective Hamiltonians. The results presented in this paper reveal, however, a very good agreement between the conductivities derived from the effective and tight binding Hamiltonians.

The paper is organized as follows. In section 2 we describe briefly the electronic states of a bilayer graphene within the tight binding Hamiltonian and also in terms of an effective Hamiltonian, which is sufficient for states in the vicinity of the Fermi level of a neutral graphene. In both cases the intrinsic spin-orbit interaction and the effect of a normal bias voltage are taken into account. In section 3 we calculate the spin Hall and spin Nernst conductivities for the tight binding Hamiltonian and compare them with those obtained from the effective Hamiltonian. In the latter case we derive analytical formulas for the spin Hall and spin Nernst conductivities. We also discuss the role of normal bias (in the framework of the effective model). Description based on a reduced low-energy effective Hamiltonian is presented in section 4. Summary and final conclusions are given in section 5.

## Ii Electronic spectrum of the bilayer graphene

A single-layer graphene is a monolayer of carbon atoms arranged in a two-dimensional honeycomb lattice which can be also considered as being composed of two nonequivalent triangular sublattices. In the absence of spin-orbit interaction the Fermi surface of a neutral single-layer graphene consists of two nonequivalent and points of the Brillouin zone, at which the valence and conduction bands touch each other. The corresponding electronic spectrum can be described by a tight binding Hamiltonian with nearest and next-nearest neighbor hopping terms. The low energy electron states near the points and can be well approximated by a conical energy spectrum (linear dispersion relations). As a result, charge carriers in the vicinity of the points and are described effectively by the relativistic Dirac equation.Geim2007 (); katsnelson () Intrinsic spin-orbit interaction opens then an energy gap at the Dirac points. kane (). The tight binding and effective Hamiltonians for a bilayer graphene are more complex, as described below.

### ii.1 Tight binding model

The bilayer graphene in the Bernal stacking (see eg [castro, ]) is described by the following tight binding Hamiltonian:

 H=∫d2kψ(k)†(HΓΓ†H)ψ(k), (1)

where

 H=(hsoSz+VS0h0S0h∗0S0−hsoSz−VS0), (2)
 Γ=(0γ1S000). (3)

The matrix elements and are defined as follows: and , where is the hopping integral between the nearest neighbors in the atomic sheets, is the next-nearest neighbor spin-orbit hopping amplitude, while with being the lattice parameter. Furthermore, is the voltage between the two atomic sheets of the bilayer (measured in energy units), denote the unit () and Pauli () matrices in the spin space, while describes coupling between the two atomic layers.

The corresponding energy eigenvalues for have then the following form:

 E1,2=∓[h2so+12(γ21+2h20−γ1√γ21+4h20)]1/2 (4)
 E3,4=∓[h2so+12(γ21+2h20+γ1√γ21+4h20)]1/2 (5)

This spectrum is shown in Fig.2(a). States near the point are shown by the solid lines in parts (b) and (c). The part (c) reveals a small energy gap created at the Dirac points by the spin-orbit interaction.

When , the inversion symmetry is broken (layers are no longer equivalent) and the degeneracy is lifted. The corresponding eigenvalues acquire then the form

 E1,2=∓[h20+h2so+V2+γ212 −12[(γ21−4Vhso)2+4(γ21+4V2)h20]1/2]1/2, (6)
 E1′,2′=∓[h20+h2so+V2+γ212 −12[(γ21+4Vhso)2+4(γ21+4V2)h20]1/2]1/2, (7)
 E3,4=∓[h20+h2so+V2+γ212 +12[(γ21−4Vhso)2+4(γ21+4V2)h20]1/2]1/2, (8)
 E3′,4′=∓[h20+h2so+V2+γ212 +12[(γ21+4Vhso)2+4(γ21+4V2)h20]1/2]1/2. (9)

The corresponding spectrum for the assumed value of is indistinguishable from the spectrum shown in Fig.2(a) for . The differences can be seen on a smaller energy scale, as in the parts (d) and (e) of Fig.2. When comparing Figs 2(b) and 2(d), one can notice a larger energy gap for . This is more clearly seen when comparing the parts (c) and (e). First, for the assumed value of the gap is wider than for . Second, the top and bottom band edges become split and shifted away from the Dirac points. As will be described later, the applied voltage between the atomic sheets first closes the gap and then opens a new one with the width increasing with .

### ii.2 Effective Hamiltonian

When only electronic states near the Fermi level (near the Dirac points) of a neutral graphene are relevant, one can make use of some effective Hamiltonians to describe the corresponding electronic spectrum. Such a Hamiltonian can be derived using the approximation. As a result, the effective Hamiltonian for states near the point of the bilayer graphene takes the form mccan (); prada ():

 HK=T0\varotimesHsK −γ12(Tx\varotimesσx\varotimesS0−Ty\varotimesσy\varotimesS0), (10)

where

 HsK=v(kxσx+kyσy)\varotimesS0 +Δsoσz\varotimesSz+Vσ0\varotimesS0. (11)

The first term on the right side corresponds to two decoupled atomic monolayers, each of them being described by the Kane Hamiltonian for a single layer graphene, . In turn, the second term describes coupling between the monolayers, with denoting respectively the unit matrix () and Pauli () matrices associated with the layer degree of freedom. In turn, () are the unit () and Pauli () matrices in the pseudo-spin (sublattice) space. Relations between the parameters of the tight binding and Kane models are: ( is the carrier velocity at the Fermi level) and .

The eigenvalues of Hamiltonian (II.2) for take the form

 E1,2=∓[k2v2+γ212+Δ2so−γ12√4k2v2+γ21]1/2 (12)

and

 E3,4=∓[k2v2+γ212+Δ2so+γ12√4k2v2+γ21]1/2. (13)

Electronic spectrum near the point K, described by the above formula, is shown in Fig.2(b,c), where it is compared with the spectrum obtained from the full tight binding Hamiltonian. Close to the K point (gap), spectra from both models coincide very well.

When , one finds

 E1,2=∓[v2k2+Δ2so+V2+γ212 −12[(γ21−4VΔso)2+4v2k2(γ21+4V2)]1/2]1/2, (14)
 E1′,2′=∓[v2k2+Δ2so+V2+γ212 −12[(γ21+4VΔso)2+4v2k2(γ21+4V2)]1/2]1/2, (15)
 E3,4=∓[v2k2+Δ2so+V2+γ212 +12[(γ21−4VΔso)2+4v2k2(γ21+4V2)]1/2]1/2, (16)
 E3′,4′=∓[v2k2+Δ2so+V2+γ212 +12[(γ21+4VΔso)2+4v2k2(γ21+4V2)]1/2]1/2. (17)

The above spectrum is shown in Fig.3(d,e), where it is compared with the corresponding spectrum obtained in the tight binding model. As before, spectra from tight-binding and effective models coincide near the K point. Note, the band splitting due to is well resolved only in part (e).

Separation of the bands and in the effective model described above, as well as in the tight binding model, is much larger than the separation of the bands and , see Fig.2. Therefore, when the electronic states close to the band edges are relevant and sufficient to describe transport properties (eg. when the Fermi level is in the gap), one may restrict considerations to the bands and . This leads to a further simplification of the effective Hamiltonian, as described in more details in section 4.

## Iii Spin Hall and spin Nernst effects

Spin Hall and spin Nernst effects correspond to transversal spin currents induced by electric field and temperature gradient, respectively. By analogy to the usual Hall and Nernst effects one may write the density of spin current due to electric field and temperature gradient as

 Jsni=∑j[σsnijEj+αsnij(−∂jT)] (18)

where (for ) is the spin Hall conductivity with being the -th component () of electron spin, while denotes the thermoelectric spin Nernst conductivity. The two conductivities are not independent and obey some general relations. Our objective is to find first the zero-temperature spin Hall conductivity, and then to calculate the low-temperature thermoelectric spin Nernst conductivity from these relations, as described below.

The quantum-mechanical operator of spin current density may be defined as

 Jsn=12[v,sn]+, (19)

where denotes the anticommutator of any two operators and , while is the velocity operator (). The latter operator can be easily found from the corresponding Hamiltonian [Eqs (1) and (10)]. In the linear response theory, the frequency-dependent spin Hall conductivity is then given by the formula dyrdal (),

 σszxy(ω)=eℏ2ωTr∫dε2πd2k(2π)2[vx,sz]+Gk(ε+ω)vyGk(ε), (20)

where is the Green function corresponding to the appropriate Hamiltonian of the system. When we restrict considerations to the topological contribution to the spin Hall current in the d.c. limit, this formula gives exactly the same result as that based on the Berry phase calculations berry () in momentum space.

It has been shown that the Berry phase leads to an additional term in the general expression for the orbital magnetization xiao2005 (). This correction gives rise to some contributions to the charge and spin currents xiao2007 (); chuu (), and also allows to write the relationship between intrinsic spin Nernst conductivity and intrinsic zero-temperature spin Hall conductivity, which in the low-temperature regime takes the form chuu ()

 αszxy=π2k2B3eTdσszxydε∣∣∣ε=μ, (21)

where stands for temperature and denotes the Boltzman constant. The latter equation is the spin analog of the Mott relation for charge transport, and will be used to calculate the low-temperature spin Nernst conductivity from the zero-temperature spin Hall conductivity. The derivative in Eq.(21) is taken at the Fermi level . The latter can be tuned by an external gate voltage.

Thus, we need to calculate the spin Hall conductivity first. The relevant derivation depends on the model applied to describe the corresponding electronic spectrum. Below we present derivation of the conductivity for the effective Hamiltonian, where analytical results are available. These results will be compared with those obtained numerically for the tight binding model.

### iii.1 The limit of V=0

Assume first the limit of . To find the spin Hall conductivity we start from Eq.(20) and write it in the form

 σszxy(ω)=e2ω∫dε2π∫d2k(2π)2D(ε+ω,ε) ×4∏n=1[ε−En+ω+μ+iδsign(ε)]−2 ×4∏m=1[ε−Em+μ+iδsign(ε)]−2. (22)

Here, is defined as

 D(ε+ω,ε)=Tr{[vx,sz]+gk(ε+ω)vygk(ε)}, (23)

where denotes the nominator of the corresponding Green function . Taking the first two terms of the expansion of with respect to , one finds

 D(ε+ω,ε)≃iωχ(ε), (24)

with

 χ(ε)=8v2Δso{v2k2[v2k2+2(Δ2−(ε+μ)2)] +(Δ2so−(ε+μ)2)(γ21+Δ2so−(ε+μ)2)}2 ×{(Δ2so−(ε+μ)2)(γ21+Δ2so−(ε+μ)2) +v2k2[v2k2+2(γ21+Δ2so−(ε+μ)2)]}. (25)

Thus, in the limit of one finds the following expression for the spin Hall conductivity

 σszxy=ie2∫dε2π∫d2k(2π)2F(ε), (26)

where

 F(ε)=χ(ε)∏4n=1[(ε−En+μ+iδsign(ε)]4. (27)

Integrating now over one finds

 ∫dεF(ε)=2πi∑nRnf(En), (28)

where () are the residua associated with the corresponding selfenergies (electron bands), and is the Fermi distribution function (here for zero temperature). These residua are equal:

 (29)

and

 R3,4=±8√2v2ΔsoLξ3(2v2k2+γ21+γ1ξ+2Δ2so)5/2 (30)

with

 L=−v2k2(3γ31+γ21ξ+4γ1Δ2so−2ξΔ2so) +2v4k4(γ1+ξ) −γ1(γ41+γ31ξ+2γ21Δ2so+2γ1ξΔ2so+2Δ4so), (31)

where . Thus, the conductivity may be written in the form

 σszxy=−e4π∑n∫kRnf(En)dk. (32)

Taking into account the integrals

 ∫kR1dk=−√2(γ1+ξ)Δsoξ(2v2k2+γ21−γ1ξ+2Δ2so)1/2, (33)
 ∫kR3dk=√2(γ1−ξ)Δsoξ(2v2k2+γ21+γ1ξ+2Δ2so)1/2, (34)

and then assuming the appropriate limits of the integration, one finds the final expressions for the spin Hall conductivity as presented below.

When the chemical level is inside the gap, , the spin Hall conductivity is equal to

 σszxy=−2e4π. (35)

When ,

 σszxy=−2(γ1+√μ2−Δ2so)2√μ2−Δ2so+γ1Δso|μ|e4π, (36)

while for one finds

 σszxy=−2(μ2−Δ2so)−γ214(μ2−Δ2so)−γ214Δso|μ|e4π. (37)

The spin Hall conductivity inside the gap is now twice as large as that in the case of a single-layer graphene. The general behavior of the conductivity with position of the Fermi level, shown in Fig.3 by the solid dark (solid red) line, is qualitatively similar to that for a single-layer graphene, i.e., outside the gap the spin Hall conductivity tends to zero with increasing , while it remains constant and quantized inside the gap. This behavior is reasonable as the spin Hall conductivity is due to spin-orbit coupling, which is the same for both atomic monolayers. Note, the formula derived correspond to one Dirac point, while the figures include contributions from both Dirac points.

The corresponding spin Nernst conductivity is given by the following formulas:

When ,

 αszxy=±π6k2BΔsoμ2T ×−γ21−2μ2−4γ1μ2√μ2−Δ2so+Δ2so(2+3γ1√μ2−Δ2so)4γ1√μ2−Δ2so+4μ2+γ21−4Δ2so. (38)

For ,

 αszxy=∓π3Δsok2BT ×8Δ4so+6Δ2soγ21+γ41−2(8Δ2so+5γ21)μ2+8μ4μ2(4Δ2so+γ21−4μ2)2. (39)

In turn, when is in the gap, , the spin Nernst conductivity vanishes, . The signs and in the above formulas correspond to the case when the chemical potential is negative or positive, respectively.

Variation of the spin Nernst conductivity with the chemical level is shown in Fig. 3 by the solid gray (solid blue) line. Similarly as in a single-layer grapheneczhang2009 (), the spin Nernst conductivity vanishes for the Fermi level inside the gap, when the system is in the insulating phase, and becomes nonzero for the Fermi level inside the valence or conduction bands, when the temperature gradient generates a longitudinal charge current. Note, the spin Nernst conductivity becomes divergent as approaches edges of the energy gap.

Conductivity in the tight binding model can be obtained in a similar way, although the corresponding formulas are cumbersome and will not be presented here. Instead of this we present some numerical results, which in Fig.3 are shown by the dotted lines. Note, the spin Hall as well as spin Nernst conductivities in the effective model coincide very well with the results obtained from the tight binding model.

### iii.2 The case of V≠0

Let us consider now the case of . The procedure presented above for the effective model with can be easily extended to a nonzero vertical bias, . The difference is that now the degeneracy of the bands is lifted and we have 8 different bands, , which have to be taken into account. Thus, instead of Eq.(27) we have now

 F(ε)=χ(ε)∏8n=1[ε−En+μ+iδsign(ε)]2 (40)

with adequate . Following the procedure described above for , one can derive the corresponding analytical formula. These formula, however, will not be presented here as they are rather cumbersome, so we present only numerical results. Moreover, since the results in the tight binding model coincide with those obtained with the effective Hamiltonian, as shown above, we restrict the analysis below to the effective Hamiltonian.

In Fig.4 we show the spin Hall conductivity as a function of the Fermi level and vertical bias . For we recover the quantized conductivity in the gap. As increases, however, the range of quantized spin Hall conductivity shrinks and at a certain value of (indicated by the dashed line in Fig.4) there is a transition (at ) from to . This behavior is explicitly shown in Fig.5, where several cross-sections of Fig.4 along constant values of are presented. The above transition is clearly evident for the curve corresponding to .

The transition from topological insulating phase at small voltages to the normal insulating behavior at large voltages is also clearly visible in Fig.6, which presents some cross-sections of Fig.4 along constant values of . This figure shows how the range of the quantized value of changes with increasing . As increases starting from , width of the range where is quantized shrinks, and at a certain critical value of width of this range goes to zero. The spin Hall conductivity at changes then from at voltages smaller then the critical one to at higher voltages. This clearly reveals a transition from the topological insulating phase to the normal insulating behavior (more information on the topological insulating phases in graphene can be found eg. in Ref. [vozmediano2010, ]). From Figs 4 to 6 one could conclude that the gap diminishes with increasing , becomes totally suppressed at the critical value of , and then becomes open again at larger voltages. Indeed, this is the case as shown in Fig.7, where the spectrum near the gap is plotted for several values of . This figure clearly shows that the gap becomes closed at the critical value of , and then is open again at larger values of . The spin Hall conductivity in the gap above the critical voltage is however suppressed.

The phase transition from the topological spin Hall insulating phase to the conventional insulator becomes revealed in the spin Nernst conductivity, too. This is presented in Fig.8, where the low-temperature spin Nernst conductivity is shown as a function of the Fermi energy for the same values of as in Fig.6. The range of zero spin Nernst conductivity decreases with increasing , goes to zero at the critical value of , and then becomes nonzero again for larger values of .

## Iv Low-energy effective Hamiltonian

As we have already mentioned above, separation of the bands and (or and ) is much larger than separation of the bands and (or and ). The latter determines the energy gap induced by the spin-orbit coupling (see Fig.2). When only the electron states near the Fermi level are relevant and the Fermi level is in the gap or close to it, one may further reduce the effective Hamiltonian to include explicitly the bands and , and the other bands only via effective parameters of the corresponding reduced effective model. The relevant reduced low-energy effective Hamiltonian takes the form mccan ():

 HrK=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Δso+V0−ℏ2k2−2m00−Δso+V0−ℏ2k2−2m−ℏ2k2+2m0−Δso−V00−ℏ2k2+2m0Δso−V⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (41)

where and is the effective electron mass. In the following we consider some special cases.

### iv.1 The case of V=0

The corresponding eigenvalues of the Hamiltonian (41) are then equal to

 E1,2=∓[Δ2so+(ℏ2k22m)2]1/2, (42)

see the inset in Fig.9. Using the notation introduced in the preceding section we find

 σszxy=ie2∫dε2π∫d2k(2π)2χ(ε)∏n=2n=1[ε−En+μ+iδsgn(ε)]4, (43)

where

 χ(ε)=4Δsoℏ4k2m2[Δ2so−(ε+μ)2+(ℏ2k22m)2]2. (44)

Performing the integration over and then over , one arrives at the following analytical formulas for the spin Hall conductivity:

 σszxy=−2Δso|μ|e4π (45)

for , and

 σszxy=−2e4π (46)

when the chemical level is inside the gap, .

The corresponding low-temperature spin Nernst conductivity is then given as follows. For one finds

 αszxy=∓π6k2BΔsoμ2T, (47)

while for inside the gap

The above results for both spin Hall and spin Nernst conductivities are shown by the dotted lines in Fig.9, where they are compared with the corresponding results obtained from the effective Hamiltonian (10) (solid lines). There is a nice agreement between the results. We note that the divergency of the spin Nernst conductivity when the Fermi level approaches the band edges, observed in the tight-binding and effective Hamiltonian, is not reproduced by the reduced Hamiltonian.

### iv.2 The case of V≠0

When the eigenvalues of the Hamiltonian (41) take the form,

 E1,2=∓[(V+Δso)2+(ℏ2k22m)2]1/2, (48)
 E1′,2′=∓[(V−Δso)2+(ℏ2k22m)2]1/2. (49)

The nonzero leads to splitting of the electron bands, as already discussed above.

When , the spin Hall conductivity is then given by

 σszxy=−e4π2Δso|μ| (50)

When ,

 σszxy=−e4π(1−V−Δso|μ|) (51)

Finally, when is inside the gap,

 σszxy=0forV>Δso (52) σszxy=−2e4πforV<Δso. (53)

The corresponding low-temperature spin Nernst conductivity is given by

 αszxy=π6k2BTΔsoμ|μ| (54)

for , and

 αszxy=−π12k2BT1|μ|μ(V−Δso) (55)

for . In turn, the spin Nernst conductivity vanishes inside the energy gap.

Behavior of spin Hall and spin Nernst conductivities with position of the Fermi level and bias voltage almost coincides with that presented in Figs 4 to 6 and 8, and therefore will not be present here. As in the model, one observes the same transition between the spin Hall insulator and classical insulator as increases. As already mentioned above, there is no divergency of the spin Nernst conductivity at the band edge for the low energy effective model considered here.

## V Summary

We have calculated analytically as well as numerically the spin Hall and spin Nernst conductivities in a bilayer graphene. To describe the relevant electronic spectrum we have assumed the tight binding model as well as some simplified effective Hamiltonians relevant for states close to the Dirac points. Both spin Hall and spin Nernst effects consist in transverse spin accumulation (spin current). However, as the spin Hall effect is due to external electric field, the Nernst effect is due to a temperature gradient. Assuming intrinsic spin orbit interaction, we have found the intrinsic contributions to both effects. Generally, the spin Hall conductivity in the energy gap of a bilayer graphene is twice as large as that for a single-layer graphene. When external voltage is applied between the two atomic sheets, we found a transition from the spin Hall insulator phase to the conventional insulator behavior. The energy gap at the transition point is closed.

We have compared results obtained from the tight-binding model with those derived from the effective Hamiltonians. From this comparison we arrived at the conclusion, that both spin Hall and spin Nernst conductivities are well described by the effective Hamiltonians, which allow derivation of some analytical results.

### Acknowledgment

This work has been supported in part by the European Union under European Social Fund ’Human - best investment’ (PO KL 4.1.1) and in part by funds of the Ministry of Science and Higher Education as a research project in years 2010-2013 (No. N N202 199239). The authors acknowledge valuable discussions with V.K. Dugaev.

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