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Similarity between the superconductivity in the graphene with the spin transport in the two-dimensional antiferromagnet in the honeycomb lattice
Physica B 507 (2017) 164–169
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Physica B journal homepage: www.elsevier.com/locate/physb
Similarity between the superconductivity in the graphene with the spin transport in the two-dimensional antiferromagnet in the honeycomb lattice
MARK
L.S. Lima Departamento de Física e Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000 Belo Horizonte, MG, Brazil
A BS T RAC T We have used the Dirac's massless quasi-particles together with the Kubo's formula to study the spin transport by electrons in the graphene monolayer. We have calculated the electric conductivity and verified the behavior of the AC and DC currents of this system, that is a relativistic electron plasma. Our results show that the AC conductivity tends to infinity in the limit ω → 0 , similar to the behavior obtained for the spin transport in the two-dimensional frustrated antiferromagnet in the honeycomb lattice. We have made a diagrammatic expansion for the Green's function and we have not gotten significative change in the results.
1. Introduction A general theory of relativistic hydrodynamics of the electronic plasma of the graphene has been derived since some time ago [1,2]. It is often said that the Dirac fluid in graphene is a quantum critical system in two spatial dimensions, and exhibits behavior analogous to the quantum critical regime at finite temperatures above the superfluid-insulator transition with some technical differences [2]. From a general way, the graphene is an allotropic form of carbon which is a lot researched recently. It has semiconductor properties with low-lying excitations obeying the massless Dirac's equation [3]. The interplay between the antiferromagnetic and Kekule valence bond solid ordering in the zero energy levels of neutral monolayer and bilayer graphene has been studied in Ref. [4]. For electronic transport in matter, strong interactions can lead to a breakdown of Fermi liquid paradigm of coherent quasi-particles scattering of impurities. In such situations, provided that certain conditions are met, the complex microscopic dynamics can be coarse-grained to a hydrodynamics description of momentum, energy, and charge transport on long length and time scales [2,5]. The transport properties in graphene has been studied a lot in the literature using the Boltzmann's equation formalism [6,7]. Graphene is an interesting material for spintronics, showing long spin relaxation lengths even at room temperature. For future spintronic devices it is important to understand the behavior of the spins and the limitations for the spin transport in structures where the dimensions are smaller than the spin relaxation length [8]. The electron spin lifetime in carbon materials is expected to be very long because of the very large natural abundance of the spinless nuclear isotope 12C and the small size of spin orbit coupling. This even led to propose graphene as an optimal material to store quantum information in the spin of confined electrons. However, most of the experiments show that the spin
lifetimes are in the range of nanoseconds, much shorter than expected from these considerations, which lies at the heart of the design of devices where graphene is used as a passive component to carry spin currents. The Kubo formalism to calculate of transport coefficients has been many employed in the literature to study the spin transport properties by ions in the lattice in magnetic spin systems. For example, Sentef et al. [9] has analyzed the spin transport in the easy-axis Heisenberg antiferromagnetic model in two and three dimensions, at T=0. Damle and Sachdev [10] treated the two-dimensional case using the nonlinear sigma model in the gapped phase. Pires and Lima [11–13] treated the one and two-dimensional easy plane Heisenberg antiferromagnetic model. Lima and Pires [14] studied the spin transport in the two-dimensional anisotropic XY model using the SU(3) Schwinger boson theory in the absence of impurities, Lima [15] studied the case of the Heisenberg antiferromagnetic model in one and two dimensions with Dzyaloshinskii-Moriya interaction. Zewei Chen et. al. [16] analyzed the effect of spatial and spin anisotropy on spin conductivity for the S = 1/2 Heisenberg model on a square lattice and more recently, Lima at.al has studied the spin transport in the two-dimensional frustrated Heisenberg model in the easy plane using the self consistent harmonic approximation and the SU(3) Schwinger boson theory [17– 23]. Recently, The electron transport properties of zigzag graphene nanoribbons with upright standing carbon chains were investigated using first-principles calculations. The calculated results show a significant odd-even dependence [24]. From an experimental point of view, there is an intense research about spin transport by electrons in graphene where it has been investigated the quantum Hall effect for spins and the magnon spintronics [25–29]. In the studies of these effects often only the sign differences between related quantities like magnetic fields and generated spin and charge currents are deter-
http://dx.doi.org/10.1016/j.physb.2016.12.001 Received 2 June 2016; Received in revised form 10 October 2016; Accepted 1 December 2016 Available online 02 December 2016 0921-4526/ © 2016 Elsevier B.V. All rights reserved.
Physica B 507 (2017) 164–169
L.S. Lima
→ → ω ( k ) ≈ =vF | k |,
mined. The spin injection and transport in single layer graphene can be investigated using nonlocal magnetoresistance (MR) measurements [30]. The aim of this paper is to verify the similarity between the electron transport in the graphene monolayer using Dirac fermions formalism with the spin transport in quantum frustrated two-dimensional antiferromagnet in the honeycomb lattice. The graphene consist in a fermion system with a relativistic Dirac spectrum where the energy vanishes linearly at isolated points in the Brillouin zone. Dirac's fermions are provided by numerous new experimental realizations. These include D-wave superconductors and topological insulators [31]. This work is divided in the following way. In Section 2, we discuss the Kubo's formalism for electron transport, in Section 3 we present the results into a mean field approximation, in Section 4 we treat the interactions electron-electron and exciton-exciton respectively using the diagrammatic expansion for the Green's function and in the last Section 5, we present our conclusions and final remarks.
where in the massless (gapless) limit, the spectrum is linear hence, the time and space scale the same way T ∼ L as required by relativistic invariance [32]. There are local interactions such as (ψ γμ ψ )2 and (ψ ψ ), where ψ = ψ †γ 0 . The action of the free Dirac's field is
S=
→⎞ ⎛ ⎞ ⎛ γ 0 = −i ⎜ 0 1 ⎟ → γ = −i ⎜ 0→ − σ ⎟ . ⎝1 0⎠ ⎝− σ 0 ⎠
{γμ, γν} = 2gμν ,
∫ d 2x (ψ (x ) ψ (x ))2
(5)
γ5 = iγ0 γ1.
(6)
gμν is the Minkowski's metric tensor ⎛1 0 0 0 ⎞ ⎜ 0 ⎟. gμν = ⎜ 0 − 1 0 ⎟ ⎜0 0 − 1 0 ⎟ ⎝0 0 0 − 1⎠
(7)
In the theory of free massless Dirac's fermions there are a fixed point in the renormalization group [32]. 3. Mean field approximation
(1)
where ψα (x ) with α , β = 1, 2 denote the two components of the Fermi field in D = 1 + 1 and D = 2 + 1 space dimensions and vF is the Fermi's velocity. We have considered the unities where vF=1 and = = 1. The interaction term has the form up to the irrelevant additive constant
/ int = −2Jγ
(4)
1 is the unit 2×2 matrix, and σ are the components of Pauli's matrices where the Dirac γ-matrix, γ0, γ1 and γ5 satisfy the algebra
The model of relativistic free-fermion particles of the graphene in D = 1 + 1 dimension is described by the following Hamiltonian density [32]
∫ d 2xψα† (x ) iσ3αβ ∂xψβ (x ),
∫ d 2xψα (x ) iγαβμ ∂μψβ (x )
where γ are the 4×4 Dirac's matrix.
2. The Dirac fermion model
/ = JvF
(3)
We use the low energy approach Dirac fermion [6,32] to determine the regular part of the electron conductivity (AC conductivity) or continuum conductivity. An electron current appears if there is an ⎯→ ⎯ electric field given by the Ohm' Law 1 = σ E . In a similar way, a spin ⎯→ ⎯ current appears as a response to a magnetic field 1S = σ ∇ B , through the system, where it plays the role of a chemical potential for spins. If one connect a low dimensional magnet with two bulk ferromagnetic, they can act as reservoirs for spins [28,29]. Then, one has a flow of a ⎯→ ⎯ spin current if there is a difference, Δ B , between the magnetic fields at the two ends of the sample. In the Kubo formalism [33,34,9,11] the conductivity is given by:
(2)
which is the interaction term of the (1+1)-dimensional Gross-Neveu→ ⎛ 1 1⎞ , ⎟, model. The model is depicted in Fig. 1, where d1 = ⎜ ⎝2 3 2⎠ ⎞ → ⎛ 1 → → → → → ⎛ 1 1⎞ , − 2 ⎟. and → a2 = d3 − d1 and d2 = ⎜ − , 0⎟, d3 = ⎜ a1 = d2 − d3, → ⎠ ⎝2 3 ⎝ 3 ⎠ → → → a3 = d1 − d2 . Assuming that electrons in graphene are non interacting, the standard band theory calculations on the honeycomb lattice in two spatial dimensions with nearest-neighbor hopping give us two species of Dirac fermions with single-particle spectrum in the relativistic form [5]
→ 〈2 〉 + Λ ( k , ω) , i (ω + i 0+) k →0
σ (ω) = →lim
(8) q , ω) is the current-current where 〈2 〉 is the kinetic energy and Λ (→ correlation function defined by ∞ → → → i Λ ( k , ω) = dteiωt 〈[1 ( k , t ), 1 (− k , 0)]〉. (9) =N 0 → + Λ ( k , ω + i 0 ) is analytic in the upper half of the complex plane and the extrapolation along the imaginary axis can be reliably done. The current operator for graphene is given by
∫
1 = ψ γμ ψ .
(10)
The real part of σ (ω), σ ′(ω), can be written in a standard form as [34]
σ ′(ω) = σ0 (ω) + σ reg (ω),
(11)
where σ0 (ω) is the DC contribution, given by σ0 (ω) = DS δ (ω). DS is the Drude's weight
→ → DS = π [〈2 〉 + Λ′( k = 0, ω → 0 )].
(12)
σ reg (ω), the regular part of σ ′(ω), is given as [34] σ reg (ω) = → → → Fig. 1. The 2D honeycomb lattice of graphene. d1, d2 , d3 connect the two triangular → → → sublattices and a1, a2 and a3 are the next-nearest-neighbor displacement vectors.
→ Λ″( k = 0, ω) . ω
(13)
It represents the AC contribution to the conductivity. In Eqs. (12) and (13), Λ′ and Λ″ stand for the real and imaginary part of Λ. 165
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Fig. 3. Representation of the flow of a spin current in the x direction in the twodimensional anisotropic frustrated Heisenberg model on a honeycomb lattice with the second neighbors and third neighbors, J1, J2 and J3.
The anisotropic two-dimensional antiferromagnet in the honeycomb lattice is depicted in Fig. 3 and is defined by the following Hamiltonian Fig. 2. Behavior of σ* for the value of T = 1.0 J . We have that σ reg (ω) =
(gμb )2 σ* (ω) . =
/ = How
ij
the Ac conductivity tends to infinity in the limit ω → 0 , therefore we must have an ideal conductor in this limit.
∑∫ α
π
0
d 2k 2 0 ∼0 vF Gα (ω1) Gα (ω + ω1) (2π )2
(14)
as
Gα0 (ω) =
1 , iωn − ωk
∼0 Gα (ω) =
−1 . iωn + ωk
(15)
From the analytical continuation, we obtain the regular part of the conductivity as
σ reg (ω) = (gμB )2
π
∫0 ∫0
π
d 2k vF2 f (ωk )[1 − f (ωk )] δ (ω − 2ωk ), (2π )2 ωk
(16)
1/(e βωk
+ 1) is the Fermi-Dirac distribution. We have where f (ωk ) = solved the Eq. (19) analytically as
2π 2 σ* (ω) = f (ω /2)[1 − f (ω /2)]. ω
(18)
i
where Jij = J1 for the nearest neighbors interaction, Jij = J2 means second neighbors interaction and Jij = J3 for third neighbors interaction. For values of anisotropy sufficiently strong D, the system is a quantum paramagnet [36,37]. The anisotropy forces each spin to stay in the nonmagnetic state. The ground state has no long-range magnetic order and there are finite gap spin excitations. Decreasing D, the energy of the gap decreases and goes to zero at a critical Dc, where a quantum phase transition takes place. For D ≤ Dc and positive, the system is in a gapless phase, that is ordered at T=0 in the non frustrated case. The honeycomb lattice is a frustrated spin system where the frustration arises from competing interactions rather than geometric constraints, and the frustration is enhanced by strong quantum fluctuations due to the low coordination number [35,37]. Moreover, experimental data are available for several compounds such as the family of compounds BaM2 (XO4 )2 , where M=Co, Ni and X=P. The magnetic ions M are arranged in weakly coupled frustrated honeycomb lattice. However, most of the compounds lies in the Néel ordered region. The model with J1, J2 and J3 interactions was studied recently by D.C. Cabra et al. [38,39] or S=1/2 along the line J1 = J3 using the linear spin wave theory (LSWT) and SU(2) mean field Schwinger boson approximation. J. B. Fouet [40] studied the same model using exact diagonalization and LSWT, Li et al. [41] treated the same system using the coupled cluster method. The spin superfluidity is a phenomenon arising due to spontaneous breaking of the U (1) symmetry, which in a spin system case is represented by the symmetry group SU (2) of spin rotations about direction of the magnetic field. The Bose-Einstein condensation (BEC) of quasiparticles such as magnons and excitons whose number is not conserved is presently one of the most debated phenomena of condensed matter physics [43]. In atomic BEC and helium superfluids the symmetry breaking leads to a non-zero value of the superfluid rigidity where the superfluid density ρS which enters the non-dissipative supercurrent of particles and thus to magnon supercurrent. Using the Matsubara's Green's function method and the SU(3) Schwinger boson theory [36,44], we obtain the spin current σ (ω) for this model at T=0 given as
Using the Matsubara's method, we obtain the Green's function at finite temperature for the model Eq. (18) as
Gj (ω) =
∑ Jij Si·Sj + D ∑ (Siz )2 .
(17)
In Fig. 2, we present the behavior of σ* (ω) with ω. Since the system is gapless we have obtained a behavior of the AC conductivity tending to infinity when ω → 0 , that correspond to the DC limit. However, the fact the electric conductivity tending to infinity, is not taken as the true definition of superconductivity. The fundamental proof that superconductivity occurs in a given material is the demonstration of the Meissner-Ochsenfeld effect. This effect is the fact that metals in the superconductor state are perfect diamagnets and hence expels a weak external magnetic field. We find the behavior of the electric conductivity in the graphene gotten is similar to one recently obtained for the spin transport in the quantum two-dimensional Heisenberg antiferromagnet in the honeycomb lattice [22]. This behavior for the spin transport in magnetic spin systems is a characteristics of systems that does not present gap in the excitation spectrum. As in the Dirac's fermion model of the graphene there is not gap in the spectrum, we will not have electrons to form one current for all values of ω of the spectrum, in the same way as happen in the frustrated antiferromagnet in the honeycomb lattice. Moreover, as there is not scattering mechanism, as in this model, it is expected that the conductivity to be divergent. The reason is that spin-spin scattering is not treated properly in a mean field approach.
σ reg (ω) =
Λ (k = 0, ω) π = (gμB )2 3 ω ω
π
∫0 ∫0
π
d 2k 2 σαα (2π )2 2
[sin kx + η sin( 3 kx ) + α sin(2 2 kx )] × [δ (ω − ω1k ) + δ (ω − ω 2k )],
(19)
where σαβ is a matrix [37]. We can integrate analytically the Eq. (19) to 166
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q∼ = (iqn , → q ) flowing out from it to the left. We can draw diagrams for the current-current correlation function using the Feynman's rules. → → → → G ( k , t , k′ , 0) = −〈; [ψ ( k , τ ) ψ ( k ′, 0)]〉,
(25)
where τ = it . ; is the time ordering operator. The Green's function can be expanded as [45,46]
→ → G ( k , t , k′ , 0) =
∞
∑∫ n =0
∑
0
β
dτ1 ⋯
∫0
β
dτn
〈0 Tr [C † (τ1) C (τ1)⋯C † (τn ) C (τn )] 0〉 (26)
t1⋯ tn
where Cc = ∑k αk . We can collect all terms in the expansion of the Green's function as
→ → → G ( k , t , k′ , 0) = G (0) ( k , c, τ − τ′) δk , k ′ δσ , σ ′ +
∫0
β
dτ1
∫0
β
→ dτ2 G (0) ( k , τ
→ → → − τ1) Σ ( k , τ1, k′ , τ2 ) G (0) ( k , τ2 − τ′),
(27)
where τ = it is the imaginary time and β = 1/ T . with the solution
→ G ( k , ik n ) =
1 , → ik n − ζk − Σ ( k , ik n ) (28) → where the self-energy Σ ( k , ik n ) appears as a direct additive renormalization of the bare energy ζk = εk − μ, and μ is the chemical potential. The Feynman's rules therefore, lead to the following expression for the self-energy diagram.
Fig. 4. Behavior of σ* (ω) for the following values: Dc=0, η = 0.4 , α = 0.11 (dashed line). For large value of Dc, Dc=8, η = 0.4 , α = 0.6 α = 0.8 (dotted line) and next to the quantum the phase transition point: Dc=0.35 and η = 0.25 and α = 0.6 (solid line). The behavior of the spin conductivity varies little with η, α parameters. We have
σ reg (ω) =
(gμb )2 σ* (ω) . =
How we obtain the AC conductivity tending to infinity in the limit
ω → 0 thus, we have an ideal spin conductor in the DC limit.
obtain
η2 3 π ⎡1 2⎤ ⎥, σ* (ω) = 3 ⎢ − η sin 3 + α sin 2 2 + + α2 ω ⎣4 12 16 ⎦
where W gives the electron-electron interaction (20)
W=
and
σ reg (ω) =
(gμB =
)2
3J12 δγγ ′ → → Σ (2) ( k γτ1, k′ γ′τ2 ) = 16
∑ k1
1 . ik n − ζk1
(32)
3J12 π [d γ (ζk )] δγγ ′ → → . −IΣ R (2) ( k γ , k′ γ′, ik n ) = 16
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
(22)
(33)
1 β
β
k1, k 2
0
β
→ → dτ3 G (0) ( k1 γ1, τ1 − τ3) G (0) ( k2 γ2, τ3) (34)
We proceed by performing the τ3 integral followed by a Fourier transformation and the result is
∑ 1 ∑ 〈;1 (→ q , iql ) iql
∑∫
sgn (τ1 − τ3),
In the frequency domain it is
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
iqm
(23)
∑ k1, k 2
1 + 2n (ζk ) 1 . ζk1 − ζk 2 ik n − ζk1
(35)
In this case, the imaginary part of the retarded self-energy is given
We conveniently rewrite it using the four-vector notation [45] q∼ = (iqn , → q ).
1 〈1 (q∼) 1 (−q∼)〉. βN2
(31)
The third-order contribution reads
1 Πxx (→ q , τ − τ′) = − 2 〈;1 (→ q , τ ) 1 (−→ q , τ′)〉. N
Πxx (q∼) −
k1
The important information about the scattering mechanism lies in the imaginary part of the retarded self-energy
The retarded Matsubara current-current correlation function is
1 (−→ q , iqm )〉e−iql τ e−iqm τ ′.
→
∑ G(0) ( k1 γ1, τ1 − τ2).
3J12 δγγ ′ → → Σ (2) ( k γ , k′ γ′, ik n ) = 16
4.1. Electron-electron interaction
d (τ − τ′) eiqn (τ − τ ′)
(30)
1)−1.
In the frequency domain it becomes
4. Vertex corrections for the Green's function
∫0
ψν1 ψν2 ψν4 ψν3,
+ f (ζk ) is the Fermi-Dirac distribution f (ζk ) = The second-order term of the self-energy is given by
(21)
In Fig. 4, we present the behavior of σ reg (ω) with ω. Due to the hexagonal crystal lattice, we must have a spin current in zig-zag over the material as depicted in Fig. 3. Since the system is gapless we have obtained a behavior of the AC spin conductivity tending to infinity when ω → 0 , that corresponds to the DC limit. Consequently, we obtain a superconductor behavior for the spin transport in the DC limit. This behavior is similar one recently obtained for the two-dimensional ferroquadrupolar model [21] and to triangular lattice [23]. This is a characteristics of spin systems without gap in the excitation spectrum.
1 Πxx (→ q , iqn ) = − 2 N
∑ ν1ν2 ν3 ν4
(e βζk
2 σαα σ (ω). 4π 2 *
β
1 2
by
3πJ1 [J1 d γ (ζk )]2 δγγ ′ → → −IΣ (3) ( k γ , k′ γ′, ik n ) = 16
(24)
D
∫−D 1 +ω 2−nζ(ζ ) dζ,
(36)
where D is a specified cut-off. This is logarithmically divergent when
The vertex conserves four-momentum, and thus has a momentum 167
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→ → → G ( k , t , k′ , 0) = G (0) ( k , c, τ − τ′) δk , k ′ δσ , σ ′ +
the temperature or the frequency goes to zero. Collecting the second and third order terms, one have for the low temperature case
⎡ ⎛ 3δγγ ′ [d γ (ζk )]2 π ⎢ → → D 1 + 2d γ (ζk )ln ⎜⎜ −IΣ (R) ( k γ , k′ γ , ω) ≈ ⎢⎣ 16 ⎝ ω 2 + (kB T )2
1 β
∑ ikn
1 N2
(37)
→ → → → → → 1 = R ∑ Γ0, x ( k , k )[GA ( k , 0) GR ( k , 0) ΓxRA ( k , k ; 0, 0)]. πN2 → k
W=
, → ik n − ζk − Σ ( k , ik n )
1 2
2
= 2W
ikn
∫ (2dπk)2 n (ζk )
(45)
ψν†1 ψν†2 ψν4 ψν3,
∑
(46)
ν1ν2 ν3 ν4
3J12 δγγ ′ → → Σ (2) ( k γτ1, k′ γ′τ2 ) = 16
(38)
1)−1.
→
∑ G(0) ( k1 γ1, τ1 − τ2). → k1
(47)
In the frequency domain it becomes
3J12 δγγ ′ → → Σ (2) ( k γ , k′ γ′, ik n ) = 16
(39)
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
∑ → k1
1 ik n − ζk1
(48)
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
⎛ 1 1 ⎞ † + ⎜ ⎟ ψ σαα ψk + q. ⎝ ω1k ω 2k ⎠ k + q
n =0
∑ t1⋯ tn
0
dτ1 ⋯
∫0
1 + 2n (ζk ) 1 . ζk1 − ζk 2 ik n − ζk1
(50)
D
∫−D 1 +ω 2−nζ(ζ ) dζ,
1 β
∑ ikn
1 N2
(51)
→ → → k ) G ( k , ik n )
∑ Γ0,x ( k , → k
→ → → G ( k , ik n + iqn ) Γx ( k , k ; ik n + iqn , ik n ),
(52)
The Matsubara sum over ikn is performed in the usual way by a contour integration over z = ik n . The final expression for the real part of the spin conductivity is given by reg σxx =
(41)
β
∑
→→ k1, k 2
where D is a specified cut-off [45]. We obtain the current-current function in this case, given as
→ → → → → → 1 R ∑ Γ0, x ( k , k )[GA ( k , 0) GR ( k , 0) ΓxRA ( k , k ; 0, 0)]. πN2 → k
The Green's function can be expanded in this case as [45,46] ∞
(49)
3πJ1 [J1 d γ (ζk )]2 δγγ ′ → → −IΣ (3) ( k γ , k′ γ′, ik n ) = 16
3 (kx + qx )) + α sin(2 2 (kx + qx ))]
∑ (−J1)n ∫
→ → dτ3 G (0) ( k1 γ1, τ1 − τ3) G (0) ( k2 γ2, τ3)
by
In this case we have the spin current operator 1x (q∼) in the fourvector notation as
→ k
β
In this case, the imaginary part of the retarded self-energy is given
Πxx (0, iqn ) = −
∑ [sin kx + η sin(
∑∫ 0
→→ k1, k 2
Proceeding further by performing the τ3 integral followed by a Fourier transformation, the result is
(40)
4.2. Exciton-Exciton interaction
→ → G ( k , t , k′ , 0) =
ikn η
− and n (ζk ) is the Bose-Einstein distribution n (ζk ) = The second-order contribution for the self-energy is
→ where τ0* is the renormalized lifetime and A ( k , 0) = Zδ (ζk ). Z is called the renormalization constant and it is a measure of the quasi-particle reg = Cτ0*, where C is another weight. Performing the integral we have σxx renormalized constant. Finally we have for the renormalized lifetime [τ0*]−1 = Z [τ0]−1. We define ballistic transport as transport where the mean free path ls of the electrons is limited by the sample dimensions. The electronelectron interaction introduces so a finite mean free that if related to the relaxation time by ls = vF τ0 , where vF is the Fermi velocity.
J1 2
2
∫ (2dπk)2 ∑ ikne − ζk
sgn (τ1 − τ3).
k
1=−
(44)
and the third-order contribution reads
→ 1 k2 R ∑ 2 τ0* A ( k , 0), πN2 m →
t2
(43)
(e βζk
At low temperatures it is valid to assume that the imaginary part of the → self-energy due to electron-electron interaction IΣ ( k ) is small. Hence, we have in this limit reg σxx =
→ dτ2 G (0) ( k , τ
where W now gives the exciton-exciton interaction
The Matsubara sum over ikn is performed in the usual way by a contour integration over z = ik n . The final expression for the real part of the spin conductivity is given then by reg σxx
β
1
→ 2W Σ ( k , ik n ) = β
→ → → k ) G ( k , ik n )
→ → → G ( k , ik n + iqn ) Γx ( k , k ; ik n + iqn , ik n ),
∫0
The Feynman's rules, in this case, lead to the following expression for the self-energy diagram
∑ Γ0,x ( k , → k
dτ1
The solution is
→ G ( k , ik n ) =
From this result, we have an enhanced scattering in third order of the exchange coupling [45]. Having the expressions for both the single-particle Green's function and the vertex function Γ, we can obtain a general formula for the conductivity making a summation over the internal Matsubara frequency. If we drop the four-vector notation in favor of the standard q = 0 , the current-current notation, and furthermore treat the case → function is
Πxx (0, iqn ) = −
β
→ → → − τ1) Σ ( k , τ1, k′ , τ2 ) G (0) ( k , τ2 − τ′).
⎞ ⎟ ⎟ ⎠
⎤ + ⋯⎥ . ⎥⎦
∫0
(53)
At low temperatures it is valid to assume that the imaginary part of the → self-energy, due to exciton-exciton interaction IΣ ( k ), is small. Therefore, we have in this limit
β
dτn
reg σxx =
〈0 Tr [C † (τ1) C (τ1)⋯C † (τn ) C (τn )] 0〉
→ 1 k2 R ∑ 2 τ0* A ( k , 0), πN2 → m
(54) → where τ0* is again the renormalized lifetime and A ( k , 0) = Zδ (ζk ). Z is called the renormalization constant and it gives a measure of the quasi-
(42)
k
where Cc = ∑k αk . We can collect all terms in the expansion of the Green's function as 168
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[5] J. Crossno, J.K. Shi, Ke Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T.A. Ohki, K.C. Fong, Science 351 (2016) 1058. [6] M. Muller, H.C. Nguyen, New J. Phys. 13 (2011) 035009. [7] M. Sentef, M. Kollar, A.P. Kampf arXiv:1203.2216v1 (2012). [8] M.H.D. Guimaraes, J.J. van den Berg, I.J. Vera-Marun, P.J. Zomer, B.J. van Wees, Phys. Rev. B 90 (2014) 235428. [9] M. Sentef, M. Kollar, A.P. Kampf, Phys. Rev. B 75 (2007) 214403. [10] K. Damle, S. Sachdev, Phys. Rev. B 56 (1997) 8714. [11] A.S.T. Pires, L.S. Lima, Phys. Rev. B 79 (2009) 064401. [12] A.S.T. Pires, L.S. Lima, J. Phys.: Condens. Matter 21 (2009) 245502. [13] A.S.T. Pires, L.S. Lima, J. Magn. Magn. Mater. 322 (2010) 668. [14] L.S. Lima, A.S.T. Pires, Eur. Phys. J. B 70 (2009) 335. [15] L.S. Lima, Phys. Status Solidi B 249 (2012) 1613. [16] Zewei Chen, Trinanjan Datta, Dao-Xin Yao, Eur. Phys. J. B 86 (2013) 63. [17] L.S. Lima, A.S.T. Pires, B.V. Costa, J. Magn. Magn. Mater. 371 (2014) 89. [18] L.S. Lima, J. Magn. Magn. Mater. 422C (2017) 412. [19] L.S. Lima, J. Magn. Magn. Mater. 405 (2016) 332. [20] L.S. Lima, J. Magn. Magn. Mater. 411 (2016) 108. [21] L.S. Lima, Solid State Commun. 228 (2016) 6. [22] L.S. Lima, Solid State Commun. 237 (2016) 19. [23] L.S. Lima, Solid State Commun. 239 (2016) 5. [24] Jing-Jing Hea, Xiao-Hong Yana, Yan-Dong Guoa, Chun-Sheng Liua, Yang Xiaoc, Lan Menga, Solid State Communications, 2015, (in press). [25] E.B. Sonin, Adv. Phys. 59 (2010) 181. [26] A.V. Chumak, V.I. Vasyuchka, A.A. Serga, B. Hillebrands, Nat. Phys. 11 (2015) 453. [27] V. Lauer, D.A. Bozhko, T. Bracher, P. Pirro, V.I. Vasyuchka, A.A. Serga, M.B. Jungfleisch, M. Agrawal, Yu.V. Kobljanskyj, G.A. Melkov, C. Dubs, B. Hillebrands, A.V. Chumak, arXiv:1508.07517v1 (2015). [28] F. Meier, D. Loss, Phys. Rev. Lett. 90 (2003) 167204. [29] Yuri V. Kobljanskyj, Gennadii A. Melkov, Alexander A. Serga, Andrei N. Slavin, Burkard Hillebrands, Phys. Rev. Appl. 4 (2015) 014014. [30] Wei Han, K.M. McCreary, K. Pi, W.H. Wang, Yan Li, H. Wen, J.R. Chen, R.K. Kawakami, J. Magn. Magn. Mater. 324 (2012) 369. [31] S. Sachdev, Quantum Phase Transitions, second ed., Cambridge University Press, 2011. [32] E. Fradkin, Field Theories of Condensed Matter Physics, second edition, (2013) Cambridge, U.K. [33] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer-Verlag, New York, 1985. [34] G.D. Mahan, Many Particles Physics, Plenum, New York, 1990. [35] D.E. Mc Nally, J.W. Simonson, J.J. Kistner-Morris, G.J. Smith, J.E. Hassinger, L. DeBeer-Schmitt, A.I. Kolesnikov, I.A. Zaliznyak, M.C. Aronson, Phys. Rev. B 91 (2015) 180407. [36] H.T. Wang, Y. Wang, Phys. Rev. B 71 (2005) 104429. [37] A.S.T. Pires, J. Magn. Magn. Mater. 412 (2016) 217. [38] D.C. Cabra, C.A. Lamas, H.D. Rosales, Phys. Rev. B 83 (2011) 094506. [39] D.C. Cabra, C.A. Lamas, H.D. Rosales, Phys. Lett. B 25 (2011) 891. [40] J.B. Fouet, P. Sindzingre, C. Lhuillier, Eur. Phys. J. B 20 (2001) 241. [41] P.H.Y. Li, R.F. Bishop, D.J.J. Farnell, C.E. Campbell, Phys. Rev. B 86 (2012) 144404. [43] M.Yu Bunkov, G.E. Volovik, Novel Superfluids, in: K.H. Bennemann, J.B. Ketterson (Eds.), Oxford University Press, 2013 (arXiv:1003.4889v3). [44] N. Papanicolaou, Nucl. Phys. B 305 (1988) 367. [45] H. Bruus, K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics An Introduction, Oxford Graduate Texts, Copenhagen, (2004). [46] Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press, 2004. [47] A.V. Sologubenko, T. Lorenz, H.R. Ott, A. Freimuth, J. Low. Temp. Phys. 147 (2007) 387.
reg = Cτ0*, where C is particle weight. Performing the integral we have σxx another renormalized constant. Finally we have for the renormalized lifetime [τ0*]−1 = Z [τ0]−1. It is well known that magnetic defects are identified as one of the important sources for the scattering of magnetic excitations. The experiments also suggest that spin-phonon scattering is very effective in reducing the energy flow, especially at high temperatures [47]. We can define the ballistic spin transport as the transport where the mean free path ls of the magnetic excitations is limited by the sample dimensions. The exciton-exciton interaction introduces so a finite mean free that if related to the relaxation time by ls = vs τ0 , where vs is the velocity of the magnetic excitations.
5. Conclusions and final remarks In summary, we have verified one similarity between the transport properties in two different quantum models. the Dirac's fermion model of the graphene monolayer, which is a quantum paramagnet and the quantum frustrated two-dimensional antiferromagnet in the honeycomb lattice. We have obtained the same behavior for the electric conductivity of the graphene and the spin conductivity of the anisotropic two-dimensional Heisenberg antiferromagnet in the honeycomb lattice. Both presenting the AC conductivity tending to infinity in the DC limit (ω → 0). However, due to the Meissner-Ochsenfeld effect, the fact the electric conductivity tends to infinity, is not taken as the true definition of superconductivity. This effect is the fact that a superconductor expels a weak external magnetic field not being valid for the case of spin transport. We have obtained also that the inclusion of the electron-electron and exciton-exciton interactions do not change the results obtained into a mean field approximation. Finally, we have that through the mapping of spin systems in other models such as the Dirac fermion model can be a tentative to measure in a way easier the spin transport in magnetic systems. Acknowledgements This work was partially supported by the Brazilian agencies FAPEMIG and CEFET-MG. References [1] S.A. Hartnoll, P.K. Kovtun, M. Muller, S. Sachdev, Phys. Rev. B 76 (2007) 144502. [2] A. Lucas, J. Crossno, K.C. Fong, P. Kim, S. Sachdev, Phys. Rev. B 93 (2016) 075426. [3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [4] J. Lee, S. Sachdev, Phys. Rev. Lett. 114 (2015) 226801.
169
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Similarity between the superconductivity in the graphene with the spin transport in the two-dimensional antiferromagnet in the honeycomb lattice
MARK
L.S. Lima Departamento de Física e Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000 Belo Horizonte, MG, Brazil
A BS T RAC T We have used the Dirac's massless quasi-particles together with the Kubo's formula to study the spin transport by electrons in the graphene monolayer. We have calculated the electric conductivity and verified the behavior of the AC and DC currents of this system, that is a relativistic electron plasma. Our results show that the AC conductivity tends to infinity in the limit ω → 0 , similar to the behavior obtained for the spin transport in the two-dimensional frustrated antiferromagnet in the honeycomb lattice. We have made a diagrammatic expansion for the Green's function and we have not gotten significative change in the results.
1. Introduction A general theory of relativistic hydrodynamics of the electronic plasma of the graphene has been derived since some time ago [1,2]. It is often said that the Dirac fluid in graphene is a quantum critical system in two spatial dimensions, and exhibits behavior analogous to the quantum critical regime at finite temperatures above the superfluid-insulator transition with some technical differences [2]. From a general way, the graphene is an allotropic form of carbon which is a lot researched recently. It has semiconductor properties with low-lying excitations obeying the massless Dirac's equation [3]. The interplay between the antiferromagnetic and Kekule valence bond solid ordering in the zero energy levels of neutral monolayer and bilayer graphene has been studied in Ref. [4]. For electronic transport in matter, strong interactions can lead to a breakdown of Fermi liquid paradigm of coherent quasi-particles scattering of impurities. In such situations, provided that certain conditions are met, the complex microscopic dynamics can be coarse-grained to a hydrodynamics description of momentum, energy, and charge transport on long length and time scales [2,5]. The transport properties in graphene has been studied a lot in the literature using the Boltzmann's equation formalism [6,7]. Graphene is an interesting material for spintronics, showing long spin relaxation lengths even at room temperature. For future spintronic devices it is important to understand the behavior of the spins and the limitations for the spin transport in structures where the dimensions are smaller than the spin relaxation length [8]. The electron spin lifetime in carbon materials is expected to be very long because of the very large natural abundance of the spinless nuclear isotope 12C and the small size of spin orbit coupling. This even led to propose graphene as an optimal material to store quantum information in the spin of confined electrons. However, most of the experiments show that the spin
lifetimes are in the range of nanoseconds, much shorter than expected from these considerations, which lies at the heart of the design of devices where graphene is used as a passive component to carry spin currents. The Kubo formalism to calculate of transport coefficients has been many employed in the literature to study the spin transport properties by ions in the lattice in magnetic spin systems. For example, Sentef et al. [9] has analyzed the spin transport in the easy-axis Heisenberg antiferromagnetic model in two and three dimensions, at T=0. Damle and Sachdev [10] treated the two-dimensional case using the nonlinear sigma model in the gapped phase. Pires and Lima [11–13] treated the one and two-dimensional easy plane Heisenberg antiferromagnetic model. Lima and Pires [14] studied the spin transport in the two-dimensional anisotropic XY model using the SU(3) Schwinger boson theory in the absence of impurities, Lima [15] studied the case of the Heisenberg antiferromagnetic model in one and two dimensions with Dzyaloshinskii-Moriya interaction. Zewei Chen et. al. [16] analyzed the effect of spatial and spin anisotropy on spin conductivity for the S = 1/2 Heisenberg model on a square lattice and more recently, Lima at.al has studied the spin transport in the two-dimensional frustrated Heisenberg model in the easy plane using the self consistent harmonic approximation and the SU(3) Schwinger boson theory [17– 23]. Recently, The electron transport properties of zigzag graphene nanoribbons with upright standing carbon chains were investigated using first-principles calculations. The calculated results show a significant odd-even dependence [24]. From an experimental point of view, there is an intense research about spin transport by electrons in graphene where it has been investigated the quantum Hall effect for spins and the magnon spintronics [25–29]. In the studies of these effects often only the sign differences between related quantities like magnetic fields and generated spin and charge currents are deter-
http://dx.doi.org/10.1016/j.physb.2016.12.001 Received 2 June 2016; Received in revised form 10 October 2016; Accepted 1 December 2016 Available online 02 December 2016 0921-4526/ © 2016 Elsevier B.V. All rights reserved.
Physica B 507 (2017) 164–169
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→ → ω ( k ) ≈ =vF | k |,
mined. The spin injection and transport in single layer graphene can be investigated using nonlocal magnetoresistance (MR) measurements [30]. The aim of this paper is to verify the similarity between the electron transport in the graphene monolayer using Dirac fermions formalism with the spin transport in quantum frustrated two-dimensional antiferromagnet in the honeycomb lattice. The graphene consist in a fermion system with a relativistic Dirac spectrum where the energy vanishes linearly at isolated points in the Brillouin zone. Dirac's fermions are provided by numerous new experimental realizations. These include D-wave superconductors and topological insulators [31]. This work is divided in the following way. In Section 2, we discuss the Kubo's formalism for electron transport, in Section 3 we present the results into a mean field approximation, in Section 4 we treat the interactions electron-electron and exciton-exciton respectively using the diagrammatic expansion for the Green's function and in the last Section 5, we present our conclusions and final remarks.
where in the massless (gapless) limit, the spectrum is linear hence, the time and space scale the same way T ∼ L as required by relativistic invariance [32]. There are local interactions such as (ψ γμ ψ )2 and (ψ ψ ), where ψ = ψ †γ 0 . The action of the free Dirac's field is
S=
→⎞ ⎛ ⎞ ⎛ γ 0 = −i ⎜ 0 1 ⎟ → γ = −i ⎜ 0→ − σ ⎟ . ⎝1 0⎠ ⎝− σ 0 ⎠
{γμ, γν} = 2gμν ,
∫ d 2x (ψ (x ) ψ (x ))2
(5)
γ5 = iγ0 γ1.
(6)
gμν is the Minkowski's metric tensor ⎛1 0 0 0 ⎞ ⎜ 0 ⎟. gμν = ⎜ 0 − 1 0 ⎟ ⎜0 0 − 1 0 ⎟ ⎝0 0 0 − 1⎠
(7)
In the theory of free massless Dirac's fermions there are a fixed point in the renormalization group [32]. 3. Mean field approximation
(1)
where ψα (x ) with α , β = 1, 2 denote the two components of the Fermi field in D = 1 + 1 and D = 2 + 1 space dimensions and vF is the Fermi's velocity. We have considered the unities where vF=1 and = = 1. The interaction term has the form up to the irrelevant additive constant
/ int = −2Jγ
(4)
1 is the unit 2×2 matrix, and σ are the components of Pauli's matrices where the Dirac γ-matrix, γ0, γ1 and γ5 satisfy the algebra
The model of relativistic free-fermion particles of the graphene in D = 1 + 1 dimension is described by the following Hamiltonian density [32]
∫ d 2xψα† (x ) iσ3αβ ∂xψβ (x ),
∫ d 2xψα (x ) iγαβμ ∂μψβ (x )
where γ are the 4×4 Dirac's matrix.
2. The Dirac fermion model
/ = JvF
(3)
We use the low energy approach Dirac fermion [6,32] to determine the regular part of the electron conductivity (AC conductivity) or continuum conductivity. An electron current appears if there is an ⎯→ ⎯ electric field given by the Ohm' Law 1 = σ E . In a similar way, a spin ⎯→ ⎯ current appears as a response to a magnetic field 1S = σ ∇ B , through the system, where it plays the role of a chemical potential for spins. If one connect a low dimensional magnet with two bulk ferromagnetic, they can act as reservoirs for spins [28,29]. Then, one has a flow of a ⎯→ ⎯ spin current if there is a difference, Δ B , between the magnetic fields at the two ends of the sample. In the Kubo formalism [33,34,9,11] the conductivity is given by:
(2)
which is the interaction term of the (1+1)-dimensional Gross-Neveu→ ⎛ 1 1⎞ , ⎟, model. The model is depicted in Fig. 1, where d1 = ⎜ ⎝2 3 2⎠ ⎞ → ⎛ 1 → → → → → ⎛ 1 1⎞ , − 2 ⎟. and → a2 = d3 − d1 and d2 = ⎜ − , 0⎟, d3 = ⎜ a1 = d2 − d3, → ⎠ ⎝2 3 ⎝ 3 ⎠ → → → a3 = d1 − d2 . Assuming that electrons in graphene are non interacting, the standard band theory calculations on the honeycomb lattice in two spatial dimensions with nearest-neighbor hopping give us two species of Dirac fermions with single-particle spectrum in the relativistic form [5]
→ 〈2 〉 + Λ ( k , ω) , i (ω + i 0+) k →0
σ (ω) = →lim
(8) q , ω) is the current-current where 〈2 〉 is the kinetic energy and Λ (→ correlation function defined by ∞ → → → i Λ ( k , ω) = dteiωt 〈[1 ( k , t ), 1 (− k , 0)]〉. (9) =N 0 → + Λ ( k , ω + i 0 ) is analytic in the upper half of the complex plane and the extrapolation along the imaginary axis can be reliably done. The current operator for graphene is given by
∫
1 = ψ γμ ψ .
(10)
The real part of σ (ω), σ ′(ω), can be written in a standard form as [34]
σ ′(ω) = σ0 (ω) + σ reg (ω),
(11)
where σ0 (ω) is the DC contribution, given by σ0 (ω) = DS δ (ω). DS is the Drude's weight
→ → DS = π [〈2 〉 + Λ′( k = 0, ω → 0 )].
(12)
σ reg (ω), the regular part of σ ′(ω), is given as [34] σ reg (ω) = → → → Fig. 1. The 2D honeycomb lattice of graphene. d1, d2 , d3 connect the two triangular → → → sublattices and a1, a2 and a3 are the next-nearest-neighbor displacement vectors.
→ Λ″( k = 0, ω) . ω
(13)
It represents the AC contribution to the conductivity. In Eqs. (12) and (13), Λ′ and Λ″ stand for the real and imaginary part of Λ. 165
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Fig. 3. Representation of the flow of a spin current in the x direction in the twodimensional anisotropic frustrated Heisenberg model on a honeycomb lattice with the second neighbors and third neighbors, J1, J2 and J3.
The anisotropic two-dimensional antiferromagnet in the honeycomb lattice is depicted in Fig. 3 and is defined by the following Hamiltonian Fig. 2. Behavior of σ* for the value of T = 1.0 J . We have that σ reg (ω) =
(gμb )2 σ* (ω) . =
/ = How
ij
the Ac conductivity tends to infinity in the limit ω → 0 , therefore we must have an ideal conductor in this limit.
∑∫ α
π
0
d 2k 2 0 ∼0 vF Gα (ω1) Gα (ω + ω1) (2π )2
(14)
as
Gα0 (ω) =
1 , iωn − ωk
∼0 Gα (ω) =
−1 . iωn + ωk
(15)
From the analytical continuation, we obtain the regular part of the conductivity as
σ reg (ω) = (gμB )2
π
∫0 ∫0
π
d 2k vF2 f (ωk )[1 − f (ωk )] δ (ω − 2ωk ), (2π )2 ωk
(16)
1/(e βωk
+ 1) is the Fermi-Dirac distribution. We have where f (ωk ) = solved the Eq. (19) analytically as
2π 2 σ* (ω) = f (ω /2)[1 − f (ω /2)]. ω
(18)
i
where Jij = J1 for the nearest neighbors interaction, Jij = J2 means second neighbors interaction and Jij = J3 for third neighbors interaction. For values of anisotropy sufficiently strong D, the system is a quantum paramagnet [36,37]. The anisotropy forces each spin to stay in the nonmagnetic state. The ground state has no long-range magnetic order and there are finite gap spin excitations. Decreasing D, the energy of the gap decreases and goes to zero at a critical Dc, where a quantum phase transition takes place. For D ≤ Dc and positive, the system is in a gapless phase, that is ordered at T=0 in the non frustrated case. The honeycomb lattice is a frustrated spin system where the frustration arises from competing interactions rather than geometric constraints, and the frustration is enhanced by strong quantum fluctuations due to the low coordination number [35,37]. Moreover, experimental data are available for several compounds such as the family of compounds BaM2 (XO4 )2 , where M=Co, Ni and X=P. The magnetic ions M are arranged in weakly coupled frustrated honeycomb lattice. However, most of the compounds lies in the Néel ordered region. The model with J1, J2 and J3 interactions was studied recently by D.C. Cabra et al. [38,39] or S=1/2 along the line J1 = J3 using the linear spin wave theory (LSWT) and SU(2) mean field Schwinger boson approximation. J. B. Fouet [40] studied the same model using exact diagonalization and LSWT, Li et al. [41] treated the same system using the coupled cluster method. The spin superfluidity is a phenomenon arising due to spontaneous breaking of the U (1) symmetry, which in a spin system case is represented by the symmetry group SU (2) of spin rotations about direction of the magnetic field. The Bose-Einstein condensation (BEC) of quasiparticles such as magnons and excitons whose number is not conserved is presently one of the most debated phenomena of condensed matter physics [43]. In atomic BEC and helium superfluids the symmetry breaking leads to a non-zero value of the superfluid rigidity where the superfluid density ρS which enters the non-dissipative supercurrent of particles and thus to magnon supercurrent. Using the Matsubara's Green's function method and the SU(3) Schwinger boson theory [36,44], we obtain the spin current σ (ω) for this model at T=0 given as
Using the Matsubara's method, we obtain the Green's function at finite temperature for the model Eq. (18) as
Gj (ω) =
∑ Jij Si·Sj + D ∑ (Siz )2 .
(17)
In Fig. 2, we present the behavior of σ* (ω) with ω. Since the system is gapless we have obtained a behavior of the AC conductivity tending to infinity when ω → 0 , that correspond to the DC limit. However, the fact the electric conductivity tending to infinity, is not taken as the true definition of superconductivity. The fundamental proof that superconductivity occurs in a given material is the demonstration of the Meissner-Ochsenfeld effect. This effect is the fact that metals in the superconductor state are perfect diamagnets and hence expels a weak external magnetic field. We find the behavior of the electric conductivity in the graphene gotten is similar to one recently obtained for the spin transport in the quantum two-dimensional Heisenberg antiferromagnet in the honeycomb lattice [22]. This behavior for the spin transport in magnetic spin systems is a characteristics of systems that does not present gap in the excitation spectrum. As in the Dirac's fermion model of the graphene there is not gap in the spectrum, we will not have electrons to form one current for all values of ω of the spectrum, in the same way as happen in the frustrated antiferromagnet in the honeycomb lattice. Moreover, as there is not scattering mechanism, as in this model, it is expected that the conductivity to be divergent. The reason is that spin-spin scattering is not treated properly in a mean field approach.
σ reg (ω) =
Λ (k = 0, ω) π = (gμB )2 3 ω ω
π
∫0 ∫0
π
d 2k 2 σαα (2π )2 2
[sin kx + η sin( 3 kx ) + α sin(2 2 kx )] × [δ (ω − ω1k ) + δ (ω − ω 2k )],
(19)
where σαβ is a matrix [37]. We can integrate analytically the Eq. (19) to 166
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q∼ = (iqn , → q ) flowing out from it to the left. We can draw diagrams for the current-current correlation function using the Feynman's rules. → → → → G ( k , t , k′ , 0) = −〈; [ψ ( k , τ ) ψ ( k ′, 0)]〉,
(25)
where τ = it . ; is the time ordering operator. The Green's function can be expanded as [45,46]
→ → G ( k , t , k′ , 0) =
∞
∑∫ n =0
∑
0
β
dτ1 ⋯
∫0
β
dτn
〈0 Tr [C † (τ1) C (τ1)⋯C † (τn ) C (τn )] 0〉 (26)
t1⋯ tn
where Cc = ∑k αk . We can collect all terms in the expansion of the Green's function as
→ → → G ( k , t , k′ , 0) = G (0) ( k , c, τ − τ′) δk , k ′ δσ , σ ′ +
∫0
β
dτ1
∫0
β
→ dτ2 G (0) ( k , τ
→ → → − τ1) Σ ( k , τ1, k′ , τ2 ) G (0) ( k , τ2 − τ′),
(27)
where τ = it is the imaginary time and β = 1/ T . with the solution
→ G ( k , ik n ) =
1 , → ik n − ζk − Σ ( k , ik n ) (28) → where the self-energy Σ ( k , ik n ) appears as a direct additive renormalization of the bare energy ζk = εk − μ, and μ is the chemical potential. The Feynman's rules therefore, lead to the following expression for the self-energy diagram.
Fig. 4. Behavior of σ* (ω) for the following values: Dc=0, η = 0.4 , α = 0.11 (dashed line). For large value of Dc, Dc=8, η = 0.4 , α = 0.6 α = 0.8 (dotted line) and next to the quantum the phase transition point: Dc=0.35 and η = 0.25 and α = 0.6 (solid line). The behavior of the spin conductivity varies little with η, α parameters. We have
σ reg (ω) =
(gμb )2 σ* (ω) . =
How we obtain the AC conductivity tending to infinity in the limit
ω → 0 thus, we have an ideal spin conductor in the DC limit.
obtain
η2 3 π ⎡1 2⎤ ⎥, σ* (ω) = 3 ⎢ − η sin 3 + α sin 2 2 + + α2 ω ⎣4 12 16 ⎦
where W gives the electron-electron interaction (20)
W=
and
σ reg (ω) =
(gμB =
)2
3J12 δγγ ′ → → Σ (2) ( k γτ1, k′ γ′τ2 ) = 16
∑ k1
1 . ik n − ζk1
(32)
3J12 π [d γ (ζk )] δγγ ′ → → . −IΣ R (2) ( k γ , k′ γ′, ik n ) = 16
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
(22)
(33)
1 β
β
k1, k 2
0
β
→ → dτ3 G (0) ( k1 γ1, τ1 − τ3) G (0) ( k2 γ2, τ3) (34)
We proceed by performing the τ3 integral followed by a Fourier transformation and the result is
∑ 1 ∑ 〈;1 (→ q , iql ) iql
∑∫
sgn (τ1 − τ3),
In the frequency domain it is
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
iqm
(23)
∑ k1, k 2
1 + 2n (ζk ) 1 . ζk1 − ζk 2 ik n − ζk1
(35)
In this case, the imaginary part of the retarded self-energy is given
We conveniently rewrite it using the four-vector notation [45] q∼ = (iqn , → q ).
1 〈1 (q∼) 1 (−q∼)〉. βN2
(31)
The third-order contribution reads
1 Πxx (→ q , τ − τ′) = − 2 〈;1 (→ q , τ ) 1 (−→ q , τ′)〉. N
Πxx (q∼) −
k1
The important information about the scattering mechanism lies in the imaginary part of the retarded self-energy
The retarded Matsubara current-current correlation function is
1 (−→ q , iqm )〉e−iql τ e−iqm τ ′.
→
∑ G(0) ( k1 γ1, τ1 − τ2).
3J12 δγγ ′ → → Σ (2) ( k γ , k′ γ′, ik n ) = 16
4.1. Electron-electron interaction
d (τ − τ′) eiqn (τ − τ ′)
(30)
1)−1.
In the frequency domain it becomes
4. Vertex corrections for the Green's function
∫0
ψν1 ψν2 ψν4 ψν3,
+ f (ζk ) is the Fermi-Dirac distribution f (ζk ) = The second-order term of the self-energy is given by
(21)
In Fig. 4, we present the behavior of σ reg (ω) with ω. Due to the hexagonal crystal lattice, we must have a spin current in zig-zag over the material as depicted in Fig. 3. Since the system is gapless we have obtained a behavior of the AC spin conductivity tending to infinity when ω → 0 , that corresponds to the DC limit. Consequently, we obtain a superconductor behavior for the spin transport in the DC limit. This behavior is similar one recently obtained for the two-dimensional ferroquadrupolar model [21] and to triangular lattice [23]. This is a characteristics of spin systems without gap in the excitation spectrum.
1 Πxx (→ q , iqn ) = − 2 N
∑ ν1ν2 ν3 ν4
(e βζk
2 σαα σ (ω). 4π 2 *
β
1 2
by
3πJ1 [J1 d γ (ζk )]2 δγγ ′ → → −IΣ (3) ( k γ , k′ γ′, ik n ) = 16
(24)
D
∫−D 1 +ω 2−nζ(ζ ) dζ,
(36)
where D is a specified cut-off. This is logarithmically divergent when
The vertex conserves four-momentum, and thus has a momentum 167
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L.S. Lima
→ → → G ( k , t , k′ , 0) = G (0) ( k , c, τ − τ′) δk , k ′ δσ , σ ′ +
the temperature or the frequency goes to zero. Collecting the second and third order terms, one have for the low temperature case
⎡ ⎛ 3δγγ ′ [d γ (ζk )]2 π ⎢ → → D 1 + 2d γ (ζk )ln ⎜⎜ −IΣ (R) ( k γ , k′ γ , ω) ≈ ⎢⎣ 16 ⎝ ω 2 + (kB T )2
1 β
∑ ikn
1 N2
(37)
→ → → → → → 1 = R ∑ Γ0, x ( k , k )[GA ( k , 0) GR ( k , 0) ΓxRA ( k , k ; 0, 0)]. πN2 → k
W=
, → ik n − ζk − Σ ( k , ik n )
1 2
2
= 2W
ikn
∫ (2dπk)2 n (ζk )
(45)
ψν†1 ψν†2 ψν4 ψν3,
∑
(46)
ν1ν2 ν3 ν4
3J12 δγγ ′ → → Σ (2) ( k γτ1, k′ γ′τ2 ) = 16
(38)
1)−1.
→
∑ G(0) ( k1 γ1, τ1 − τ2). → k1
(47)
In the frequency domain it becomes
3J12 δγγ ′ → → Σ (2) ( k γ , k′ γ′, ik n ) = 16
(39)
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
∑ → k1
1 ik n − ζk1
(48)
3J13 δγγ ′ → → Σ (3) ( k γτ1, k′ γ′τ2 ) = − 16
⎛ 1 1 ⎞ † + ⎜ ⎟ ψ σαα ψk + q. ⎝ ω1k ω 2k ⎠ k + q
n =0
∑ t1⋯ tn
0
dτ1 ⋯
∫0
1 + 2n (ζk ) 1 . ζk1 − ζk 2 ik n − ζk1
(50)
D
∫−D 1 +ω 2−nζ(ζ ) dζ,
1 β
∑ ikn
1 N2
(51)
→ → → k ) G ( k , ik n )
∑ Γ0,x ( k , → k
→ → → G ( k , ik n + iqn ) Γx ( k , k ; ik n + iqn , ik n ),
(52)
The Matsubara sum over ikn is performed in the usual way by a contour integration over z = ik n . The final expression for the real part of the spin conductivity is given by reg σxx =
(41)
β
∑
→→ k1, k 2
where D is a specified cut-off [45]. We obtain the current-current function in this case, given as
→ → → → → → 1 R ∑ Γ0, x ( k , k )[GA ( k , 0) GR ( k , 0) ΓxRA ( k , k ; 0, 0)]. πN2 → k
The Green's function can be expanded in this case as [45,46] ∞
(49)
3πJ1 [J1 d γ (ζk )]2 δγγ ′ → → −IΣ (3) ( k γ , k′ γ′, ik n ) = 16
3 (kx + qx )) + α sin(2 2 (kx + qx ))]
∑ (−J1)n ∫
→ → dτ3 G (0) ( k1 γ1, τ1 − τ3) G (0) ( k2 γ2, τ3)
by
In this case we have the spin current operator 1x (q∼) in the fourvector notation as
→ k
β
In this case, the imaginary part of the retarded self-energy is given
Πxx (0, iqn ) = −
∑ [sin kx + η sin(
∑∫ 0
→→ k1, k 2
Proceeding further by performing the τ3 integral followed by a Fourier transformation, the result is
(40)
4.2. Exciton-Exciton interaction
→ → G ( k , t , k′ , 0) =
ikn η
− and n (ζk ) is the Bose-Einstein distribution n (ζk ) = The second-order contribution for the self-energy is
→ where τ0* is the renormalized lifetime and A ( k , 0) = Zδ (ζk ). Z is called the renormalization constant and it is a measure of the quasi-particle reg = Cτ0*, where C is another weight. Performing the integral we have σxx renormalized constant. Finally we have for the renormalized lifetime [τ0*]−1 = Z [τ0]−1. We define ballistic transport as transport where the mean free path ls of the electrons is limited by the sample dimensions. The electronelectron interaction introduces so a finite mean free that if related to the relaxation time by ls = vF τ0 , where vF is the Fermi velocity.
J1 2
2
∫ (2dπk)2 ∑ ikne − ζk
sgn (τ1 − τ3).
k
1=−
(44)
and the third-order contribution reads
→ 1 k2 R ∑ 2 τ0* A ( k , 0), πN2 m →
t2
(43)
(e βζk
At low temperatures it is valid to assume that the imaginary part of the → self-energy due to electron-electron interaction IΣ ( k ) is small. Hence, we have in this limit reg σxx =
→ dτ2 G (0) ( k , τ
where W now gives the exciton-exciton interaction
The Matsubara sum over ikn is performed in the usual way by a contour integration over z = ik n . The final expression for the real part of the spin conductivity is given then by reg σxx
β
1
→ 2W Σ ( k , ik n ) = β
→ → → k ) G ( k , ik n )
→ → → G ( k , ik n + iqn ) Γx ( k , k ; ik n + iqn , ik n ),
∫0
The Feynman's rules, in this case, lead to the following expression for the self-energy diagram
∑ Γ0,x ( k , → k
dτ1
The solution is
→ G ( k , ik n ) =
From this result, we have an enhanced scattering in third order of the exchange coupling [45]. Having the expressions for both the single-particle Green's function and the vertex function Γ, we can obtain a general formula for the conductivity making a summation over the internal Matsubara frequency. If we drop the four-vector notation in favor of the standard q = 0 , the current-current notation, and furthermore treat the case → function is
Πxx (0, iqn ) = −
β
→ → → − τ1) Σ ( k , τ1, k′ , τ2 ) G (0) ( k , τ2 − τ′).
⎞ ⎟ ⎟ ⎠
⎤ + ⋯⎥ . ⎥⎦
∫0
(53)
At low temperatures it is valid to assume that the imaginary part of the → self-energy, due to exciton-exciton interaction IΣ ( k ), is small. Therefore, we have in this limit
β
dτn
reg σxx =
〈0 Tr [C † (τ1) C (τ1)⋯C † (τn ) C (τn )] 0〉
→ 1 k2 R ∑ 2 τ0* A ( k , 0), πN2 → m
(54) → where τ0* is again the renormalized lifetime and A ( k , 0) = Zδ (ζk ). Z is called the renormalization constant and it gives a measure of the quasi-
(42)
k
where Cc = ∑k αk . We can collect all terms in the expansion of the Green's function as 168
Physica B 507 (2017) 164–169
L.S. Lima
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reg = Cτ0*, where C is particle weight. Performing the integral we have σxx another renormalized constant. Finally we have for the renormalized lifetime [τ0*]−1 = Z [τ0]−1. It is well known that magnetic defects are identified as one of the important sources for the scattering of magnetic excitations. The experiments also suggest that spin-phonon scattering is very effective in reducing the energy flow, especially at high temperatures [47]. We can define the ballistic spin transport as the transport where the mean free path ls of the magnetic excitations is limited by the sample dimensions. The exciton-exciton interaction introduces so a finite mean free that if related to the relaxation time by ls = vs τ0 , where vs is the velocity of the magnetic excitations.
5. Conclusions and final remarks In summary, we have verified one similarity between the transport properties in two different quantum models. the Dirac's fermion model of the graphene monolayer, which is a quantum paramagnet and the quantum frustrated two-dimensional antiferromagnet in the honeycomb lattice. We have obtained the same behavior for the electric conductivity of the graphene and the spin conductivity of the anisotropic two-dimensional Heisenberg antiferromagnet in the honeycomb lattice. Both presenting the AC conductivity tending to infinity in the DC limit (ω → 0). However, due to the Meissner-Ochsenfeld effect, the fact the electric conductivity tends to infinity, is not taken as the true definition of superconductivity. This effect is the fact that a superconductor expels a weak external magnetic field not being valid for the case of spin transport. We have obtained also that the inclusion of the electron-electron and exciton-exciton interactions do not change the results obtained into a mean field approximation. Finally, we have that through the mapping of spin systems in other models such as the Dirac fermion model can be a tentative to measure in a way easier the spin transport in magnetic systems. Acknowledgements This work was partially supported by the Brazilian agencies FAPEMIG and CEFET-MG. References [1] S.A. Hartnoll, P.K. Kovtun, M. Muller, S. Sachdev, Phys. Rev. B 76 (2007) 144502. [2] A. Lucas, J. Crossno, K.C. Fong, P. Kim, S. Sachdev, Phys. Rev. B 93 (2016) 075426. [3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [4] J. Lee, S. Sachdev, Phys. Rev. Lett. 114 (2015) 226801.
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