Galvano- and thermo-magnetic effects at low and high temperatures within non-Markovian quantum Langevin approach
I.B. Abdurakhmanov, G.G. Adamian, N.V. Antonenko, Z. Kanokov
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Galvano- and thermo-magnetic effects at low and hightemperatures within non-Markovian quantum Langevin approach
I.B. Abdurakhmanov , G.G. Adamian , N.V. Antonenko , and Z. Kanokov , , Curtin Institute for Computation, Department of Physics,Astronomy and Medical Radiation Sciences,Curtin University, Perth, WA 6845, Australia Joint Institute for Nuclear Research, 141980 Dubna, Russia National University, 700174 Tashkent, Uzbekistan Institute of Nuclear Physics, 702132 Tashkent, Uzbekistan (Dated: June 6, 2018)
Abstract
The quantum Langevin formalism is used to study the charge carrier transport in a two-dimensional sample. The center of mass of charge carriers is visualized as a quantum particle,while an environment acts as a heat bath coupled to it through the particle-phonon interaction.The dynamics of the charge carriers is limited by the average collision time which takes effectivelyinto account the two-body effects. The functional dependencies of particle-phonon interaction andaverage collision time on the temperature and magnetic field are phenomenologically treated. Thegalvano-magnetic and thermo-magnetic effects in the quantum system appear as the result of thetransitional processes at low temperatures.
PACS numbers: 09.37.-d, 03.40.-a, 03.65.-w, 24.60.-kKeywords: Classical and Quantum Hall effect; Shubnikov-De Haas effect; cyclotron frequency; frictioncoefficients; Langevin formalism; non-Markovian dynamics; electric and magnetic field . INTRODUCTION The behavior of solid matter under the influence of external fields at low temperatureis one of the interesting topics in solid-state physics [1–3]. The external field may be anelectric field, magnetic field, optical signal, or temperature gradient. These fields modifysome electronic properties, such as the carrier concentration and the carrier mobility. Besidesthe carrier mobility, the electric current is also affected by the magnetic field which deflects itsdirection and leads to a nonzero cross voltage (the classical Hall effect) linearly proportionalto the field strength [1–4]. The oscillator nature of the longitudinal magneto-resistance ofbismuth sample at low temperature, known as the Shubnikov-De Haas effect, has been alsoobserved in the presence of very intense magnetic fields [1, 3, 5, 6]. The effect is morepronounced at low temperatures where the amplitude of oscillations is significantly larger.The first experimental study of the influence of electric fields of the order of 100 mV cm − onthe Shubnikov-de Haas magneto-resistance oscillations in n -InSb sample has been reportedin Ref. [5]. In addition to the shift of the extremes to higher magnetic fields, a decreaseof oscillation amplitudes with electric field has been observed. The Integer Quantum HallEffect (IQHE) in the GaAs-Al . Ga . As heterostructure has been discovered [7, 8] at strongexternal electromagnetic fields and very low temperature. The quantization of conductivitysurprisingly occurred in a certain two-dimensional electron gas (2DEG) under the influenceof a strong magnetic field. In the IQHE, the Hall conductance σ xy has a stepwise dependence(the appearance of plateau) on the strong external magnetic field. At these plateau, σ xy isquantized as σ xy = ie / (2 π ~ ), i = 1 , , ... , while the longitudinal conductivity σ xx nearlyvanishes. The vanishing σ xx implies the absence of dissipation. This is another hallmark ofthe IQHE. Also, the Fractional Quantum Hall Effect (FQHE), where the Hall conductivityis quantized in fractional multiples of e / (2 π ~ ), has been discovered [9, 10].The theoretical models in Refs. [11–15] for describing the IQHE and FQHE have beendeveloped. The combination of a random potential created by impurities in a sample andstrong magnetic field gives rise to the special coexistence of localized and extended electronstates. As known, the Fermi level lies in the energy gap (mobility gap) free from the extendedstates and the change of the electron density or the magnetic field can only result in differentoccupations of localized states which do not affect the conductivity. Based on these findingsthe appearance of the conductivity quantization has been explained. The general approach,2hich explains the quantization as well as the integer quantized values, has been developedlater with the scaling theory. This approach also describes correctly the regions wherethe conductivity is not quantized [11]. For the theoretical explanation of the FQHE, thewave functions have been introduced [12] to describe the incompressible quantum states andexplain the small but experimentally prominent class of fractions 1 / odd. It turned out thatthe quasiparticle excitations are the charge/fluxcomposites with a fractional charge andstatistics, also known as dubbed anions. The special properties of charge/fluxcompositeshave been used in Refs. [13–15] to construct two so-called hierarchies, sets of Hall fractionsfor which the incompressible ground states could be found. These hierarchies are ableto reproduce all fractions observed, but also yield many fractions that have never beenmeasured. The striking universality in the manifestation of the quantum Hall effect attractedlarge attention, not only in solid state physics but also in high energy physics. The extensivemathematical methods of topological field theory [16] and infinite dimensional algebras [17–19] have been applied to the IQHE and FQHE. As found in several independent works, thedescription of incompressible quantum states exploits the theory of chiral edge currents [17].The Quantum Hall ground states and quasiparticle excitations have been described in termsof representations of the infinite-dimensional algebras [17–19].The aim of the present work is to treat the classic and quantum Hall effect as well asthe Shubnikov-De Haas effect within the same model. The basic idea of our model is thefollowing. In the electric current, we determine the time-dependent number of electrons withgiven momentum at a certain location. We consider the center of mass of charge carrierswith a positive charge e = | e | as a quantum particle coupled to the environment (heat bath)through the particle-phonon interactions. Solving the second order Heisenberg equations forthe heat bath degrees of freedom, the generalized non-Markovian Langevin equations areexplicitly obtained for a quantum particle. The memory effects in these equations resultsfrom the coupling to the environment. The dynamics of the charge carriers is restrictedby the average collision time. The functional form of the particle-phonon coupling strengthand the average collision time on the temperature and magnetic field are phenomenologicallytreated.The paper is organized as follows. In Sec. II, we introduce the Hamiltonian of the systemand solve the generalized non-Markovian Langevin equations for a quantum particle. Theelectric and thermal conductivities are derived in two-dimensional systems. Note that the3uantum Langevin approach or the density matrix formalism has been widely applied tofind the effects of fluctuations and dissipation in macroscopic systems [20–47]. The mainassumptions of the model are discussed. The model developed is used in Sec. III to describethe experimental data on the classic and quantum Hall and Shubnikov-De Haas effects. Asummary is given in Sec. IV. II. NON-MARKOVIAN LANGEVIN EQUATIONS WITH EXTERNAL MAG-NETIC AND ELECTRIC FIELDSA. Derivation of quantum Langevin equations
Let us consider two-dimensional motion of a quantum charge particle in the presence ofheat bath and external constant electric E = ( E x , ,
0) and magnetic fields B = (0 , , B ).The total Hamiltonian of this system is H = H c + H b + H cb . (1)The Hamiltonian H c describes the collective subsystem (quantum particle) with effectivemass tensor and charge e = | e | in electric and magnetic fields: H c = 12 m x [ p x − eA x ( x, y )] + 12 m y [ p y − eA y ( x, y )] + eE x x = π x m x + π y m y + eE x x. (2)Here, m x and m y are the components of the effective mass tensor, R = ( x, y,
0) and p =( p x , p y ,
0) are the coordinate and canonically conjugated momentum, respectively, A =( − yB, xB,
0) is the vector potential of the magnetic field, and the electric field E x actsin x direction. For simplicity, in Eq. (2) we introduce the notations π x = p x + 12 m x ω cx y, π y = p y − m y ω cy x with frequencies ω cx = eBm x and ω cy = eBm y . The cyclotron frequency is ω c = √ ω cx ω cy = eB √ m x m y .The second term in Eq. (1) represents the Hamiltonian of the phonon heat bath H b = X ν ~ ω ν b + ν b ν , (3)where b + ν and b ν are the phonon creation and annihilation operators of the heat bath. Thecoupling between the heat bath and collective subsystem is described by H cb = X ν V ν ( R )( b + ν + b ν ) + X ν ~ ω ν V ν ( R ) . (4)4he first term in Eq. (4) corresponds to the energy exchange between the collective sub-system and heat bath. We introduce the counterterm (second term) in H cb to compensatethe coupling-induced renormalization of the collective potential. Naturally, it can be alwaysseparated from eE x x in Eq. (2). In general case, V ν ( R ) depends on the strength of magneticfield and an impact of B is entered into the dissipative kernels and random forces.The equations of two-dimensional motion are˙ x ( t ) = i ~ [ H, x ] = π x ( t ) m x , ˙ y ( t ) = i ~ [ H, y ] = π y ( t ) m y , ˙ π x ( t ) = i ~ [ H, π x ] = π y ( t ) ω cy − eE x − X ν V ′ ν,x ( R )( b + ν + b ν ) − X ν V ν ( R ) V ′ ν,x ( R ) ~ ω ν , ˙ π y ( t ) = i ~ [ H, π y ] = − π x ( t ) ω cx − X ν V ′ ν,y ( R )( b + ν + b ν ) − X ν V ν ( R ) V ′ ν,y ( R ) ~ ω ν , (5)and ˙ b + ν ( t ) = i ~ [ H, b + ν ] = iω ν b + ν ( t ) + i ~ V ν ( R ) , ˙ b ν ( t ) = i ~ [ H, b ν ] = − iω ν b ν ( t ) − i ~ V ν ( R ) . (6)The solution of Eqs. (6) are b + ν ( t ) + b ν ( t ) = f + ν ( t ) + f ν ( t ) − V ν ( R ) ~ ω ν + 2 ~ ω ν t Z dτ ˙ V ν ( R ( τ )) cos( ω ν [ t − τ ]) ,b + ν ( t ) − b ν ( t ) = f + ν ( t ) − f ν ( t ) + 2 i ~ ω ν t Z dτ ˙ V ν ( R ( τ )) sin( ω ν [ t − τ ]) , (7)where f ν ( t ) = [ b ν (0) + 1 ~ ω ν V ν ( R (0))] e − iω ν t . Substituting (7) into (5) and eliminating the bath variables from the equations of motionfor the collective subsystem, we obtain the set of nonlinear integro-differential stochasticdissipative equations˙ x ( t ) = π x ( t ) m x , ˙ y ( t ) = π y ( t ) m y , ˙ π x ( t ) = π y ( t ) ω cy − eE x − t Z dτ { K xx ( t, τ ) , ˙ x ( τ ) } + − t Z dτ { K xy ( t, τ ) , ˙ x ( τ ) } + + F x ( t ) , ˙ π y ( t ) = − π x ( t ) ω cx − t Z dτ { K yy ( t, τ ) , ˙ y ( τ ) } + − t Z dτ { K yx ( t, τ ) , ˙ y ( τ ) } + + F y ( t ) . (8)5he dissipative kernels and random forces in (8) are K xx ( t, τ ) = X ν ~ ω ν { V ′ ν,x ( R ( t )) , V ′ ν,x ( R ( τ )) } + cos( ω ν [ t − τ ]) ,K xy ( t, τ ) = X ν ~ ω ν { V ′ ν,x ( R ( t )) , V ′ ν,y ( R ( τ )) } + cos( ω ν [ t − τ ]) ,K yx ( t, τ ) = X ν ~ ω ν { V ′ ν,y ( R ( t )) , V ′ ν,x ( R ( τ )) } + cos( ω ν [ t − τ ]) ,K yy ( t, τ ) = X ν ~ ω ν { V ′ ν,y ( R ( t )) , V ′ ν,y ( R ( τ )) } + cos( ω ν [ t − τ ]) (9)and F x ( t ) = X ν F νx ( t ) = − X ν V ′ ν,x ( R ( t ))[ f + ν ( t ) + f ν ( t )] ,F y ( t ) = X ν F νy ( t ) = − X ν V ′ ν,y ( R ( t ))[ f + ν ( t ) + f ν ( t )] , (10)respectively. Here, we use the notations: V ′ ν,x = ∂V ν /∂x , V ′ ν,y = ∂V ν /∂y , and { Z , Z } + = Z Z + Z Z . Following the usual procedure of statistical mechanics, we identify the oper-ators F νx and F νy as fluctuations because of the uncertainty of the initial conditions for thebath operators. To specify the statistical properties of fluctuations, we consider an ensembleof initial states in which the fluctuations have the Gaussian distribution with zero averagevalue ≪ F νx ( t ) ≫ = ≪ F νy ( t ) ≫ = 0 . (11)The symbol ≪ ... ≫ denotes the average over the bath with the Bose-Einstein statistics ≪ f + ν ( t ) f + ν ′ ( t ′ ) ≫ = ≪ f ν ( t ) f ν ′ ( t ′ ) ≫ = 0 , ≪ f + ν ( t ) f ν ′ ( t ′ ) ≫ = δ ν,ν ′ n ν e iω ν [ t − t ′ ] , ≪ f ν ( t ) f + ν ′ ( t ′ ) ≫ = δ ν,ν ′ ( n ν + 1) e − iω ν [ t − t ′ ] , (12)where the occupation numbers n ν = [exp( ~ ω ν /T ) − − for phonons depend on temperature T given in energy units.Using the properties (11) and (12) of random forces, we obtain the following symmetrized6orrelation functions ϕ νkk ′ ( t, t ′ ) = ≪ F νk ( t ) F νk ′ ( t ′ ) + F νk ′ ( t ′ ) F νk ( t ) ≫ ( k, k ′ = x, y ): ϕ νxx ( t, t ′ ) = [2 n ν + 1] { V ′ ν,x ( R ( t )) , V ′ ν,x ( R ( t ′ )) } + cos( ω ν [ t − t ′ ]) ,ϕ νyy ( t, t ′ ) = ϕ νxx ( t, t ′ ) | x → y ,ϕ νxy ( t, t ′ ) = [2 n ν + 1] { V ′ ν,x ( R ( t )) , V ′ ν,y ( R ( t ′ )) } + cos( ω ν [ t − t ′ ]) ,ϕ νyx ( t, t ′ ) = ϕ νxy ( t, t ′ ) | x → y . (13)The quantum fluctuation-dissipation relations read X ν ϕ νxx ( t, t ′ ) tanh[ ~ ω ν T ] ~ ω ν = K xx ( t, t ′ ) , X ν ϕ νyy ( t, t ′ ) tanh[ ~ ω ν T ] ~ ω ν = K yy ( t, t ′ ) , X ν ϕ νxy ( t, t ′ ) tanh[ ~ ω ν T ] ~ ω ν = K xy ( t, t ′ ) , X ν ϕ νyx ( t, t ′ ) tanh[ ~ ω ν T ] ~ ω ν = K yx ( t, t ′ ) . (14)The validity of the fluctuation-dissipation relations means that we have properly identifiedthe dissipative terms in the non-Markovian dynamical equations of motion. The quantumfluctuation-dissipation relations differ from the classical ones and are reduced to them in thelimit of high temperature. B. Solution of Non-Markovian Langevin equations
In order to solve the equations of motion (8) for the collective variables, we apply theLaplace transformation. It significantly simplifies the solution of the problem. After theLaplace transformation, the equations of motion read x ( s ) s = x (0) + π x ( s ) m x , y ( s ) s = y (0) + π y ( s ) m y ,π x ( s ) s + π x ( s ) m x ( K xx ( s ) + K xy ( s )) = π x (0) + ω cy π y ( s ) − s eE x + F x ( s ) ,π y ( s ) s + π y ( s ) m y ( K yy ( s ) + K yx ( w )) = π y (0) − ω cx π x ( s ) + F y ( s ) . (15)Here, K xx ( s ), K yy ( s ), K xy ( s ), K yx ( s ) and F x ( s ), F y ( s ) are the Laplace transforms of thedissipative kernels and random forces, respectively. To solve these equations, one should7nd the roots of the determinant D = s ( m x m y ω c + [ K xx ( s ) + K xy ( s ) + m x s ][ K yy ( s ) + K yx ( s ) + m y s ]) = 0 . (16)The explicit solutions for the originals are x ( t ) = x (0) + A ( t ) π x (0) + A ( t ) π y (0) − A ( t ) eE x + I x ( t ) + I ′ x ( t ) ,y ( t ) = y (0) + B ( t ) π y (0) − B ( t ) π x (0) + B ( t ) eE x − I y ( t ) + I ′ y ( t ) ,π x ( t ) = C ( t ) π x (0) + C ( t ) π y (0) − C ( t ) eE x + I π x ( t ) + I ′ π x ( t ) ,π y ( t ) = D ( t ) π y (0) − D ( t ) π x (0) + D ( t ) eE x − I π y ( t ) + I ′ π y ( t ) , (17)where I x ( t ) = Z t A ( τ ) F x ( t − τ ) dτ, I ′ x ( t ) = Z t A ( τ ) F y ( t − τ ) dτ,I y ( t ) = Z t B ( τ ) F x ( t − τ ) dτ, I ′ y ( t ) = Z t B ( τ ) F y ( t − τ ) dτ,I π x ( t ) = Z t C ( τ ) F x ( t − τ ) dτ, I ′ π x ( t ) = Z t C ( τ ) F y ( t − τ ) dτ,I π y ( t ) = Z t D ( τ ) F x ( t − τ ) dτ, I ′ π y ( t ) = Z t D ( τ ) F y ( t − τ ) dτ with the following time-dependent coefficients: A ( t ) = ˆ L − (cid:20) K yy ( s ) + K yx ( s ) + m y sD (cid:21) = B ( t ) | x ↔ y ,A ( t ) = m y ω cy ˆ L − (cid:20) D (cid:21) = B ( t ) | x ↔ y ,A ( t ) = ˆ L − (cid:20) K yy ( s ) + K yx ( s ) + m y ssD (cid:21) , B ( t ) = m x ω cx ˆ L − (cid:20) sD (cid:21) ,C ( t ) = m x ˆ L − (cid:20) s ( K yy ( s ) + K yx ( s ) + m y s ) D (cid:21) = D ( t ) | x ↔ y ,C ( t ) = m x m y ω cy ˆ L − h sD i = D ( t ) | x ↔ y ,C ( t ) = m x ˆ L − (cid:20) K yy ( s ) + K yx ( s ) + m y sD (cid:21) , D ( t ) = m x m y ω cx ˆ L − (cid:20) D (cid:21) . (18)Here, ˆ L − denotes the inverse Laplace transformation. The exact solutions of x ( t ), y ( t ), π x ( t ), and π y ( t ) in terms of roots s i are given by the residue theorem.For the system with linear coupling in coordinate, the coupling term is written as H cb = X ν ( α ν x + β ν y )( b + ν + b ν ) + X ν ~ ω ν ( α ν x + β ν y ) , (19)8here α ν and β ν are the real coupling constants. Here, we again introduce the counterterm which depends on the coordinates of collective system and is treated as a part of thepotential. The operators of random forces and dissipative kernels in Eqs. (8) are F x ( t ) = − X ν α ν ( f + ν + f ν ) , F y ( t ) = − X ν β ν ( f + ν + f ν )and K xx ( t − τ ) = X ν α ν ~ ω ν cos( ω ν [ t − τ ]) ,K yy ( t − τ ) = X ν β ν ~ ω ν cos( ω ν [ t − τ ]) , (20)respectively. We assume that there are no correlations between F νx and F νy , so that K xy = K yx = 0. If the coupling constants α ν and β ν depend on magnetic field, then the dissipativekernels K xx and K yy are the functions of B .It is convenient to introduce the spectral density D ω of the heat bath excitations to replacethe sum over different oscillators by an integral over the frequency: P ν ... → ∞ R dωD ω ... . Thisreplacement is accompanied by the following replacements: α ν → α ω , β ν → β ω , ω ν → ω ,and n ν → n ω . Let us consider the following spectral functions [28] D ω | α ω | ~ ω = α π γ γ + ω , D ω | β ω | ~ ω = β π γ γ + ω , (21)where the memory time γ − of the dissipation is inverse to the phonon bandwidth of the heatbath excitations which are coupled to a quantum particle. This is the Ohmic dissipationwith the Lorentian cutoff (Drude dissipation) [20–25, 28, 36].Using the spectral functions (21), we obtain the dissipative kernels and their Laplacetransforms in the convenient form K xx ( t ) = m x λ x γe − γ | t | , K yy ( t ) = m y λ y γe − γ | t | ,K xx ( s ) = m x λ x γs + γ , K yy ( s ) = m y λ y γs + γ , (22)where λ x = ~ α = 1 m x Z ∞ K xx ( t − τ ) dτ, λ y = ~ β = 1 m y Z ∞ K yy ( t − τ ) dτ A ( t ) = ˙ A ( t ) , A ( t ) = ˙ B ( t ) | x ↔ y ,A ( t ) = 1 m x ( λ y λ x λ y + ω c t + ω c ( γ − λ y ) − λ y ( γ − λ x ) γ ( λ x λ y + ω cx ω cy ) + X i =1 b i e s i t ( γ + s i )( γλ y + s i ( γ + s i )) s i ) ,B ( t ) = ˙ A ( t ) | x ↔ y , B ( t ) = ˙ B ( t ) ,B ( t ) = ω cx m y tλ x λ y + ω cx ω cy + 2 λ x λ y − γ ( λ x + λ y ) γ ( λ x λ y + ω cx ω cy ) + X i =1 b i e s i t ( γ + s i ) s i ! ,C ( t ) = m x ¨ A ( t ) , C ( t ) = m x ¨ B ( t ) , C ( t ) = m x ˙ A ( t ) ,D ( t ) = C ( t ) | x ↔ y , D ( t ) = m y ¨ B ( t ) , D ( t ) = m y ˙ B ( t ) , (23)where b i = [ Q j = i ( s i − s j )] − with i, j = 1 , , , s i are the roots of the equation γλ x [ γλ y + s ( γ + s )] + ( γ + s )( s [ s + ω c ] + γ [ ω c + s ( λ y + s )]) = 0 . (24) C. Galvano-magnetic effects
In order to determine the transport coefficients, we use Eqs. (17). Averaging them overthe whole system and by differentiating in t , we obtain the system of equations for the firstmoments < ˙ x ( t ) > = < π x ( t ) >m x , < ˙ y ( t ) > = < π y ( t ) >m y ,< ˙ π x ( t ) > = ˜ ω cy ( t ) < π y ( t ) > − λ π x ( t ) < π x ( t ) > − e ˜ E xx ( t ) ,< ˙ π y ( t ) > = − ˜ ω cx ( t ) < π x ( t ) > − λ π y ( t ) < π y ( t ) > − e ˜ E xy ( t ) , (25)with the friction coefficients λ π x ( t ) = − D ( t ) ˙ C ( t ) + D ( t ) ˙ C ( t ) C ( t ) D ( t ) + C ( t ) D ( t ) ,λ π y ( t ) = − C ( t ) ˙ D ( t ) + C ( t ) ˙ D ( t ) C ( t ) D ( t ) + C ( t ) D ( t ) , (26)and renormalized cyclotron frequencies˜ ω cx ( t ) = D ( t ) ˙ D ( t ) − D ( t ) ˙ D ( t ) C ( t ) D ( t ) + C ( t ) D ( t ) , ˜ ω cy ( t ) = C ( t ) ˙ C ( t ) − C ( t ) ˙ C ( t ) C ( t ) D ( t ) + C ( t ) D ( t ) , (27)10hile the components of the electric field read:˜ E xx ( t ) = E x [ D ( t )˜ ω cy ( t ) + C ( t ) λ π x ( t ) + ˙ C ( t )] , ˜ E xy ( t ) = E x [ C ( t )˜ ω cx ( t ) − D ( t ) λ π y ( t ) − ˙ D ( t )] . (28)As seen, the dynamics is governed by the non-stationary coefficients. As found, the externalmagnetic field generates the flow of charge carriers and electric field in the cross direction(the classical Hall effect). It should be noted that the cross component ˜ E xy ( t ) of the electricfield is initially absent and appears during the non-Markovian evolution of the collectivesubsystem.Let us consider the magneto-transport process in the two-dimensional system with currentdensity defined as [1–3] J i = X j =1 σ ij ( B ) E j . (29)Here, σ ij ( B ) is the electric conductivity tensor which depends on the magnitude and directionof the magnetic field B . One can also define the current density by using the expression forthe collective momentum (17) and the fact that J = − ne ˙ R because J x = ne m j C ( t ) E x ( t ) , J y = − ne m y D ( t ) E x ( t ) . (30)If we change the direction of the external electric field ~E ( E x , ,
0) to ~E (0 , E y , J x = ne m x ˜ D ( t ) E y ( t ) , J y = ne m y ˜ C ( t ) E y ( t ) , (31)where˜ C ( t ) = m y L − (cid:20) K xx ( s ) + K xy ( s ) + m x sD (cid:21) , ˜ D ( t ) = m x m y ω cy L − (cid:20) D (cid:21) . Comparing (35) with (30) and (31), one can write the expression for conductivity tensorat time t = τ σ ( τ ) = ne C ( τ ) m x − D ( τ ) m y ˜ D ( τ ) m x ˜ C ( τ ) m y , (32)while its inverse transformation yields the specific resistance tensor ρ ( τ ) = 1 ne [ C ( τ ) ˜ C ( τ ) + D ( τ ) ˜ D ( τ )] m x ˜ C ( τ ) m x D ( τ ) − m y ˜ D ( τ ) m y C ( τ ) . (33)11he non-diagonal elements of the specific magneto-resistance tensor have the meaning ofthe Hall resistance ρ H ( τ ) = m x D ( τ ) ne [ C ( τ ) ˜ C ( τ ) + D ( τ ) ˜ D ( τ )] = m y ˜ D ( τ ) ne [ C ( τ ) ˜ C ( τ ) + D ( τ ) ˜ D ( τ )] . (34)In the case of two charge carriers, the model is generalized in Appendix A. D. The main assumptions of the model
Here, we list the main assumptions of the model which allow us to proceed with thecalculations for real systems. We suppose that in each collision the charge carriers losetheir ordered motion and their velocities vanish. As in the kinetic theory of gases, in ourmodel we assume that the lengths and times t = τ of free path are the same for all chargecarriers and all collisions. So, we introduce the time limit t = τ in the conductivity tensor(32) or resistance tensor (33). In our model there are three different characteristic timesdescribing the dynamics of charge carriers: 1) the relaxation time τ r = λ − ( λ = λ x = λ y ),2) the average time between two collisions τ , and 3) the memory time γ − of the heat bathexcitations. The values of τ r and τ are related with one-body (mean-field) and two-bodyeffects (dissipations). So, by introducing the time parameter τ , we take effectively intoconsideration the two-body collisions of charge carriers. The mean free time τ is related tothe thermodynamic equilibrium properties of the material, whereas the relaxation time τ r relates to the thermal and electrical transport properties (see Fig. 1). The relaxation time τ r of electrons is the characteristic time for a distribution of charge carriers in a solid toapproach or ”relax” to equilibrium after the disturbance is removed. A familiar example isthe relaxation of current to zero equilibrium value after the external electric field is turnedoff. Highly conductive materials have relatively long relaxation and free motion times. At τ ≫ τ r , the one-body (mean-field) dissipation dominates. If these times are comparable,then the process has a transitional behavior. Note that in general the values of τ r and τ depend on temperature T and the strength of magnetic field B . E. Axial symmetric system
One can obtain clearer physical picture of the process, if the space-symmetric system isconsidered. In this system m x = m y = m , λ x = λ y = λ , and ω cx = ω cy = ω c . So, Eqs. (24),12 t r B FIG. 1: Schematic presentation of the scattering process and different time scales in a two-dimensional magneto-transport. which defines the poles, is simplified:( s + ω c )( γ + s ) + 2 γλs ( γ + s ) + λ γ = 0 . (35)This equation has the roots s = − (cid:16) γ + iω c + p ( γ − iω c ) − γλ (cid:17) , s = s ∗ ,s = − (cid:16) γ + iω c − p ( γ − iω c ) − γλ (cid:17) , s = s ∗ . In order to split the real and imaginary parts of the roots, we expand them up to the firstorder in λ/γ : s = − λγ γ + ω c − i ω c + γ + λγγ + ω c ω c ,s = − γ γ + ω c − γλγ + ω c + i λγω c γ + ω c .
13n this approximation, the components of conductivity (32) at t = τ are σ xx ( τ ) = σ xx ω c τ r ) − σ xx p ω c τ r ) exp[ − γ γ + ω c ττ r ] cos[ ( γ + γ/τ r + ω c ) ω c τγ + ω c + arctan( 1 ω c τ r )] ,σ xy ( τ ) = σ xx ω c τ r ω c τ r ) − σ xx p ω c τ r ) exp[ − γ γ + ω c ττ r ] cos[ ( γ + γ/τ r + ω c ) ω c τγ + ω c − arctan( ω c τ r )] . (36)As seen, the expressions for macroscopically observable values such as the cross and longi-tudinal components of conductivity (resistance) contain the non-oscillatory and oscillatoryparts. In very strong magnetic fields, when the cyclotron frequency is much larger thanthe friction of the system ω c ≫ τ − r , the longitudinal and transverse components of conduc-tivity oscillate in antiphase. Depending on the ratio between τ r and τ , the oscillatory ornon-oscillatory term of (36) has a major role. At τ ≫ τ r or τ → ∞ , the oscillatory termvanishes and we obtain the Drude conductivity σ = σ xx ω c τ r ) − ω c τ r ω c τ r , (37)where σ xx = ne τ r m is the Drude conductivity at B = 0. As seen, the conductivity (36) differs from the Drudeone at B = 0 by additional oscillatory term. F. Thermomagnetic effects
Here, we assume that the electric and thermal currents are carried by the same particlesand find the relation between the electric and thermal conductivities. If the electric energygradient eE in the Hamiltonian (2) is substituted by the temperature gradient dTdx (thetemperature is in energy units), the generating force of particle motion is changed fromthe electric to thermal potential, giving rise to the thermomagnetic effects. Taking intoconsideration the expression for the heat flux Q = nε kin ˙ R , χ ( τ ) = 1 e ε kin ( τ ) σ ( τ ) , (38)where ε kin is the kinetic energy of charge carriers. The kinetic energy is defined throughthe variances Σ π x π x and Σ π y π y (Appendix B) and mean values < π x > and < π y > . In thequasi-equilibrium high temperature limit ( τ → ∞ ), the kinetic energy ε kin ( ∞ ) = T (39)is defined by the equipartition theorem (the equilibrium variances Σ π x π x ( ∞ ) = m x T andΣ π y π y ( ∞ ) = m y T ). Finally, one can rederive the classical Wiedemann-Franz law [1–3] L = χT σ = 1 e = const, where L is the Lorentz number which reflects the fact that the ability of the carriers to carrya charge is the same as to transport heat. III. CALCULATED RESULTS AND DISCUSSIONS
In the calculations we set λ = λ x = λ y (or λ π = λ π x = λ π y ) and m = m x = m y (or ω c = ω cx = ω cy ). In order to turn to the observable values, all parameters τ − , τ − r , ω c , and γ in the expressions are multiplied by me : τ − → me τ − , (40) τ − r → me τ − r = µ − ,ω c → me ω c = B,γ → me γ = Γ . As a result, instead of the friction coefficient λ , cyclotron frequency ω c , and inverse responsetime γ of the system we have the inverse reciprocal mobility µ − of charge carriers, intensityof the magnetic field B , and new parameter Γ connected to the memory time. The mobilityof a charge carrier µ = µ ( B = 0) in the absence of magnetic field is the measurable value.The value of B is set by the experimental condition.15 .0 0.4 0.8 1.2 1.60.000.050.100.15 E xy / E xx B (V s/ m ) T=20.4 K ~~~~ E xy / E xx T=4.22 K
FIG. 2: The experimental [4] (symbols) and theoretical dependencies (lines) of the tangent of theHall angle on magnetic field B for zinc at temperatures T indicated. In addition, one can also study the magnetic moment of the system. It should be notedthat in our model the influence of the magnetic field on the coupling between quantumparticle and heat-bath is neglected. The impact of magnetic field is entered into the dissi-pative kernels. However, there are solids with constant resistance in the wide spectrum ofmagnetic field. Their properties can be described by neglecting the effect of magnetic field16n the coupling term.
A. Classical Hall effect
The classical case corresponds τ ≫ τ r or τ → ∞ . To demonstrate the capabilities ofthe model, we calculate the tangent of the Hall angle, tan[Θ H ] = ˜ E xy ( ∞ ) / ˜ E xx ( ∞ ), for thesample of Zn settled in the increasing external magnetic field at two temperatures. Manyexperiments were performed to measure this value in several materials. We choose Zn [4]because it has one type of charge carriers, and consequently, the technique of implementationof the model can be easily understood. The calculated and experimental characteristics ofZn are listed in Table I. The calculations performed with the values of mobility µ at B = 0are in a good agreement with the experimental data (Fig. 2), especially at high strength ofmagnetic field. TABLE I: Experimental (asterisks) [4] and theoretical characteristics of Zn at two temperatures.The value of B ∗ max corresponds to the position of the maximum of experimental non-diagonalcomponent of electric field as a function of magnetic field.Temperature ∗ , Resistance ∗ , ρ xx , Mobility, Γ = B ∗ max , Max. Hall T (K) × − (Ω · m) µ (m / V · s ) (Tesla) angle ∗ , Θ H . ◦ . ◦ B. Integer Hall effect
The IQHE has been observed in the heterostructure GaAs-Al . Ga . As at the low tem-perature of T = 50 mK [8]. According to the experiment, the concentration of the artificiallyprepared two-dimensional sample GaAs-Al . Ga . As is n = 3 . · m − . There are largeintervals of B where the longitudinal conductivity has its minimum, while the Hall conduc-tivity is quantized with immense precision in integer multiples of e / (2 π ~ ). The calculatedcomponents of the resistivity tensor are shown in Fig. 3 for wide range of magnetic field.17 B (Tesla) xx ( k xy ( k ... ( m / ( V s )) i=3=1/(0.001+0.005096 B) . FIG. 3: Theoretical plots of the components of resistivity tensor and mobility of charge carriersin GaAs-Al . Ga . As at T = 50 mK. The experimental concentration, n = 3 . · m − , is fromRef. [8] and the mobility has the functional form µ ( B ) = 1 / (0 .
001 + 0 . B ) indicated on theplot. The ratio of mean collision time per relaxation time is set to be ττ r = 2, and Γ = 100 /µ . The observed increase of the width of plateau in ρ xy with the field is explained by the de-crease of the mean collision time of charge carriers. The external magnetic field effects thecoupling between the collective system and heat-bath. This coupling linearly rises with themagnetic field which induces the reciprocal decrease of τ r . Such a decrease in relaxationtime effectively changes the mobility of charge carriers (Fig. 3). The drastic decrease ofmobility can be attributed to the effect of localization under the influence of magnetic field.For the formation of the step-wise nature of ρ xy , the ratio τ /τ r is required to be constantat whole magnetic field spectrum. Thus, both the mean collision time and relaxation timefall down inversely with increasing magnetic field, as in the experiment [1]. However, theratio between them remains constant. As seen in Fig. 3, in the region between two plateausfor ρ xy the longitudinal resistivity has the maximum, while at the center of plateau it isminimal. This phenomenon is explained by the π/ ρ xx and18 ( i=1/3i=1/3i=1 i=2 (T ) (( )) FIG. 4: The experimental (black) [9] and theoretical (red) curves of magnetic field dependenciesof the components of resistivity tensor at different temperatures T =0.48, 1.00, 1.65, 4.15 K in theorder from top to bottom. ρ xy which is clearly visible in the approximate formulas for the axial symmetric system atvery strong magnetic fields ( ω c τ r ≫ ρ xx ρ xx = 1 − µB exp (cid:20) − Γ Γ + B ττ r (cid:21) cos (cid:20) (Γ + Γ /µ + B )( e/m ) τ B Γ + B (cid:21) ,ρ xy ρ xx = µB (cid:18) − exp (cid:20) − Γ Γ + B ττ r (cid:21) sin (cid:20) (Γ + Γ /µ + B )( e/m ) τ B Γ + B (cid:21)(cid:19) , where ρ xx = 1 /σ xx . 19 . Fractional Hall effect !" !" $ %& !" &" !"($ ’ ( ) ! !"($)*’ !" $ %& (cid:37)(cid:3)(cid:11)(cid:55)(cid:72)(cid:86)(cid:79)(cid:68)(cid:12) ) ! !"%$(*’ (cid:37)(cid:3)(cid:11)(cid:55)(cid:72)(cid:86)(cid:79)(cid:68)(cid:12) FIG. 5: The calculated magnetic moment as a function of magnetic field at indicated temperatures.
The Hall plateau at strong magnetic field has been discovered in Ref. [9] (Fig. 4) andcorresponds to the fractional value of the filling factor i = 1 /
3. The experiment has beencarried out at four temperatures below the helium temperature for the sample of GaAs-Al . Ga . As with the 2D concentration n = 1 . · m − and carrier mobility µ = 9 m /(V · s). The step-wise appearance of the Hall resistivity becomes smoother with increasingtemperature. The purity of the sample is so high that the electrons move ballistically, i.e.without scattering against impurity atoms, over relatively long distances. In our calculations(Fig. 4), we take the experimental values of mobility µ and 2D concentration n and Γ =100 /µ (Table II). As in the case of the integer Hall effect, the relaxation time and meancollision time of charge carriers decrease inversely with increasing magnetic field and theirratio remains constant (Table II).We also calculate the magnetic moment M ( τ ) = neL z ( τ )2 m , where L z ( τ ) = < x ( τ ) π y ( τ ) − y ( τ ) π x ( τ ) > = m ~ γ π Z ∞ Z τ Z τ dωdtdt ′ ω coth (cid:2) ~ ω T (cid:3) ω + γ cos( ω [ t − t ′ ]) × { λ x [ B ( t ) C ( t ′ ) − A ( t ) D ( t ′ )] + λ y [ A ( t ) D ( t ′ ) − B ( t ) C ( t ′ )] } (41)20 ! " "! !" ! ’’ & ( ! & * &&$ + (cid:35)(cid:1)(cid:9)(cid:53)(cid:70)(cid:84)(cid:77)(cid:66)(cid:10) (cid:10)(cid:9) FIG. 6: The calculated dependence of transverse component of resistivity and the mean free timeon magnetic field. The blue line corresponds to the experiment [10]. is the z -component of angular momentum. The calculations were performed with the pa-rameters from Table II. As seen in Fig. 5, the magnetic moment approaches to ne m ~ at alltemperatures considered. At low temperatures T =0.48 and 1 K, small oscillations of themagnetic moment are observed in the region of weak magnetic field. These oscillations aregetting smoother as the temperature increases and disappear at sufficiently high tempera-tures.In Fig. 6, one can see the experimental curve (blue line) obtained for the sampleGaAs/AlGaAs at lower temperature 85 mK. The measured concentration of the 2DEG21 ABLE II: The experimental (asterisks) [9] and theoretical parameters used in the calculations ofthe FQHE. MF denotes a magnetic field.Temperature ∗ , Mobility in the absence Ratio Functional form of T (K) of MF, µ ∗ (m / V · s) τ /τ r µ ( B ) = τ r ( B ) e/m .
11 + 0 . B ) − .
11 + 0 . B ) − .
11 + 0 . B ) − .
11 + 0 . B ) − created in this sample is n = 3 × m − and the mobility of charge carriers at B = 0 is µ = 100 m /V · s. According to this figure the width of the plateau increases up to 10 Teslaand then suddenly decreases. It starts to increase again from 19 Tesla to 30 Tesla. It isobvious that above 10 Tesla the properties of the system drastically change. In our modelthis behavior is explained by the abrupt change of the functional form of the mean collisiontime. Initially being em τ = 801 . B below 19 Tesla (red line), it changes to em τ = 4 B − /µ . It should be notedthat the mobility µ = eτ r /m = µ at T = 85 mK remains constant in a whole range ofmagnetic field.As shown, the non-oscillatory term of the conductivity (resistance) plays a key role athigh temperature (the classical Hall effect), whereas an oscillatory part mainly contributesto the resistance at low temperature (the quantum Hall and Shubnikov-De Haas effects),where the mobility of charge carriers is sufficiently large and the values of relaxation timeand average collision time are comparable. One should stress that the Shubnikov-De Haas,integer and fractional quantum Hall effects are the results of the transitional processes. Notethat for the integer and fractional quantum Hall effects, the values of relaxation time andaverage collision time should be comparable.22 . Shubnikov-De Haas effect Theory xx / xx xx / xx Magnetic Field (kG) xx / xx -6 Magnetic Field (kG)
15 K12118.57.54.2
Experiment
FIG. 7: The experimental [6] and theoretical dependencies of the oscillatory part of the longitudinalmagneto-resistance on magnetic field for various temperatures.
Let us consider the experiment performed in Ref. [6] with the n -InSb sample which hasbeen prepared from the n -type single crystal of InSb having a charge carrier concentration n = 5 . · cm − (see Fig. 7). According to the experiment, the mobility of charge carriersin the absence of magnetic field varies within 5% with increasing temperature from 4.2 to15 K. However, this value does not show a variation with increasing magnetic field. Thepoints of intersection between the zero axis and the resistance curves are the same at alltemperatures. This implies that the period of oscillations does not depend on temperature.Moreover, the equal increase in period of oscillations with the magnetic field is observedat all temperatures in the experiment. So, contrary to the cases of integer and fractionalHall effects, the mobility µ or relaxation time τ r of charge carriers in the sample does notchange with magnetic field while the mean free time decreases similarly as the magnetic fieldincreases at any temperature. Thus, the ratio of the mean collision time and the relaxationtime, which remains constant in the quantum Hall regimes, now decreases with increasingfield (see Table III and Fig. 7). No oscillations have been detected in the magnetic fielddependence of transverse resistance in the experiment. Though there are oscillations in ourapproach (Fig. 8), their amplitudes are negligible with respect to the non-oscillatory partof the resistance. In order to detect them, the non-oscillatory part should be subtractedfrom the measured value as ∆ ρ xy = ρ xy − Bne . The calculations of the magnetic moment donot show the oscillations in Fig. 9. The absolute value of magnetic moment decreases with23 ! " ! " !" $ ! (cid:3) (cid:3)(cid:11)(cid:78)(cid:42)(cid:12) %&%%%&% %&%" " (&$ ) ! " !" % & $ *+,-./ (cid:9) (cid:1)(cid:1)(cid:1)(cid:1) (cid:10)(cid:9) (cid:1)(cid:1)(cid:1) (cid:10) FIG. 8: The dependence of the calculated cross resistance on magnetic field at various temperatures.TABLE III: The experimental (asterisks) [6] and theoretical parameters used in the calculationsof Shubnikov-De Haas effect. MF denotes a magnetic field.Temperature ∗ , T (K) 4.2 7.5 8.5 11 12 15Mobility in the absence of MF, µ ∗ (m / V · s) 9.5 9.1 8.8 8.4 8.3 8Functional form of τ ( B ) e/m µ increasing magnetic field.In Fig. 10, the dependencies of the longitudinal thermal resistance k xx = χ xx χ xx + χ xy on24 !" ! !" $ %& (cid:37) (cid:11)(cid:55)(cid:72)(cid:86)(cid:79)(cid:68)(cid:12) FIG. 9: The calculated magnetic field dependence of the magnetic moment. The curves from topto bottom correspond to the calculations at temperatures T =4.2, 7.5, 8.5, 11, 12, and 15 K. ! " (cid:15) &$%& (cid:15) &"%& (cid:15) & & (cid:15) &&& (cid:15) & !" ’()*+,-./0-+12/3456 (cid:18) (cid:743)(cid:743) (cid:18) FIG. 10: The calculated dependencies of the oscillatory part of the longitudinal thermal-resistanceon magnetic field for various temperatures. magnetic field are presented at different temperatures. Comparing Figs. 7 and 10, one can25otice the correlations between magneto- and thermal-resistances.
IV. SUMMARY
Using the non-Markovian Langevin approach and coupling between the charge carriersand environment, the behavior of the generated flow of charge carriers under the influence ofexternal magnetic field was investigated for the two-dimensional case. The model developedwas applied to the case where the collective coordinates are linearly coupled to the heat-bathcoordinates. In order to average the influence of environment on the collective system, weapplied the spectral function of heat-bath excitations which describes the Drude dissipationwith Lorenzian cutoffs. The dynamics of charge carriers was limited by the average collisiontime as in the kinetic theory of gases. In this way, the two-body effects were taken effectivelyinto consideration. The functional dependencies of the average collision time and couplingstrength between the charge carriers and environment on temperature and magnetic fieldwere phenomenologically treated. One can say that we solve the inverse problem by findingsuitable coupling strengths and the average collision times for describing the experimentaldata. As shown, the galvano- and thermo-magnetic effects strongly depend on the ratiobetween the relaxation time (the inverse friction coefficient) and average collision time.The explicit expressions were obtained for the macroscopically observable values such astransverse and longitudinal components of conductivity (resistance) and the Hall angle. Itwas concluded that the non-oscillatory term of conductivity (resistance) plays a key roleat high temperature (the classical Hall effect) whereas the oscillatory part of conductivity(resistance) mainly contributes to the resistance at low temperature (quantum Hall andShubnikov-De Haas effects), where the mobility of charge carriers is sufficiently large and thevalues of relaxation time and average collision time are comparable. Thus, the Shubnikov-DeHaas, integer and fractional quantum Hall effects are the results of the transitional processes.The Shubnikov-De Haas effect has been observed both in the two-dimensional and three-dimensional samples, whereas the integer and fractional quantum Hall effects have beendetected only in the two-dimensional samples so far. However, our model also predicts theirexistence in the three-dimensional samples. The model was applied to the thermomagneticprocesses as well. The oscillations of the thermal coefficients were predicted in the quantumHall and Shubnikov-De Haas regimes. The experimental observation of such oscillations26ould be a good criteria for the justification of the present model.The model developed can be extended further by taking the spin of electrons and non-stationary external fields into consideration.
Acknowledgments
This work was partially supported by the Russian Foundation for Basic Research(Moscow) and DFG (Bonn). The IN2P3(France)-JINR(Dubna) Cooperation Programmeis gratefully acknowledged.
Appendix A: The case of two charge carriers
In the case of two kinds of charge carriers, for example electrons and holes, in the currenttwo-band model, we should solve the equations of motion (8) for each kind of charge carriers.The total conductivity tensor consists of the sum of conductivity tensors of transmissionelectrons and holes σ ( τ ) = e C e ( τ ) n e m ex + C h ( τ ) n h m hx − D h ( τ ) n h m hy + D e ( τ ) n e m ey ˜ D h ( τ ) n h m hx − ˜ D e ( τ ) n e m ex ˜ C e ( τ ) n e m ey + ˜ C h ( τ ) n h m hy , (A1)where n e ( m ex,y ) and n h ( m hx,y ) are the concentrations (the components of the effective masstensor) of electrons and holes, respectively. Performing the inverse operation on σ ( τ ), wefind the magneto-resistance tensor: ρ ( τ ) = 1∆( τ ) × (A2) × m ex m hx [ m ey n h ˜ C h ( τ ) + m hy n e ˜ C e ( τ )] m ex m hx [ m ey n h D h ( τ ) − m hy n e D e ( τ )] − m ey m hy [ m ex n h ˜ D h ( τ ) − m hx n e ˜ D e ( τ )] m ey m hy [ m ex n h C h ( τ ) + m hx n e C e ( τ )] , where ∆( τ ) = e ([ m ex n h C h ( τ ) + m hx n e C e ( τ )][ m ey n h ˜ C h ( τ ) + m hy n e ˜ C e ( τ )]+[ m ey n h D h ( τ ) − m hy n e D e ( τ )][ m ex n h ˜ D h ( τ ) − m hx n e ˜ D e ( τ )]) . The Hall resistance takes the following form ρ H ( τ ) = m ex m hx [ m hy n e D e ( τ ) − m ey n h D h ( τ )]∆( τ )= m ey m hy [ m hx n e ˜ D e ( τ ) − m ex n h ˜ D h ( τ )]∆( τ ) . (A3)27 ppendix B: Variances The equations for the second moments (variances),Σ q i q j ( t ) = 12 < q i ( t ) q j ( t ) + q j ( t ) q i ( t ) > − < q i ( t ) >< q j ( t ) >, where q i = x, y, π x , or π y ( i =1-4), are˙Σ xx ( t ) = 2Σ xπ x ( t ) m x , ˙Σ yy ( t ) = 2Σ yπ y ( t ) m y , ˙Σ xy ( t ) = Σ xπ y ( t ) m y + Σ yπ x ( t ) m x , ˙Σ xπ y ( t ) = − λ π y ( t )Σ xπ y ( t ) − ˜ ω cx ( t )Σ xπ x ( t ) + Σ π x π y ( t ) m x + 2 D xπ y ( t ) , ˙Σ xπ x ( t ) = − λ π x ( t )Σ xπ x ( t ) + ˜ ω cy ( t )Σ xπ y ( t ) + Σ π x π x ( t ) m x + 2 D xπ x ( t ) , ˙Σ yπ x ( t ) = − λ π x ( t )Σ yπ x ( t ) + ˜ ω cy ( t )Σ yπ y ( t ) + Σ π x π y ( t ) m y + 2 D yπ x ( t ) , ˙Σ yπ y ( t ) = − λ π y ( t )Σ yπ y ( t ) − ˜ ω cx ( t )Σ yπ x ( t ) + Σ π y π y ( t ) m y + 2 D yπ y ( t ) , ˙Σ π y π y ( t ) = − λ π y ( t )Σ π y π y ( t ) − ω cx ( t )Σ π x π y ( t ) + 2 D π y π y ( t ) , ˙Σ π x π x ( t ) = − λ π x ( t )Σ π x π x ( t ) + 2˜ ω cy ( t )Σ π x π y ( t ) + 2 D π x π x ( t ) , ˙Σ π x π y ( t ) = − ( λ π x ( t ) + λ π y ( t ))Σ π x π y ( t ) + ˜ ω cy ( t )Σ π y π y ( t ) − ˜ ω cx ( t )Σ π x π x ( t ) + 2 D π x π y ( t ) . (B1)So, we obtain the Markovian-type (local in time) equations for the first and second moments,but with the transport coefficients depending explicitly on time. The time-dependent diffu-28ion coefficients D q i q j ( t ) are determined as D xx ( t ) = D yy ( t ) = D xy ( t ) = 0 ,D π x π x ( t ) = λ π x ( t ) J π x π x ( t ) − ˜ ω cy ( t ) J π x π y ( t ) + 12 ˙ J π x π x ( t ) ,D π y π y ( t ) = λ π y ( t ) J π y π y ( t ) + ˜ ω cx ( t ) J π x π y ( t ) + 12 ˙ J π y π y ( t ) ,D π x π y ( t ) = − h − ( λ π x ( t ) + λ π y ( t )) J π x π y ( t ) + ˜ ω cy ( t ) J π y π y ( t ) − ˜ ω cx ( t ) J π x π x ( t ) − ˙ J π x π y ( t ) i ,D xπ y ( t ) = − (cid:20) − λ π y ( t ) J xπ y ( t ) − ˜ ω cx ( t ) J xπ x ( t ) + J π x π y ( t ) m x − ˙ J xπ y ( t ) (cid:21) ,D yπ x ( t ) = − (cid:20) − λ π x ( t ) J yπ x ( t ) + ˜ ω cy ( t ) J yπ y ( t ) + J π x π y ( t ) m y − ˙ J yπ x ( t ) (cid:21) ,D xπ x ( t ) = − (cid:20) − λ π x ( t ) J xπ x ( t ) + ˜ ω cy ( t ) J xπ y ( t ) + J π x π x ( t ) m x − ˙ J xπ x ( t ) (cid:21) ,D yπ y ( t ) = − (cid:20) − λ π y ( t ) J yπ y ( t ) − ˜ ω cx ( t ) J yπ x ( t ) + J π y π y ( t ) m y − ˙ J yπ y ( t )] (cid:21) . (B2)Here, ˙ J q i q j ( t ) = dJ q i q j ( t ) /dt and J xx ( t ) = ≪ I x ( t ) I x ( t ) + I ′ x ( t ) I ′ x ( t ) ≫ , J yy ( t ) = J xx ( t ) | x → y ,J xy ( t ) = ≪ I x ( t ) I y ( t ) + I ′ x ( t ) I ′ y ( t ) ≫ , J π x π y ( t ) = J xy ( t ) | x → π x ,y → π y ,J xπ x ( t ) = ≪ I x ( t ) I π x ( t ) + I ′ x ( t ) I ′ π x ( t ) ≫ , J yπ x ( t ) | = J xπ x ( t ) | x → y ,J xπ y ( t ) = ≪ I x ( t ) I π y ( t ) + I ′ x ( t ) I ′ π y ( t ) ≫ , J yπ y ( t ) = J xπ y ( t ) | x → y ,J π x π x ( t ) = ≪ I π x ( t ) I π x ( t ) + I ′ π x ( t ) I ′ π x ( t ) ≫ , J π y π y ( t ) = J π x π x ( t ) | pi x → π y . (B3)The explicit expressions for J q i q j ( t ) are J xx ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x A ( t ′ ) A ( t ′′ ) + λ y A ( t ′ ) A ( t ′′ )] cos( ω [ t ′′ − t ′ ]) ,J xy ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x A ( t ′ ) B ( t ′′ ) + λ y A ( t ′ ) B ( t ′′ )] cos( ω [ t ′′ − t ′ ]) ,J π x π x ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x C ( t ′ ) C ( t ′′ ) + λ y C ( t ′ ) C ( t ′′ )] cos( ω [ t ′′ − t ′ ]) , π x π y ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x C ( t ′ ) D ( t ′′ ) + λ y C ( t ′ ) D ( t ′′ )] cos( ω [ t ′′ − t ′ ]) ,J xπ x ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x A ( t ′ ) C ( t ′′ ) + λ y A ( t ′ ) C ( t ′′ )] cos( ω [ t ′′ − t ′ ]) ,J xπ y ( t ) = m ~ γ π Z ∞ dω Z t dt ′ Z t dt ′′ ω coth (cid:2) ~ ω T (cid:3) ω + γ × [ λ x A ( t ′ ) D ( t ′′ ) + λ y A ( t ′ ) D ( t ′′ )] cos( ω [ t ′′ − t ′ ]) . 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