PHYSICAL REVIEW E 83,012103(2011)

Comparison of the Kubo formula, the microscopic response method, and the Greenwood formula

Mingliang Zhang and D. A. Drabold Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA (Received 6 November 2010; revised manuscript received 6 December 2010; published 21 January 2011) For a mechanical perturbation, the microscopic response method is equivalent to and more convenient to use than the Kubo formula. When the gradient of the carrier density is small, the current density reduces to that used by Greenwood.

DOI: 10.1103/PhysRevE.83.012103 PACS number(s): 05.60.Gg, 72.10.Bg, 72.20.Dp

Linear response theory [1,2]isarigorousandcomplicated In this Brief Report, we prove that for mechanical pertur- procedure used to compute transport coefficients. It constructs bations, the MRM is equivalent to the KF. We will use the the macroscopic response by averaging the many-body wave-function representation which is equivalent of a collective variable over the of system [3,4]. to the frequently used second quantized representation [16,17]. However, it is difficult to apply the Kubo formula to systems To prove the equivalence between the KF and the MRM, we which have several types of elementary excitations and for first write out the observable macroscopic current density to which the interactions among elementary excitations are first and second order in external field in the Kubo formulation. strong. For example, the complexity of the imaginary time Then, the same procedures are carried out with the MRM integral prevents researchers from computing all the important [11,14]. We see that the macroscopic response calculated in the contributions in conductivity and Hall mobility of small po- two methods is the same. We discuss the connection between larons [5–8]. Amorphous semiconductors and semiconducting density matrix and transition amplitudes at different orders polymers require approximations beyond small polarons: the of perturbation. The contribution for transport coefficient low-lying excited states often contain both localized states and from each transport process can be easily read off from a extended states [9], and the electron-phonon interactions in diagrammatic rule [11,14]. We show that the current density localized states are much stronger than those in extended states used by Greenwood is justified only when the gradient of [10]. Even for the lowest order self-consistent approximation the carrier density is small in addition to invoking the single- [11], one cannot easily include all important contributions for particle approximation. conductivity or Hall mobility. In this work, we use the Schrodinger¨ picture. Consider a One hopes to find some alternative method for computing system with N electrons and nuclei in an electromagnetic transport coefficients for various concrete external distur- field with potentials (A,φ), atN time t,themany-electronstate bances, which should be simpler than the original linear of system is described by ! (r1,r2, ,rN ; t). To save space, ! ··· response theory [1]andstillasrigorousaspossible.To we will not write out the nuclear coordinates explicitly. !! calculate the viscoelasticity of fluid, Evans et al. [12,13] satisfies the Schrodinger¨ equation introduced a nonequilibrium Hamiltonian to replace the nonequilibrium free energy, which significantly reduced the ih¯ ∂!!/∂t H !!!,H! H V (t), (1) labor required in the linear response theory for a thermal = = + disturbance [2]. Recently we have designed the microscopic where V (t)istheinteractionbetweenthesystemandexternal response method (MRM) to compute the conductivity and field. The time dependence in V (t!)comesfromtheexternal Hall mobility for complex systems with topological and field. H is the Hamiltonian of the system without external field. thermal disorder [11,14], which is more convenient than the We use m or !m and Em to denote the mth stationary state and Kubo formula (KF) for a “mechanical perturbation” [1]. A the corresponding| " eigenvalue of the N electrons nuclei + N mechanical perturbation such as the coupling with an external system: H m Em m .Ifthesystemisinathermalbathat field can be expressed via additional terms in the Hamiltonian temperature| T",thenbeforeintroducing= | " V (t), the equilibrium [1,2], and the many-body !! of system in density operator is an external field at a later moment is determined by its βEm initial value and the time-dependent Schrodinger¨ equation. The ρ m Pm m ,Pm e− /Z,(2) microscopic response can be expressed in terms of the changes = m | " # | = in the many-body wave function induced by the external field " ! [11,14]. The ensemble and coarse-grained average needed to where Z e βEn is the partition function. In an external = n − compute the macroscopic response (the transport coefficient) electromagnetic field, the velocity operator vi for the ith can then be carried out at the final stage. Thus for a mechanical particle is [18#] perturbation, we are able to avoid the imaginary time integral in the KF [11,14]. The Greenwood formula (GF) [15]has 1 vi m− ih¯ ri eA(ri ; t) , (3) been implemented in many ab initio codes to calculate the = − ∇ − $ % dielectric function and AC conductivity. However, the GF is where ri is the position operators of the ith electron and e is based on a simplified expression for the current density, which the charge of electron. Because velocity and position cannot be is borrowed from the kinetic theory of gases. simultaneously measured, one has to symmetrize the velocity

1539-3755/2011/83(1)/012103(4)012103-1 ©2011 American Physical Society BRIEF REPORTS PHYSICAL REVIEW E 83,012103(2011) and position operators in the current density operator. The To write out the macroscopic response in the KF, we notice current density operator at point r is [16]: that the time evolution for a system involving mixed states is N included in the density matrix. The basis set should be a group e of wave functions without time dependence [20]. In the m j(r) [vi δ(r ri ) δ(r ri )vi ]. (4) = 2 − + − | " i 1 representation, the matrix elements of j(r)are "= In the! MRM [11,14], we avoided j(r). Now we show that Neih¯ n j(r) m dτ![!m !r!∗ !∗ r!m] Eq. (4)leadstoapropermicroscopiccurrentdensity[11, # | | "= 2m ∇ n − n ∇ & 14]. Because a mechanical perturbation! can be expressed with 2 ! Ne additional terms in Hamiltonian, the states at time t can be A(r; t) dτ!!∗!m, (7) − m n described by a wave function which is determined by the initial & conditions. The microscopic current density at time t and point where the arguments of !m and !n are (r,r2, ,rN ). With ρ ··· r in state !!(r1,r2, ,rN ; t)is in Eq. (2)astheinitialcondition,onecanuseperturbation ··· theory to solve the Liouville equation to any order in (1) jm(r; t) dτ!!∗j(r)!!, (5) V (t). The density matrix at time t is ρ!(t) ρ ρ (t) = (2) = +(1) + & ρ (t) . To first order in V (t), the deviation ρ (t)from +··· where dτ dr1dτ! and dτ! dr!2 drN . Integrating by ρ is [1,4] parts, one has= = ··· m ρ(1)(t) n e2N # | | t " jm(r; t) A(r; t) dτ!!!∗ r!! 1 i(t t!)(En Em)/h¯ =− m ∇ dt!e − − m V (t!) n (Pn Pm). (8) & = ih¯ # | | " − iheN¯ &−∞ dτ!(!! r!!∗ !!∗ r!!), (6) + 2m ∇ − ∇ The conductivity can be read off from the macroscopic current & density: where the arguments of ! in Eq. (6)are(r,r2, ,rN ; t). ! ··· Equation (6)hasbeenindependentlyderivedfromtheprinciple j(1)(r,t) m ρ(1)(t) n n j(r) m . (9) of virtual work [19], the continuity equation [14], and the = mn # | | "# | | " polarization density [11]. The current operator given in Eq. (4) " will bridge the KF and the MRM. To second order in V (t), the deviation ρ(2)!(t)is

m ρ(2)(t) n # | | " 1 t t! t t!! it(En Em)/h¯ it!!(Ek En)/h¯ it!(Em Ek )/h¯ e − Pn dt! dt!! Pm dt!! dt! e − e − m V (t!) k k V (t!!) n =−h2 + # | | "# | | " ¯ k '( ) " &−∞ &−∞ &−∞ &−∞ t t! t t!! it!!(Ek En)/h¯ it!(Em Ek )/h¯ dt! dt!! dt!! dt! e − e − m V (t!) k Pk k V (t!!) n , (10) − + # | | " # | | " (&−∞ &−∞ &−∞ &−∞ ) *

The second-order macroscopic response j(2)(r,t)isobtained where from Eq. (9)byreplacing m ρ(1)(t) n with m ρ(2)(t) n .We # | | " # | (2) | " (0) iEnt/h¯ are going to compare Eqs. (8), (9), and (10)andj (r,t)with a (n,t) e− (13) the corresponding quantities in the MRM. = In the MRM, the macroscopic response is given by [11,14]: and

i t (1) iEmt/h¯ i(Em En)t!/h¯ j(r,t) Pn !! (t) j(r) !! (t) . (11) a (mn,t) e− dt!e − m V (t!) n = # n | | n " =−h¯ # | | " n &−∞ " (14) ! If the initial state is !n,thenthestate!n! (t)ofasystemat time t in an external field can be determined [20]byapplying and perturbation theory to Eq. (1): 1 t (2) iEmt/h¯ i(Em Ek )t /h¯ a (mn,t) e− dt!e − ! m V (t!) k =−h2 # | | " ¯ k (0) (1) (2) "&−∞ !n! (t) a (n,t)!n a (mn,t)!m a (mn,t)!m, t! = + + i(Ek En)t /h¯ m m dt!!e − !! k V (t!!) n . (15) " " × # | | " (12) &−∞

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To first order in V (t), the macroscopic current density is With these explanations, substituting Eqs. (13)and(14)into Eq. (19), one reaches Eq. (8), which was obtained from the (1) (0) (1) j (r,t) Pn !n (t) j(r) !n (t) Liouville equation. To second order in V (t), = n " $+ , , - 1 !(1)(t) j(r) !,!(0)(,t) . (16) ρ(2) a (0)(n,t)a(2)(m,t) a (2)(n,t)a(0)(m,t) + n n mn α∗ α α∗ α = α + Substituting Eqs. (12), (13+ ), and, (14)intoEq.(, -% 16), and using M " / ,! , (1) (1) the fact that V (t )andj(r) are Hermitian operators, one aα∗ (n,t)aα (m,t) . (20) ! + finds the same result as Eqs. (8)and(9). To second order In the first term of (20), the initial0 state is n ,anda(2)(m,t) in V (t), the macroscopic current! density is α is the second-order transition amplitude through| " intermediate j(2)(r,t) P !(0)(t) j(r) !(2)(t) !(2)(t) j(r) !(0)(t) states k .BymeansofEqs.(13)and(15), the first term of n n n n n | " = n + Eq. (20)isthesameasthefirsttermofEq.(10). In the second " , , , , (2) (1)$+ (1) - + - term of (20), the initial state is m , aα (n,t)isthesecond- !n (t) j(r) !,!n (,t) . ,! , (17) | " + order transition amplitude through intermediate states k to | " Substituting Eqs.+ (12,)–(15, )intoEq.(-% 17), the first term of final state n .ThesecondtermofEq.(20)isthesameas Eq. (17)isthesameasthefirsttermof,! , j(2) from Eq. (10), the second| term" of Eq. (10). In the third term of (20), two the second term of Eq. (17)isthesameasthesecondtermof final states n and m come from a common initial state k , j(2) resulting from Eq. (10). One can see that the third term all states |k " satisfy| "k n and k m can be taken as| the" Eq. (17)equalsthesumofthethirdtermandthefourthterm initial state.{| Using"} the same)= trick in)= comparing the third term of j(2) from Eq. (10), if one notices (i) three integrands are in Eq. (17)andthesumofthethirdandfourthtermsofj(2)(r,t) the same, (ii) the third term in Eq. (17)isatwo-dimensional derived from Eq. (10), we can see that the third term of (20) integral in domain [ ,t; ,t], (iii) the third term of j(2) is is the same as the sum of the third term and the fourth term −∞ t −∞ t! a successive integration dt! dt!!,(iv)thefourtermof in Eq. (10). −∞ t −∞ t Now we explain why the MRM is simpler than the KF j(2) is a successive integration dt !! dt .Theprocedure . . !! ! for mechanical perturbations. To a given order in residual is easy to carry out to any order−∞ in field.−∞ Equation (11)does interactions, various transport processes contribute to a specific not use any specific property. of electromagnetic. field; the transport coefficient. In the MRM, each transport process is procedure works for any mechanical perturbation. Introducing composed of several elementary transitions caused by the the current density operator (4)isthekeytotheproof.Inthe external field and by residual interactions [11,21]. Because original MRM, one does not need the current density operator; the microscopic response is expressed by the wave function the macroscopic response is obtained by averaging over the of the system in an external field rather than density matrix, microscopic response (6)overthecanonicaldistribution. each elementary transition appears as a transition amplitude Equations (4)and(11)establishedaconnectionbetweenthe [11]. According to Eq. (12), the state at time t is a linear two methods. The “nonequilibrium Hamiltonian” approach superposition of the various changes induced by the external [12,13]canbeviewedasaclassicalrealizationoftheMRM field. By means of Eqs. (6), (16), and (17), the gradient for velocity gradient. Introducing a suitable “non-equilibrium operator connects two components of the final state [11]. In Hamiltonian” [12]forthetemperaturegradient,onemight addition, the transition amplitude of a higher-order transition is calculate the Seebeck effect directly. constructed by first forming a product of the sequence of first- It is worthwhile to find the connection between the prob- order amplitudes of elementary transitions and then summing ability amplitudes in Eqs. (13)–(15)andthedensitymatrices over all intermediate states. We can describe each transport in Eqs. (8)and(10). The element of the density matrix is the process with a diagram, which has one line connecting average the product of two probability amplitudes over the two components of the final state, and several other lines members in an ensemble [20]: M representing the elementary transitions. To a given order of 1 residual interactions, the topology of diagrams can help us ρmn aα∗(n,t)aα(m,t), (18) classify and construct all possible transport processes [11]. = α M " In the Kubo formulation, all time dependence is included in where α is the index of a member in the canonical ensemble. density matrix, cf. Eqs. (9)and(7). To a given order in an To the first order in V (t), external field, the change in density matrix involves different members of the ensemble, cf. Eqs. (19)and(20). Besides, the (1) 1 (0) (1) 1 (1) (0) ρ a∗ (n,t)a (m,t) a∗ (n,t)a (m,t), density matrix is bilinear in transition amplitude. Therefore for mn = α α + α α α α atransportprocesswithmorethanoneelementarytransition, M " M " (19) one cannot express it as a product of propagators. It is interesting to note the semiclassical limit of Eq. (6). where a(0)(n,t)isthezero-ordertransitionamplitudefrom α Using Eq. (3)andthecommutationrelationbetweenr and v , initial state n to final state n , a(1)(m,t)isthefirst-order i i α Eq. (4)becomes transition amplitude| " from initial| " state n to final state m , (0) | " | " a (m; t)isthezero-ordertransitionamplitudefrominitial N N (1) ihe¯ state m to final state m ,anda (n,t)isthefirst-order j(r) e δ(r ri )vi r δ(r ri ) . (21) | " | " = − − 2m ∇ i − transition amplitude from initial state m to final state n . i 1 i 1 | " | " "= "= $ % !

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Averaging Eq. (21)overstate!!(r1,r2, ,rN ; t), Eq. (5) approximation and the semiclassical approximation [16], gives another expression for the microscopic··· current density: Eq. (22)isreducedtoEq.(28)ofRef.[15], the current density ihe¯ used by Greenwood. jm(r; t) rn!(r) = 2m ∇ In summary, we proved that for a mechanical perturbation 1 the microscopic response method is equivalent to and simpler eN dτ!!!∗m− [ ih¯ r eA(r; t)]!!, (22) + − ∇ − than the Kubo formula. To compute transport coefficients for & mechanical perturbations, the microscopic response method is where the arguments of ! in Eq. (22)are(r,r2, ,rN ; t), ! ··· advantageous because of the ease of obtaining expression to and n!(r) N dτ!!!∗!! is the number density of electrons a given order of residual interactions consistently. When the at point r in= external field (A,φ). The first term of Eq. (22)can . gradient of carrier density is small, the strict current density be neglected only when the gradient of the carrier density is Eq. (22)reducestothekineticexpressionofcurrentdensity small. It is consistent with qualitative reasoning based on the used by Greenwood [15]. : the smaller the gradient of the carrier density, the poorer the accuracy of carrier coordinate. If the position error of an electron in a state !! is )x,theerrorof We thank the Army Research Office for support under 1 velocity is )v m− h/¯ )x.Theerrorofcurrentdensityis MURI W91NF-06-2-0026, and the National Science Foun- 1 ∼ 1 n e)v m hn¯ /)x m h¯ rn .Underthesingle-particle dation for support under Grant No. DMR 0903225. ! ∼ − ! ∼ − ∇ !

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