Hot electron relaxation in one-dimensional models: Exact polaron dynamics versus relaxation in the presence of a Fermi sea

V. Meden, J. Fricke, C. Wöhler, and K. Schönhammer

Institut für Theoretische Physik der Universität Göttingen, Bunsenstr.9, D-37073 Göttingen, Germany (April 19, 1995)

Abstract

We present the exact solution for the time evolution of the electron and phonon momentum distribution for a one-dimensional polaron model with a linear electronic energy dispersion. The electron momentum distribution is shown to obey a Markovian quantum kinetic equation. Numerical results for the polaron model are compared to the corresponding exact results, when the negative momentum states are filled in the initial state. The presence of this Fermi sea modifies the dynamics except in the short time regime. The different long time dynamics might show up in comparison of hot electron relaxation of undoped and doped semiconductors.

PACS numbers: 71.38.+i, 78.47.+p, 72.10.Bg

Typeset using REVTEX

I. INTRODUCTION

Recently we proposed an exactly solvable one-dimensional model for the relaxation of an electron excited highly above the Fermi level due to the coupling to optical phonons [1]. The main tool used was the bosonization of the Hamiltonian and the electronic field operator well known from calculations of equilibrium properties of the Tomonaga-Luttinger model [2–4]. The exact results were compared to approximate calculations using quantum kinetic equations (QKE). Due to the existence of the electrons in the Fermi sea the inclusion of the corresponding renormalization of the phonon propagation in the QKE turned out to be important to obtain good agreement also for larger times. The model can be considered as a toy model for hot electron relaxation in doped semiconductors.

Here we present exact results for the even simpler polaron type model, where no Fermi sea is present. As in [1] we assume a linear dispersion in a single branch of “right moving” electrons. In contrast to our previous model we do not fill the Dirac sea of states with negative momenta, but consider a single electron coupled to optical phonons. As noticed previously this polaron problem simplifies due to the linear electronic dispersion. While the exact one-particle Green’s function was presented in [5] no results for the nonequilibrium distribution of the electron were discussed. Our exact results for the relaxation process to lower momenta for the polaron model can be considered as a toy model description for the hot electron relaxation in an undoped semiconductor for an appropriate time regime.

In Sec. II we present the model and the exact solution for the electron and phonon momentum distribution when an electron is injected into a system of thermal phonons. In Sec. III the numerical exact results for the polaron model are compared with the results from the model including the Fermi sea. The time regime where its presence is irrelevant is identified. In Sec. IV we show that the electron momentum distribution for the polaron model rigorously obeys a Markovian kinetic equation. The relation to approximate QKE is discussed. We summarize our results in Sec. V. In the Appendix we discuss the modifications in the exact solution to the polaron problem for the model containing branches of right and left moving electrons with a coupling to long wavelength optical phonons [6].

II. THE MODEL AND ITS SOLUTION

To describe the low energy physics of interacting electrons in one dimension the energy dispersion is usually linearized around the two Fermi points [2,3], which leads to two independent branches of fermions. In [1] we studied the coupling of electrons in a single branch with the dispersion ε_{k} = v_{F} k to optical phonons. The Dirac sea of negative energy states was assumed to be filled in the groundstate.

The Hamiltonian for a finite system of length L is given by

H = H_{e} + H_{p} + H_{ep} = H_{0} + H_{ep} ,  (1)

with

\begin{aligned}H_{e} & = & ∑_{k} ε_{k} (a_{k}^{†}a_{k} − ⟨a_{k}^{†}a_{k}⟩_{0}) , H_{p} = ∑_{q>0} ω_{q}B_{q}^{†}B_{q} , \\ H_{ep} & = & (\frac{2π}{L} )^{1/2} ∑_{q>0} g(q) (ρ_{q}^{†}B_{q} + B_{q}^{†}ρ_{q}) , \end{aligned}  (2)

where a_{k} (B_{q}) is the annihilation operator of an electron (phonon) and ρ_{q} denotes the Fourier transform of the electronic density operator (q ≠ 0)

ρ_{q} = ∑_{k} a_{k}^{†}a_{k+q} .  (3)

The electron-phonon coupling in our single branch model [1] is simplified compared to the usual Holstein type coupling [6] discussed in the Appendix and the noninteracting ground state expectation value ⟨...⟩_{0} has been subtracted in the kinetic energy of the electrons. While ⟨n_{k}⟩_{0} ≡ ⟨a_{k}^{†}a_{k}⟩_{0} = Θ(−k) was assumed in [1], presenting the infinite Dirac sea, the choice ⟨n_{k}⟩_{0} ≡ 0 is considered in the following, corresponding to the electronic vacuum state, which describes the system before an electron is injected. While the presence of the Dirac sea implies a finite anomaly [3], i. e. the properly normalized density operators ρ_{q} obey boson commutation relations, the density operators commute without the Dirac sea

[ρ_{q}, ρ_{q′}] = 0 ,  (4)

for arbitrary signs of q ≠ 0 and q′ ≠ 0. Due to this commutation relation the time evolution of ρ_{q} in the Heisenberg picture corresponding to the full Hamiltonian equals the one with the unperturbed one

ρ_{q}(t) = e^{−iqv_{F} t}ρ_{q} .  (5)

In the following we study the dynamics when a single electron is injected. Then in first quantization ρ_{q} is given by

ρ_{q} = e^{−iqx̂} ,  (6)

where x̂ is the position operator. As this is the subspace we mainly consider in this paper, we have

ρ_{q}^{†}ρ_{q} = 1  (7)

keeping in mind the restriction. It turns out that Eqs. (5) and (7) provide the key to the exact solution to our polaron problem. This can already be seen in the straightforward solution of the phonon dynamics. The Heisenberg equation of motion for B_{q}(t) reads

i\frac{d}{dt}B_{q}(t) = ω_{q}B_{q}(t) + (\frac{2π}{L} )^{1/2} g(q)ρ_{q}(t) .  (8)

As ρ_{q}(t) is given by Eq. (5) the solution to this inhomogenous equation is

B_{q}(t) = e^{−iω_{q}t} [B_{q} + (\frac{2π}{L} )^{1/2} g(q) (\frac{e^{−i(v_{f} q−ω_{q})t} − 1}{v_{F} q − ω_{q}}) ρ_{q}] .  (9)

The expectation value with the statistical operator |ϕ⟩ ⟨ϕ| ⊗ e^{−βH_{p}}/Z_{p}, Z_{p} = Tr_{p}e^{−βH_{p}}, where |ϕ⟩ is a normalized one-electron state, follows using Eq. (7) as

⟨N_{q}⟩ (t) ≡ ⟨B_{q}^{†}(t)B_{q}(t)⟩ = \frac{2π}{L} g^{2}(q) (\frac{sin [(ω_{q} − v_{F} q) t/2]}{(ω_{q} − v_{F} q) /2})^{2} + n_{B} (ω_{q}) ,  (10)

where n_{B} (ω_{q}) ≡ (e^{βω_{q}} − 1)^{−1} is the Bose function.

In order to calculate the time dependence of the electronic momentum distribution for an initial state |ϕ⟩ = |k_{0}⟩ we first consider the time evolution operator Ũ(t) ≡ e^{iH_{0}t}e^{−iHt} in the interaction representation. It obeys the Schrödinger equation

i\frac{d}{dt}Ũ(t) = (C(t) + C^{†}(t)) Ũ(t)  (11)

with

C(t) ≡ (\frac{2π}{L} )^{1/2} ∑_{q>0} g(q)e^{−i(ω_{q}−v_{F} q)t}ρ_{q}^{†}B_{q} .  (12)

We solve Eq. (11) with the ansatz

Ũ(t) = exp {−i ∫_{0}^{t} C^{†}(t′)dt′} exp {−i ∫_{0}^{t} C(t′)dt′}A(t) .  (13)

In the calculation of ˙ Ũ(t) we use [C(t_{1}), C(t_{2})] = 0 and

exp {−i ∫_{0}^{t} C^{†}(t′)dt′}C(t) exp {i ∫_{0}^{t} C^{†}(t′)dt′} =
C(t) + i ∫_{0}^{t} \frac{2π}{L} ∑_{q>0} g^{2}(q)e^{−i(ω_{q}−v_{F} q)(t−t′)}dt′1 .  (14)

The nested commutator series terminates after the second term because of Eq. (7). Insertion into Eq. (11) yields

Ũ(t) [\frac{2π}{L} ∑_{q>0} g^{2}(q)\frac{1 − e^{−i(ω_{q}−v_{F} q)t}}{ω_{q} − v_{F} q} − (A(t))^{−1}iȦ(t)] = 0 .  (15)

This shows that A(t) is a C-number. The solution of the differential equation (15) reads

A(t) = exp \{−\frac{2π}{L} ∑_{q>0} g^{2}(q)\frac{1}{(ω_{q} − v_{F} q)^{2}} [1 − i (ω_{q} − v_{F} q) t − e^{−i(ω_{q}−v_{F} q)t}]\} .  (16)

Together with Eq. (13) this provides the exact result for Ũ(t) in the one-electron sector of the Hilbert space.

Before we discuss the momentum distribution ⟨n_{k}⟩ (t) for the electron we shortly derive the exact result for the one-particle Green’s function iG_{k}(t) ≡ ⟨a_{k}(t)a_{k}^{†}⟩_{Vac,β}, where ⟨...⟩_{Vac,β}

denotes the expectation value with the statistical operator |Vac⟩ ⟨Vac| ⊗ e^{−βH_{p}}/Z_{p}. It was obtained previously by a different method [5]. Using e^{−iHt} = e^{−iH_{0}t}Ũ(t) in the one-electron subspace and e^{iHt} = e^{iH_{0}t} in the vacuum sector, we obtain

iG_{k}(t) = e^{−iv_{F} kt} ⟨a_{k}Ũ(t)a_{k}^{†}⟩_{Vac,β}
= e^{−iv_{F} kt} \frac{1}{L} ∫_{−L/2}^{L/2} dx ⟨Ũ_{x}(t)⟩_{β} ,  (17)

where Ũ_{x}(t) is obtained from Ũ(t) by replacing ρ_{q} = e^{−iqx̂} by e^{−iqx}. This operator in the phonon Hilbert space has to be averaged over e^{−βH_{p}}/Z_{p}, which we denote by ⟨...⟩_{β}. Using [7]

⟨exp {∑_{q} α_{q}B_{q}^{†}} exp {∑_{q′} μ_{q′}^{∗}B_{q′} }⟩_{β} = exp {∑_{q} α_{q}μ_{q}^{∗}n_{B} (ω_{q})}  (18)

we obtain

iG_{k}(t) = e^{−iv_{F} kt}A(t) exp {− ∑_{q>0} |f_{q}(t)|^{2} n_{B} (ω_{q})} ,  (19)

with

|f_{q}(t)|^{2} = \frac{2π}{L} g^{2}(q) (\frac{sin [(ω_{q} − v_{F} q) t/2]}{(ω_{q} − v_{F} q) /2})^{2} .  (20)

We next calculate the electronic momentum distribution when a single electron is injected in the momentum state |k_{0}⟩ at time t = 0

⟨n_{k}⟩ (t) = ⟨⟨k_{0} |e^{iHt}| k⟩ ⟨k |e^{−iHt}| k_{0}⟩⟩_{β}
= ⟨⟨k_{0} |Ũ^{†}(t)| k⟩ ⟨k |Ũ(t)| k_{0}⟩⟩_{β}
= \frac{1}{L^{2}} ∫_{−L/2}^{L/2} dxdx′e^{i(k−k_{0})(x−x′)} ⟨Ũ_{x}^{†}(t)Ũ_{x′}(t)⟩_{β} .  (21)

Using the Baker-Hausdorff formula and Eq. (18) the average of the time evolution operators is seen to depend only on x − x′. Then one integration can be performed and we obtain

⟨n_{k}⟩ (t) = \frac{1}{L} ∫_{−L/2}^{L/2} dxe^{i(k−k_{0})x}
× exp {∑_{q>0} |f_{q}(t)|^{2} [(e^{iqx} − 1) (1 + n_{B}(ω_{q})) + (e^{−iqx} − 1) n_{B}(ω_{q})]} .  (22)

In Eq. (22) ⟨n_{k}⟩ (t) is expressed as a spatial Fourier transform. As the Fourier transform can be written as a product of a loss part (involving e^{iqx}) and a gain part (involving e^{−iqx}) the result for ⟨n_{k}⟩ (t) can be written as a convolution of the loss and the gain contribution. Efficient numerical techniques to evaluate integrals of the type occuring in Eq. (22) were proposed in the context of spectral functions for the Tomonaga-Luttinger model [8]. This method will be outlined in the following section. For the two branch model discussed in the Appendix the only difference in the exact solution is the removal of the restriction of the q-sum in the exponent in Eq. (22). In this model the q-sum has to be performed over all q.

Analytical results can be obtained in the long time limit after the thermodynamic limit L → ∞ has been performed. In the thermodynamic limit the momentum distribution goes over to a function defined on IR. In order to normalize this function to one, ⟨n_{k}⟩ (t) has to be multiplied by the density of k-points, i. e. we consider the function n(k, t) ≡ (L/2π) ⟨n_{k}⟩ (t). For L → ∞ it is given by

n(k, t) = ∫_{−∞}^{∞} \frac{dx}{2π}e^{i(k−k_{0})x} exp \{∫_{0}^{∞} dqg^{2}(q) (\frac{sin [(ω_{q} − v_{F} q) t/2]}{(ω_{q} − v_{F} q) /2})^{2}
× [(e^{iqx} − 1) (1 + n_{B}(ω_{q})) + (e^{−iqx} − 1) n_{B}(ω_{q})]} .  (23)

In the long time limit [sin (xt)/x]^{2} can be replaced by 2πtδ(x), as familiar from the derivation of Fermi’s golden rule, and the q-integration in the exponent can be performed. After expanding the result for exp {...} in Eq. (23), the x-integration leads to a double sum of delta functions

n(k, t) → e^{−r_{B}t[1+2n_{B}(ω_{q_{B}})]} ∑_{n=0}^{∞} ∑_{m=0}^{∞} \frac{(r_{B}t [1 + n_{B}(ω_{q_{B}})])^{n}}{n!}
×\frac{(r_{B}tn_{B}(ω_{q_{B}}))^{m}}{m!} δ(k − k_{0} + [n − m]q_{B}) ,  (24)

where q_{B} is given by ω_{q_{B}} = v_{F} q_{B}. Alternatively one can write this expression as a single sum over delta functions with weights involving the Bessel functions of integer order I_{l}(z) as discussed in a different context in Ref. [5]. For dispersionless optical phonons ω_{q} ≡ ω_{0} the rate r_{B} is given by

r_{B} = 2πg^{2}(q_{B})/v_{F}  (25)

and q_{B} = ω_{0}/v_{F}. In this long time limit sharp energy and momentum conservation holds: The electron has emitted or absorbed an integer number of phonons. At zero temperature only emission is possible and the probability p_{n} for the emission of n phonons is given by a Poisson distribution. The average momentum loss r_{B}q_{B}t in the large time limit increases linearly with time independent of temperature and the mean square deviation increases like q_{B}\sqrt{r_{B}t [1 + 2n_{B}(ω_{q_{B}})]}. As there is no Fermi sea to stop relaxation the envelope function of n(k, t) becomes Gaussian when the average of k − k_{0} tends to large negative values as the loss and gain contribution become Gaussian separately leading to a Gaussian also for the convolution. In order to see how the large time limit in Eq. (24) is reached, the integral on the right hand side (rhs) of Eq. (22) or (23) has to be performed numerically. This is done in the following section, where exact results for very large but finite systems are compared to the corresponding results in the presence of the Fermi sea.

III. NUMERICAL RESULTS

In this section we present exact numerical results for the polaron dynamics for ω_{q} ≡ ω_{0} and a q-independent coupling constant g(q) up to a cutoff momentum q_{c} ≫ q_{B} = ω_{0}/v_{F}. For zero temperature we compare the results with the corresponding ones in the presence of the Dirac sea. The dimensionless coupling strength is given by α ≡ r_{B}/ω_{0} = 2πg^{2}/(v_{F} ω_{0}). The shape of the change of the phonon momentum distribution δ ⟨N_{q}⟩ (t) ≡ ⟨N_{q}⟩ (t) − n_{B}(ω_{q}) in Eq. (10) is independent of the coupling strength and the temperature and shows oscillatory behaviour for q ≠ q_{B}, but increases like (2π/L)g^{2}t^{2} for q = q_{B}. A 3d-plot of δ ⟨N_{q}⟩ (t) is shown in Fig. 1 for a finite system. The system size is determined by the dimensionless energy level spacing ν ≡ (\frac{2π}{L}v_{F} ) /ω_{0}, which is chosen to be ν = 1/40 in the figure. The behaviour for q ≈ q_{B} is very different from the model including the Dirac sea discussed in Ref. [1] at zero temperature. There ⟨N_{q}⟩ (t) is oscillatory for all momenta and for q = q_{B}, for example, one obtains (2π/L)[1−cos (2g\sqrt{q_{B}}t)]/(2q_{B}). This shows that for t > t_{0} ≡ 1/\sqrt{r_{B}ω_{0}} the phonon dynamics in both cases differs considerably for q ≈ q_{B}. This difference shows up also in momentum integrated quantities like the change in the total energy in the phonon system δ ⟨H_{p}⟩ (t) which in the polaron case increases linearly with time δ ⟨H_{p}⟩ (t) → ω_{0}r_{B}t for t ≫ t_{0} as shown in Fig. 2. From the exact result for ⟨n_{k}⟩ (t) in Eq. (22) it is easy to show that ⟨H_{e}⟩ (t) − v_{F} k_{0} → −ω_{0}r_{B}t, while ⟨H_{ep}⟩ (t) tends to a constant in the large time limit. For t = 0 the first derivative of the electron energy with respect to time vanishes and the second derivative is given by

\frac{d^{2}}{dt^{2}} ⟨H_{e}⟩ (t)|_{t=0} = −2\frac{2π}{L} ∑_{q>0} g^{2}(q)v_{F} q < 0 .  (26)

Therefore ⟨H_{e}⟩ (t) − v_{F} k_{0} decreases quadratically for short times. The total energy ⟨H⟩ (t) is strictly conserved.

To obtain the dynamics of the electronic distribution ⟨n_{k}⟩ (t) the integral on the rhs of Eq.

(22) has to be performed numerically. Due to the strongly oscillatory integrand we expand the terms with e^{±iqx} in a power series in exp (±i2πx/L)

exp \{∑_{n=1}^{n_{c}} |f_{n}(t)|^{2} (\frac{1}{2} ± \frac{1}{2} + n_{B}(ω_{n})) (e^{±i \frac{2π}{L} x})^{n}\} = ∑_{n=0}^{∞} a_{n,±}^{(n_{c})}(t) (e^{±i \frac{2π}{L} x})^{n} ,  (27)

where q_{c} = 2πn_{c}/L and the coefficients a_{n,±}^{(n_{c})}(t) can be determined recursively [8]. For m > 1, l ∈ IN_{0}, and i = 0, ..., m − 1 we obtain

a_{lm+i,±}^{(m)}(t) = ∑_{j=0}^{l} \frac{|f_{m}(t)|^{2j}}{j!} (\frac{1}{2} ± \frac{1}{2} + n_{B}(ω_{n}))^{j} a_{m(l−j)+i,±}^{(m−1)}(t)  (28)

with the starting values (m = 1)

a_{l,±}^{(1)}(t) = \frac{|f_{1}(t)|^{2l}}{l!} (\frac{1}{2} ± \frac{1}{2} + n_{B}(ω_{n}))^{l} .  (29)

Then the x-integration is trivial and with k_{0} = n_{0}2π/L leads to

⟨n_{n \frac{2π}{L}} ⟩ (t) = exp \{− ∑_{q≠0} |f_{q}(t)|^{2} [1 + 2n_{B}(ω_{q})]\} ∑_{m=max(0,n_{0}−n)}^{∞} a_{m,+}^{(n_{c})}(t)a_{m+n−n_{0},−}^{(n_{c})}(t) , (30)

for n ∈ ZZ . For zero temperature a_{l,−}^{(m)} = δ_{l,0} which shows that for k > k_{0}, ⟨n_{k}⟩ (t) vanishes identically for all times due to the rotating wave form of the electron-phonon interaction in the one branch Hamiltonian in Eq. (2). This is different in the two branch model discussed in the Appendix, which even at T = 0 leads to small weight also for k > k_{0}. This effect was discussed previously in the approximate description of a model with the usual nonrelativistic quadratic energy dispersion [9]. As in [1] only the one branch model was treated we restrict ourselves to this model in the discussion of the influence of the Fermi sea. Later we show explicit results for ⟨n_{k}⟩ (t) also for the two branch model.

For the polaron result Eq. (30) the initial momentum k_{0} is irrelevant as only relative momenta enter. This is different in the model including the Fermi sea. As the Fermi momentum k_{F} for this model was taken to be zero in [1], we use k_{0} > 0 in the following. The presence of the Fermi sea has two effects. First the trivial one that the long time relaxation discussed following Eq. (23) is blocked due to the Pauli principle. The more subtle one is the different dynamics on the time scale t_{0} introduced above. In order to avoid the blocking effect to influence the difference of the results we choose k_{0}/q_{B} ≫ \sqrt{α} in the following. Fig. 3 shows a 3d-plot for ⟨n_{k}⟩ (t) at T = 0 as a function of k/q_{B} and ω_{0}t in the weak coupling regime α = 0.0314. The value of ⟨n_{k_{0}}⟩ (t) is scaled by a factor of 1/5. The figure shows the sharpening of the distribution around k_{0} − nq_{B}, n ∈ IN for increasing time. In this weak coupling limit the peaks are already well resolved, when only a few phonons have been emitted. This is different for the large coupling α = 0.314 used in Fig. 4 (T = 0), where ⟨n_{k_{0}}⟩ (t) is scaled by a factor of 1/20. Now the sharpening is only prominent when the average momentum loss of the electron is much larger than q_{B}.

In the following we compare the polaron results for ⟨n_{k}⟩ (t) with the corresponding momentum distribution f(k, k_{0}, t) in the model including the Fermi sea. For the method to obtain the exact result for f(k, k_{0}, t) at T = 0 we refer to Ref. [1]. Figs. 5 - 7 show the comparison at T = 0 for k_{0} = 7.5q_{B} in the weak coupling limit (α = 0.0314) and ω_{0}t_{0}, 2ω_{0}t_{0}, and 10ω_{0}t_{0}, where the time t_{0} inferred from the deviation of the phonon dynamics is given by ω_{0}t_{0} = 5.64. While in Fig. 6 and especially Fig. 5 the difference of the electron dynamics is small there are strong deviations for 10ω_{0}t_{0} shown in Fig. 7. As the dashed line we additionally present the single branch polaron results for a finite temperature T = ω_{0} in the Figs. 5 - 7. The main differences compared to T = 0 are the much faster depletion of the initially occupied state k_{0} and the transfer of weight also to momenta k = k_{0} + nq_{B} with n ∈ IN. Due to this transfer the T ≠ 0 envelope function becomes Gaussian faster than the one at T = 0.

The T = 0 electron distributions for k_{0} = 17.5q_{B} and α = 0.314 (ω_{0}t_{0} = 1.78) are shown in Figs. 8-10. For this stronger coupling we in addition present exact results for the two branch polaron model (dashed line). For small times we find weight also at momenta k > k_{0} as discussed above. Nevertheless ⟨H_{e}⟩ (t) never exceeds the starting value v_{F} k_{0}. At t = 0 d^{2} ⟨H_{e}⟩ (t)/dt^{2} vanishes in the two branch model with the linear electron dispersion and the first nonvanishing derivative, the fourth, is negative. This is in contrast to the polaron problem with the usual quadratic electron dispersion where the second derivative is positive and the energy of the electron increases for small times [9]. For larger times the difference of ⟨n_{k}⟩ (t) between the two branch and the single branch model vanishes as can be seen in Fig.

10. For t → ∞ the two branch model shows the same asymptotic behaviour as discussed following Eq. (23).

As mentioned in the introduction the modification of the dynamics for larger times due to the presence of the Fermi sea suggests that it might manifest itself in the comparison of hot electron relaxation in undoped and doped optically excited semiconductors. We expect the difference to be less pronounced in dimensions larger than one.

IV. QUANTUM KINETIC EQUATIONS

As discussed following Eq. (23) the momentum distribution for the polaron shows sharp peaks in the long time limit. This is what one expects from an attempt to describe the polaron dynamics in the framework of a Boltzmann equation, which assumes successive completed electron-phonon processes with momentum and energy conservation in each scattering event. From the exact results presented in the preceeding section it is obvious that such a simplification does not occur at finite times. Usually one derives approximate non-Markovian QKE’s in the occupation numbers, by breaking the hierarchy of the equations of motion [10,9] or using nonequilibrium Green’s functions techniques [11–13]. For our polaron model it is surprising that the exact electron distribution obeys a much simpler differential equation. In fact differentiating Eq. (22) with respect to time leads to an exact Markovian kinetic equation for the momentum distribution

\frac{d}{dt} ⟨n_{k}⟩ (t) = 2\frac{2π}{L} ∑_{q>0} g^{2}(q)\frac{sin [(ω_{q} − v_{F} q) t]}{ω_{q} − v_{F} q} (⟨n_{k+q}⟩ (t) [1 + n_{B}(ω_{q})]
+ ⟨n_{k−q}⟩ (t)n_{B}(ω_{q}) − ⟨n_{k}⟩ (t) [1 + 2n_{B}(ω_{q})]) .  (31)

This equation is linear in the electronic occupancies and contains phonon occupation numbers only via their initial thermal averages. As we arrived at this equation using the exact solution for the ⟨n_{k}⟩ (t) it is interesting to see if it is possible to derive this equation without obtaining the exact result first. The most straightforward way to derive an approximate QKE for our model is to break the hierarchy of the equations of motion by factorizing higher expectation values [10,9]. For our model the first step is the equation for the Heisenberg operators n_{k}(t). It reads

i\frac{d}{dt}n_{k}(t) = (\frac{2π}{L} )^{1/2} ∑_{q>0} g(q) \{[a_{k}^{†}(t)a_{k−q}(t) − a_{k+q}^{†}(t)a_{k}(t)] B_{q}(t) − h.c.\}  (32)

In the equations of motion for the operators a_{k}^{†}a_{k−q}B_{q} etc. the averages of the resulting products of four operators are factorized. This yields

\frac{d}{dt} ⟨n_{k}⟩ (t) = 2\frac{2π}{L} ∑_{q>0} g^{2}(q) ∫_{0}^{t} dt′ cos [(ω_{q} − v_{F} q) (t − t′)]
× {⟨n_{k+q}⟩ (t′) [1 − ⟨n_{k}⟩ (t′)] [1 + ⟨N_{q}⟩ (t′)]
+ ⟨n_{k−q}⟩ (t′) [1 − ⟨n_{k}⟩ (t′)] ⟨N_{q}⟩ (t′)
− ⟨n_{k}⟩ (t′) [1 − ⟨n_{k−q}⟩ (t′)] [1 + ⟨N_{q}⟩ (t′)]
− ⟨n_{k}⟩ (t′) [1 − ⟨n_{k+q}⟩ (t′)] ⟨N_{q}⟩ (t′)} .  (33)

For the phonons one can proceed similarly and obtains

\frac{d}{dt} ⟨N_{q}⟩ (t) = 2\frac{2π}{L} g^{2}(q) ∫_{0}^{t} dt′ cos [(ω_{q} − v_{F} q) (t − t′)]

× ∑_{k} {⟨n_{k}⟩ (t′) [1 − ⟨n_{k−q}⟩ (t′)] [1 + ⟨N_{q}⟩ (t′)]

− ⟨n_{k−q}⟩ (t′) [1 − ⟨n_{k}⟩ (t′)] ⟨N_{q}⟩ (t′)}

= 2\frac{2π}{L} g^{2}(q) ∫_{0}^{t} dt′ cos [(ω_{q} − v_{F} q) (t − t′)]
× [1 − ∑_{k} ⟨n_{k}⟩ (t′) ⟨n_{k−q}⟩ (t′)] ,  (34)

where ∑_{k} ⟨n_{k±q}⟩ (t′) = 1 for all q and t′ has been used. In the limit L → ∞ we can linearize Eqs. (33) and (34) as ⟨n_{k}⟩ (t) ∼ O(1/L) for k ≠ k_{0} and ⟨N_{q}⟩ (t) ∼ n_{B}(ω_{q}) + O(1/L). Then Eq. (33) simplifies to

\frac{d}{dt} ⟨n_{k}⟩ (t) = 2\frac{2π}{L} ∑_{q>0} g^{2}(q) ∫_{0}^{t} dt′ cos [(ω_{q} − v_{F} q) (t − t′)]
× {⟨n_{k+q}⟩ (t′) [1 + n_{B}(ω_{q})] + ⟨n_{k−q}⟩ (t′)n_{B}(ω_{q}) − ⟨n_{k}⟩ (t′) [1 + 2n_{B}(ω_{q})]} (35)

and Eq. (34) reduces to

\frac{d}{dt} ⟨N_{q}⟩ (t) = 2\frac{2π}{L} g^{2}(q) ∫_{0}^{t} dt′ cos [(ω_{q} − v_{F} q) (t − t′)]
= 2\frac{2π}{L} g^{2}(q)\frac{sin [(ω_{q} − v_{F} q) t]}{ω_{q} − v_{F} q} .  (36)

Comparing Eq. (36) with Eq. (10) we find that the non-Markovian equation for the phonon distribution becomes exact in this limit. For large but finite L and our initial conditions the nonlinear term in Eq. (34) is small for all times and we can neglect its contribution compared to 1, i.e. the non-Markovian equation for ⟨N_{q}⟩ (t) is a good approximation to the exact result.

If we replaced the curly bracket in Eq. (35) by its value at time t, the t′-integration could be carried out to yield the exact Markovian equation (31). But from the derivation one would consider this step as an additional approximation. As this procedure is certainly not satisfying we try again to derive Eq. (31), but now building in the special assumptions under which we obtained the exact solution: An initial state with a single electron and equilibrium phonons described by a canonical ensemble. In the single electron subspace the operator a_{k}^{†}a_{k−q} − a_{k+q}^{†}a_{k} in Eq. (32) is identical to (n_{k} − n_{k+q}) ρ_{q}^{†}. The time dependence of the Heisenberg operators ρ_{q}^{†}(t) and B_{q}(t) given by Eqs. (5) and (9) can now be inserted in Eq. (32) and one obtains after averaging over an initial statistical operator ρ_{0} = |ϕ⟩ ⟨ϕ| ⊗ ρ_{p,0} denoted by ⟨...⟩ (⟨n_{k}(t)⟩ ≡ ⟨n_{k}⟩ (t))

\frac{d}{dt} ⟨n_{k}⟩ (t) = 2\frac{2π}{L} ∑_{q>0} g^{2}(q)\frac{sin [(ω_{q} − v_{F} q) t]}{ω_{q} − v_{F} q} [⟨n_{k+q}⟩ (t) − ⟨n_{k}⟩ (t)]

−i (\frac{2π}{L} )^{1/2} ∑_{q>0} g(q) \{e^{−i(ω_{q}−v_{F} q)t} ⟨[n_{k}(t) − n_{k+q}(t)] ρ_{q}^{†}B_{q}⟩ − c.c.\} .  (37)

At T = 0, i.e. the phonons in the ground state |{0}⟩, the second line on the rhs of Eq.

(37) vanishes and our second attempt to directly derive a Markovian kinetic equation seems successful. But it is obvious from this derivation that for a pure initial state for the phonons this equation holds for the ground state only. If the phonons are initially in a mixed state described by a statistical operator ρ_{p,0} again the Markovian equation holds only if ρ_{p,0} is of the canonical form ρ_{p,0} = e^{−βH_{p}}/Z_{p}. With the canonical statistical operator the expectation values in the second line on the rhs of Eq. (37) can be calculated generalizing the procedure used in Eq. (21). They yield the terms proportional to the Bose function n_{B} in Eq. (31). This again is not a derivation which completely avoids information about the exact solution. But the derivation makes it clear that Eq. (31) holds only under the special assumptions made for the initial state.

In Fig. 11 we compare the result obtained for ⟨n_{k}⟩ (t) from Eqs. (33) and (34) (dotted line) with the exact one (solid line) for the weak coupling strength α = 0.0314 and 2ω_{0}t_{0} at T = 0. We find two prominent deviations from the exact result. The weight in the first replica at k_{0}−q_{B} is too large and the shape of the second replica at k_{0}−2q_{B} is different. Both differences are solely due to the non-Markovian nature of the QKE and do not occur if the Markovian version of Eq. (33) is solved, because in Fig. 11 the time is sufficiently small to enable the linearization of Eq. (33). The first shortcoming can be cured by introducing a damping factor exp [−2γ(t − t′)] in Eq. (33). This damping factor occurs in a derivation of QKE’s within the Green’s function techniques if the renormalization of the electron propagation is taken into account [12,13,1]. For the determination of γ from the electron selfenergy we refer to Ref. [1]. The result of Eq. (33) including the damping is shown as the dashed line in Fig. 11. For larger times the non-Markovian equation without damping leads to slightly negative occupation numbers as discussed previously [12,14,1]. The inclusion of the damping suppresses such negative occupation numbers.

In the model including the Fermi sea we found at T = 0 that for short times the Markovian approximation works surprisingly well [1]. For short times there is no difference between the dynamics of the Fermi sea model and the polaron model. We found that a Markovian QKE (31), which follows by linearization from the usual Markovian version of Eq. (33), is exact for the polaron model. As the linearization is a good approximation at short times, it is understandable that also in the Fermi sea model the Markovian approximation should work very well at short times.

V. SUMMARY

We have presented the exact analytical result for the time dependence of the electronic momentum distribution for a polaron model with a linear electronic energy dispersion. The simple looking exact expression Eq. (22) for ⟨n_{k}⟩ contains qualitatively different possibilities to approach the distribution with the Gaussian envelope in the long time limit, depending on the strength of the electron-phonon coupling. The polaron results were compared to the corresponding model, where an additional electron was injected into a system with an infinite Dirac sea. The different commutation relations for the electronic momentum operators ρ_{q} lead to different dynamics of the phonons, strictly oscillatory in the presence of the Dirac sea, in contrast to the polaron dynamics in Eq. (10), shown in Fig. 1. The phonon dynamics starts to differ at times of the order t_{0} = 1/\sqrt{r_{B}ω_{0}}. At times much larger than t_{0} also the results for the electronic momentum distributions differ appreciably in both models.

The electronic momentum distribution was shown to exactly obey a Markovian kinetic equation. This surprising fact could not be understood from the usual derivation of approximate kinetic equations, which in the large time limit do not provide an adequate description. It was shown that this exact Markovian equation holds only under the special assumptions made for the initial state. The possible relevance of our results for hot electron relaxation in semiconductors was discussed.

ACKNOWLEDGMENTS

The authors would like to thank E. Runge and R. Zimmermann for stimulating discussions. Three of us (V. M., J. F., and C. W. ) are grateful to the Deutsche Forschungsgemeinschaft (SFB 345 “Festkörper weit weg vom Gleichgewicht”) for financial support.

APPENDIX A

In this Appendix we discuss the polaron dynamics for a two branch model with coupling to long wavelength optical phonons. The Hamiltonian reads [6]

H = ∑_{α=±} ∑_{k} αv_{F} ka_{k,α}^{†}a_{k,α} + ∑_{q} ω_{q}B_{q}^{†}B_{q}
+ (\frac{2π}{L} )^{1/2} ∑_{α=±} ∑_{q} g(q)ρ_{q,α} (B_{q}^{†} + B_{−q}) ,  (A1)

with ρ_{q,α} = ∑_{k} a_{k,α}^{†}a_{k+q,α}, ω_{q} = ω_{−q} and g(q) = g(−q). For q_{c} ≪ k_{F}, where q_{c} is the cutoff in the coupling function g(q), this model results from the one-dimensional model with the usual quadratic dispersion by linearizing around the two Fermi points ±k_{F} of the related model with the Fermi sea [2]. The density operators commute [ρ_{q,α}, ρ_{q′,α′}] = 0 leading to ρ_{q,α} = exp (−iαv_{F} qt)ρ_{q,α} generalizing Eq. (5). In the subspace with one right moving electron we have ρ_{q,α}^{†}ρ_{q,α′} = δ_{α,+}δ_{α′,+}1 and Eqs. (9) and (10) hold for arbitrary q. In the equation of motion for Ũ(t) for this model the corresponding operator C(t) in Eq. (11) has the same form as the operator in Eq. (12) but without the restriction of the q-sum to positive values. The remaining steps to calculate ⟨n_{k}⟩ (t) are identical to the ones in Sec. II and we obtain

⟨n_{k}⟩ (t) = \frac{1}{L} ∫_{−L/2}^{L/2} dxe^{i(k−k_{0})x}
× exp {∑_{q} |f_{q}(t)|^{2} [(e^{iqx} − 1) (1 + n_{B}(ω_{q})) + (e^{−iqx} − 1) n_{B}(ω_{q})]} .  (A2)

with |f_{q}(t)|^{2} given in Eq. (20). The exact Markovian kinetic equation for ⟨n_{k}⟩ (t) is the same as in Eq. (31) without the restriction of the q-sum.

The numerical evaluation of the integral in Eq. (A2) is the same as in Eq. (30), apart from the replacement

|f_{q}(t)|^{2} (\frac{1}{2} ± \frac{1}{2} + n_{B}(ω_{q})) → |f_{q}(t)|^{2} (\frac{1}{2} ± \frac{1}{2} + n_{B}(ω_{q}))^{+ |f_{−q}(t)|^{2} (\frac{1}{2} ∓ \frac{1}{2} + n_{B}(ω_{q})) . (A3)}

13

REFERENCES

[1] Meden, V., Wöhler, C., Fricke, J., Schönhammer, K.: submitted for publication, 1 March 1995

[2] Tomonaga, S.: Prog. Theor. Phys. 5, 544 (1950); Luttinger, J.M.: J. Math. Phys. 4, 1154 (1963)

[3] Mattis, D.C., Lieb, E.H.: J. Math. Phys. 6, 304 (1965)

[4] Haldane, F.D.M.: J. Phys. C 14, 2585 (1981)

[5] Mahan, G.D.: Many-Particle Physics. New York: Plenum Press 1981

[6] Meden, V., Schönhammer, K., Gunnarsson, O.: Phys. Rev. B 50, 11179 (1994)

[7] Mermin, N.D.: J. Math. Phys. 7, 1038 (1966)

[8] Schönhammer, K., Meden, V.: Phys. Rev. B 47, 16205 (1993) and Phys. Rev. B 48, 11521 (1993)

[9] Schilp, J., Kuhn, T., Mahler, G.: Phys. Rev. B 50, 5435 (1994)

[10] Zimmermann, R., Phys. Stat. Sol. 159, 317 (1990)

[11] Lipavský, P., Špička, V., Velický, B.: Phys. Rev. B 34, 6933 (1986)

[12] Haug, H., Phys. Stat. Sol. 173, 139 (1992); Bányai, L., Tran Thoai, D.B., Remling, C., Haug, H.: Phys. Stat. Sol. 173, 149 (1992)

[13] Schoeller, H., Ann. Phys. (NY) 229, 273 (1994)

[14] Zimmermann, R., Wauer, J., J. Lum. 58, 271 (1994)

FIGURES

FIG. 1. Change of the phonon distribution δ ⟨N_{q}⟩ (t) as a function of the momentum q/q_{B} and the time ω_{0}t for α = 0.0314 and q_{c}/q_{B} = 5 for a finite system with ν ≡ (\frac{2π}{L} v_{F} ) /ω_{0} = 1/40.

FIG. 2. Comparison between the change of the energy of the phonon system in the Fermi sea model and the polaron model. The parameters are: α = 0.0314, q_{c}/q_{B} = 5, and ν = 1/16. The Fermi sea (solid line) and the polaron (dotted line) results start to deviate at ω_{0}t ≈ ω_{0}t_{0} = 5.64.

FIG. 3. Zero temperature electron distribution ⟨n_{k}⟩ (t) for the one branch polaron model as a function of k/q_{B} and ω_{0}t for α = 0.0314, q_{c}/q_{B} = 5, and k_{0}/q_{B} = 7.5 for a finite system with ν = 1/16. The occupation at k_{0} is scaled by a factor of 1/5.

FIG. 4. Zero temperature electron distribution ⟨n_{k}⟩ (t) for the one branch polaron model as a function of k/q_{B} and ω_{0}t for α = 0.314, q_{c}/q_{B} = 5, and k_{0}/q_{B} = 7.5 for a finite system with ν = 1/16. The occupation at k_{0} is scaled by a factor of 1/20. The tiny jump at k/q_{B} = 2.5 for short times is due to the cutoff of the coupling function g(q).

FIG. 5. Comparison between the electron distribution functions of the T = 0 Fermi sea model (solid line), the T = 0 polaron model (dotted line), and the T ≠ 0 polaron model. The parameters are: α = 0.0314, k_{0}/q_{B} = 7.5, q_{c}/q_{B} = 5, and ν = 1/16, and the time is ω_{0}t = ω_{0}t_{0} = 5.64. The occupation at k_{0} is scaled by a factor 1/10. For the dashed line the temperature is T = ω_{0}.

FIG. 6. The same as in Fig. 5, but for ω_{0}t = 2ω_{0}t_{0}.

FIG. 7. The same as in Fig. 5, but for ω_{0}t = 10ω_{0}t_{0}. Here the occupation at k_{0} is unscaled.

FIG. 8. Comparison between the zero temperature electron distribution functions of the Fermi sea model (solid line), the single branch polaron model (dotted line), and the two branch polaron model (dashed line). The parameters are: α = 0.314, k_{0}/q_{B} = 17.5, q_{c}/q_{B} = 5, and ν = 1/16, and the time is ω_{0}t = ω_{0}t_{0} = 1.78. The occupation at k_{0} is scaled by a factor 1/10.

FIG. 9. The same as in Fig. 8, but for ω_{0}t = 2ω_{0}t_{0}.

FIG. 10. The same as in Fig. 8, but for ω_{0}t = 10ω_{0}t_{0}. Here the occupation at k_{0} is unscaled.

FIG. 11. Comparison between the exact zero temperature electron distribution function of the one branch polaron model (solid line) and the one calculated within the non-Markovian QKE (dotted line without damping, dashed line with damping). The parameters are: α = 0.0314, k_{0}/q_{B} = 7.5, q_{c}/q_{B} = 5, and ν = 1/16, and the time is ω_{0}t = 2ω_{0}t_{0}. The occupation at k_{0} is scaled by a factor 1/10.