# Theory of laser-induced demagnetization at high temperatures

**ABSTRACT** Laser-induced demagnetization is theoretically studied by explicitly taking

into account interactions among electrons, spins and lattice. Assuming that the

demagnetization processes take place during the thermalization of the

sub-systems, the temperature dynamics is given by the energy transfer between

the thermalized interacting baths. These energy transfers are accounted for

explicitly through electron-magnons and electron-phonons interaction, which

govern the demagnetization time scale. By properly treating the spin system in

a self-consistent random phase approximation, we derive magnetization dynamic

equations for a broad range of temperature. The dependence of demagnetization

on the temperature and pumping laser intensity is calculated in detail. In

particular, we show several salient features for understanding magnetization

dynamics near the Curie temperature. While the critical slowdown in dynamics

occurs, we find that an external magnetic field can restore the fast dynamics.

We discuss the implication of the fast dynamics in the application of heat

assisted magnetic recording.

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Page 1

arXiv:1112.2428v1 [cond-mat.mes-hall] 12 Dec 2011

Theory of laser-induced demagnetization at high temperatures

A. Manchon1,2, Q. Li1, L. Xu1, and S. Zhang1

1Department of Physics, University of Arizona, Tucson, AZ 85721, USA;

2Materials Science and Engineering, Physical Science and Engineering Division, KAUST, Saudi Arabia

(Dated: December 13, 2011)

Laser-induced demagnetization is theoretically studied by explicitly taking into account interac-

tions among electrons, spins and lattice. Assuming that the demagnetization processes take place

during the thermalization of the sub-systems, the temperature dynamics is given by the energy

transfer between the thermalized interacting baths. These energy transfers are accounted for explic-

itly through electron-magnons and electron-phonons interaction, which govern the demagnetization

time scale. By properly treating the spin system in a self-consistent random phase approximation,

we derive magnetization dynamic equations for a broad range of temperature. The dependence of

demagnetization on the temperature and pumping laser intensity is calculated in detail. In par-

ticular, we show several salient features for understanding magnetization dynamics near the Curie

temperature. While the critical slowdown in dynamics occurs, we find that an external magnetic

field can restore the fast dynamics. We discuss the implication of the fast dynamics in the application

of heat assisted magnetic recording.

PACS numbers: 75.78.Jp,75.40Gb,75.70.-i

Keywords:

I.INTRODUCTION

Laser-induced Demagnetization1,2(LID) and Heat As-

sisted Magnetization Reversal3(HAMR) constitute a

promising way to manipulate the magnetization direc-

tion by optical means. While both LID and HAMR in-

volve laser-induced magnetization dynamics of magnetic

materials, there are several important differences. LID

is usually considered as an ultrafast process where the

hot electrons excited by the laser field transfer their en-

ergy to the spin system, causing demagnization. The

demagnetization time scale ranges from 100 femtosecond

to a few picoseconds. For HAMR, the laser field is to

heat the magnetic material up to the Curie temperature

so that the large room-temperature magnetic anisotropy

is reduced to a much smaller value and consequently, a

moderate magnetic field is able to reverse the magneti-

zation. The time scale for the HAMR process is about

sub-nanosecond, three orders of magnitude larger com-

pared to LID.

LIDobservations have been carried outina

number of magnetic materials including transition

metals1,4–6, insulators7, half-metals8–10and dilute mag-

netic semiconductors11. A general consensus of the laser-

induced demagnetization process is that the high energy

non-thermal electrons generated by a laser field relax

their energy to various low excitation states of the elec-

tron, spin and lattice12. The phenomenological model for

this physical picture is referred to as three-temperature

model1,5,9where the three interacting sub-systems (elec-

trons, spins, lattice) are assumed thermalized individ-

ually at different temperatures which are equilibrated

according to a set of energy rate equations. By fitting

experimental data to the model, reasonable relaxation

times of the order of several hundred femtosecond to a

few picoseconds have been determined.

Various microscopic theories4,13–16have been proposed

to interpret these ultrafast time scales of electron-spin

and electron-lattice relaxations.

proposed that the laser field can directly excite the spin-

polarized ground states to spin-unpolarized excited states

in the presence of spin-orbit coupling, i.e., the spin-flip

transition leads to the demagnetization during the laser

pulse. In this picture, the demagnetization is instan-

taneous (≈50-150fs).

show that due to a few active ”hot spots”, the instan-

taneous demagnetization is expected for at most a few

percent of the magnetization, consistently with experi-

mental arguments18. Koopmans et al.4,5suggested that

the excited electrons lose their spins in the presence of

spin-orbit coupling and impurities or phonons, through

an ”Elliot-Yafet”-type (EY) spin-flip scattering. Recent

numerical evaluations of the EY mechanism in transition

metals14tend to support this point of view. Alterna-

tively, Battiato et al.19recently modelled such ultrafast

demagnetization in terms of superdiffusive currents. Fi-

nally, numerical simulations of the ultrafast demagneti-

zation based on the phenomenological Landau-Lifshitz-

Bloch equation have been achieved successfully20.

While these demagnetization mechanisms provide rea-

sonable estimation for the demagnetization time scales,

the theories are usually limited to the temperature much

lower than the Curie temperature and/or make no di-

rect connection to the highly successful phenomenologi-

cal three-temperature model1,5,9. As it has been recently

shown experimentally7,8, most interesting magnetization

dynamics occur near the Curie temperature.

In this paper, we propose a microscopic theory of the

laser-induced magnetization dynamics under the three-

temperature framework and derive the equations that

govern the demagnetization at arbitrary temperatures.

More specifically we predict magnetization dynamics in

the critical region.

Zhang and H¨ ubner13

Recent numerical simulations17

Page 2

2

The paper is organized as follows. In Sec. II, we pro-

pose a model for LID processes. In Sec. III, we describe

the spin system by the Heisenberg model which is solved

by using a self-consistent random phase approximation.

In Sec. IV, the central dynamic equations for the magne-

tization are derived. In Sec. V, the numerical solutions of

the equations are carried out and the connection of our

results with the experimental data of LID and HAMR is

made in Sec. VI. We conclude our paper in Sec. VII.

II.MODEL OF LID

A.Spin loss mechanisms

One of the keys to understand ultrafast demagneti-

zation is to identify the mechanisms responsible for the

spin memory loss. In the case of transition metal ferro-

magnets for example, the spin relaxation processes lead

to complex spin dynamics due to the itinerant character

of the magnetization. Elliott21first proposed that delo-

calized electrons in spin-orbit coupled bands may lose

their spin under spin-independent momentum scatter-

ing events (such as electron-electron or electron-impurity

interaction). This mechanism was later extended to

electron-phonon scattering by Yafet and Overhauser22.

Consequently, the spin relaxation time τsis directly pro-

portional to the momentum relaxation time τp. Whereas

the electron-electron relaxation time is on the order of

a few femtoseconds23(fs), the electron-impurity and

electron-phonon relaxation time is on the picoseconds

(ps) scale. In semiconductors, bulk and structural inver-

sion symmetry breaking as well as electron-hole interac-

tions lead to supplementary spin relaxation mechanisms

such as D’yakonov-Perel24and Bir-Aronov-Pikus25that

are beyond the scope of the present study.

Relaxation processes also apply to collective spin exci-

tations such as magnons. Whereas the electron-magnon

interaction conserves the angular momentum, magnon-

magnon interactions and magnon-lattice interactions in

the presence of spin-orbit coupling contribute to the total

spin relaxation. While the former occurs on the magnon

thermalization time scale26(100fs), the latter is however

at the second order in spin-orbit coupling and is con-

sidered to occur on the 100ps time scale. Therefore, in

a laser-induced demagnetization experiment, it is most

probable that all the processes mentioned above take

place during the thermalization time scale of the excited

electrons and excited magnons.

B.Demagnetization scenario

To establish our model, we first separate the LID pro-

cesses into four steps: (i) generation of non-thermal hot

electrons by laser pumping; (ii) relaxation of these hot

electrons into thermalized electrons characterized by an

electron temperature Te; (iii) energy transfer from the

thermalized hot electrons to the spin and lattice sub-

systems; (iv) heat diffusion to the environment.

In our model, to be given below, we will take steps

(i) and (ii) infinitely fast. In the step (i), a laser pump

excites a fraction of electrons below the Fermi sea to ≈1.5

eV above the Fermi level. This excitation process is of the

order of a few fs. The photo-induced electron transition is

considered spin conserving and thus does not significantly

contribute to the demagnetization although the spin-flip

electron transition could occur in the presence of the spin-

orbit coupling13.

In step (ii), the strong Coulomb interaction among

electrons relaxes these non-thermal high-energy electrons

to form a hot electron bath which may be described by

a thermalized hot electron temperature Te. During this

electron thermalization process, strong electron-electron

interaction-induced momentum scattering in the pres-

ence of spin-orbit coupling leads to the ultrafast transfer

of the spin degree of freedom to the orbital one27. In our

model, the electron thermalization is considered instan-

taneous and any possible femtosecond coherent processes

are disregarded28. Therefore, due to ultrafast (fs) mo-

mentum scattering, the thermalized hot electrons act as

a spin sink. Under this approximation, the demagnetiza-

tion itself, defined as the loss of spin angular momentum,

takes place during the thermalization of the electron bath

in the presence of (either intrinsic or extrinsic) spin-orbit

coupling.

Following the definition of the three-temperature

model, we assume that the system can be described

in term of three interacting baths composed of laser-

induced hot electrons, spin excitations of the ground

state (magnons) and lattice excitations (phonons). The

applicability of this assumption is discussed in Sec. IID.

Therefore, the magnetic signal essentially comes from the

collective spin excitation and it is assumed that the laser-

induced hot electron only contribute weakly to the mag-

netization. Consequently, under the assumption that the

spin loss occurs during the thermalization time of the

electron and spin systems, the demagnetization problem

reduces to tracking the energy transfer between the spin

bath and the electron and phonon bathd.

Our main objective is then to understand step (iii),

where the electrons at a higher temperature transfer

their energy to the spin and lattice sub-systems. Un-

der the electron-magnon interaction, the magnons spin is

transferred to the electron system, and is eventually lost

through thermalization of the electron bath. Through

interactions among electrons, spins and lattice, the en-

tire system will ultimately reach a common temperature.

Finally, a heat diffusion, step (iv), will expel the heat to

the environment; this last step will be considered via a

simple phenomenological heat diffusion equation.

To quantitatively determine the energy transfer among

electrons, spins and lattice in the step (iii), one not only

needs to know the explicit interaction, but also the distri-

bution of the densities of excitations (electrons, magnons

and phonons). Within the spirit of the three tempera-

Page 3

3

ture model, we consider that each sub-system (electron,

spin and lattice) is thermalized, i.e., one can define three

temperatures for electrons Te, spins Ts and lattice Tl.

The justification of this important assumption has been

made in the previous section and can be qualitatively

summarized: 1) For the hot electrons of the order of

1eV, the electron-electron relaxation time is τee≈ 10fs,

which is about 100 times faster than the electron-spin and

electron-phonon interactions23. 2) The lattice-lattice in-

teraction is about one order of magnitude smaller than

the electron-electron relaxation time, τll ≈ 100fs29. 3)

Multiple spin-waves processes are known to take place

in the ferromagnetic relaxation leading to so-called Suhl

instabilities26. The relaxation time is of the order of

τss∝ ¯ h/Tc≈ 100fs at least for high energy magnons26

(for long wave length magnons, the lifetime could be

significantly longer). Thus, it is reasonable to assume

that the concepts of the three temperatures are approx-

imately valid as long as the time scale is longer than

sub-picoseconds.

C.Model Hamiltonian

We now propose the following Hamiltonian for LID

ˆH =

?

µ

ˆHµ+ˆHes+ˆHel+ˆHsl, (1)

whereˆHµ (µ = e,s,l) are the electron, spin and lat-

tice Hamiltonians, andˆHµν (µ ?= ν) are the interaction

among sub-systems. In the remaining of the article, the

hat ˆ denotes an operator. Each term is explicitly de-

scribed below.

The electron system is described by a free electron

modelˆHe =

?

rium distribution is simply the Fermi distribution at Te.

The lattice HamiltonianˆHl=?

phonon creation (annihilation) operator and λ is the po-

larization of the phonon. The phonon distribution at Tl

is nkλ= [exp(¯ hωp

is modeled by the Heisenberg exchange interaction,

kǫkˆ c+

kˆ ck where ˆ c+

k(ˆ ck) represents the

electron creation (annihilation) operator. The equilib-

qλ¯ hωp

kλˆb+

kλˆbkλis mod-

kλ(ˆbkλ) is the eled by simple harmonic oscillators whereˆb+

kλ/kBTl)−1]−1. The spin Hamiltonian

ˆHs= −

?

<ij>

JijˆSi·ˆSj− gµBHex

?

i

ˆSz

i,(2)

where Jij is the symmetric exchange integral,ˆSi is the

spin operator at the site i, and Hexis the external mag-

netic field applied in z-direction. Unlike the electron and

lattice Hamiltonians, the spin Hamiltonian is not a single

particle Hamiltonian and the distribution of the spin den-

sity is neither a fermionic nor a bosonic distribution. To

describe the equilibrium distribution of the spin system

at arbitrary temperatures, we will model the equilibrium

properties of the spin system in the next section.

The electron-lattice interactionˆHelis taken as a stan-

dard form29

ˆHel =

?

k,qλ

Bqλ(ˆ c+

k+qˆ ckˆbqλ+ ˆ c+

k−qˆ ckˆb+

qλ), (3)

where the Bqλis the electron-phonon coupling constant.

For acoustic phonons, the coupling constant takes a par-

ticularly simple form29,

Bqλ=2ǫFq

3

?

¯ h

2MNωp

qλ

. (4)

Here ǫF is the electron Fermi energy and M is the mass

of the ion.

The electron-spin interactionˆHes is modeled by the

conventional exchange interaction (sd Hamiltonian):

ˆHes = −Jex

?

j,k,k′

ˆ c+

keik·rj(ˆ σ ·ˆSj)ˆ ck′e−ik′·rj, (5)

where we have assumed a constant coupling constant Jex

and ˆ σ is the electron spin. When one replaces ˆ σ ·ˆSjby

ˆ σzˆSz

effects: the first term is responsible for the spin-splitting

of the conduction bands and the second term leads to a

transfer of angular momentum between the spins of the

hot electrons and the spins of the ground state, i.e. spin-

waves generation and annihilation. While the interaction

conserves the total spin angular momentum, the ther-

malization process of each bath is not spin conserving

as mentioned above. Therefore, this interaction trans-

fers energy between the electron and spin baths, which

results in the effective demagnetization of the magnon

bath. Consequently, the generation of magnons by hot

electron is a key mechanism in our model (see also Ref.

6).

Finally, the spin-lattice interactionˆHsl has been at-

tributed to spin-orbit coupling30. The energy and the

angular momentum conservations requireˆHslcontaining

two-magnon (ˆ a+

Since the spin-orbit coupling is already treated as a

perturbation, this process is second order in the spin-

orbit coupling parameter and it is expected to be rather

small30. Thus,ˆHsl is much smaller thanˆHes andˆHel,

and we placeˆHsl= 0 throughout the rest of the paper.

To summarize our model, we consider three subsystems

(electrons, spins, and lattice) described byˆHe,ˆHs and

ˆHlrespectively. These subsystems have their individual

equilibrium temperatures Te, Ts and Tl. The heat or

energy transfer among them are given by the interaction

ˆHesandˆHel. To determine the kinetic equation for three

subsystems, we should first establish the low excitation

properties of the spin system fromˆHs and relate Ts to

the magnetization m(Ts).

j+1

2(ˆ σ−ˆS+

j+ ˆ σ+ˆS−

j), the above Hes contains two

qˆ aq′) and two-phonon operators (ˆb+

kˆbk′).

Page 4

4

D. Materials considerations

As stated in the introduction, laser-induced demagne-

tization has been observed in a wide variety of materials

presenting very diverse band structures and magnetism.

From the materials viewpoint, the present model makes

three important assumptions: (i) laser-induced hot elec-

trons, ground state spin excitations and phonons can be

treated as separate interacting sub-systems; (ii) there ex-

ists a direct interaction between hot electrons and collec-

tive spin excitations; (iii) the excited spin sub-system can

be described in terms of spin-waves.

Whereas the consideration of a separate phonon bath

is common, the separation between the electron and spin

populations may seem questionable. In systems where

the itinerant and localized electrons can be identified

(such as 4f-rare earth or carrier-mediated dilute mag-

netic semiconductors), it seems quite reasonable. How-

ever, in typical itinerant ferromagnets such as transi-

tion metals, the magnetism arises from a significant por-

tion of itinerant electrons.

model, the separation between electron and spin baths

arises from the fact the electrons we consider are laser-

induced hot electrons near Fermi level (in the range

[ǫF−kBTe,ǫF+kBTe]), whereas the spin bath describes

the magnetic behavior of electrons lying well below Fermi

level. The concept of spin waves used in the present ar-

ticle is rather general and applies to a wide range of fer-

romagnetic materials. Although energy dispersion may

vary from one material to another, it is unlikely to have

strong influence on the main conclusions of this work.

The interaction between hot electrons and magnons is

actually more restrictive since it assumes overlap between

electrons near and far below Fermi level. For example,

this approach does not apply to half-metals (electron-

magnon interaction is quenched by the 100% spin polar-

ization) or magnetic insulators. Nevertheless, in metallic

materials such as transition metals and rare-earth, this

interaction does not vanish and can lead to strong spin

wave generation, as demonstrated by Schmidt et al.31in

Fe.

We stress out that in our

III.EQUILIBRIUM PROPERTIES OF THE

SPIN SYSTEM

The Heisenberg model for the spin system, Eq. (2),

has no exact solution even in equilibrium. At low tem-

perature, the simplest approach is based on the spin-

wave approximation which predicts Bloch’s law for the

magnetization m(T) = m0− B(T/Tc)3/2where Tc is

the Curie temperature and B is a numerical constant32.

As the temperature approaches the Curie temperature,

Bloch’s law fails. Instead, one uses a molecular mean

field to model the magnetization. The resulting magne-

tization displays a critical relation near Tc, i.e., m(T) ∝

(1 − T/Tc)1/2. Since we are interested in modeling the

magnetization in the entire range of temperature, we de-

scribe below a self-consistent random phase approxima-

tion which reproduces Bloch’s law at low temperatures

and the mean field result at high temperatures.

We first recall some elementary relations of these spin

operators given below,

ˆS+

i=ˆSx

i+ iˆSy

= 2ˆSz

i,ˆS−

?ˆS±

i=ˆSx

i− iˆSy

= ∓ˆS±

i− (ˆSz

i, (6)

?ˆS+

i,ˆS−

i

?

i,

i,ˆSz

i

?

i, (7)

ˆS+

iˆS−

i= S(S + 1) +ˆSz

i)2, (8)

and the spin Hamiltonian, Eq. (2) can be rewritten as

ˆH = −

?

ij

Jij(ˆS−

iˆS+

j+ˆSz

iˆSz

j) − gµBHex

?

i

ˆSz

i.(9)

Our self-consistent random phase approximation treats

the resulting commutator,

?ˆS+

as a c-number, where m(T) is the thermal average of

ˆSz

ito be determined self-consistently. If we introduce

the Fourier transformation,ˆS±

the above commutator reads as [ˆS+

and thus by introducing ˆ a±

a standard boson commutator relation [ˆ aq,ˆ a+

Similarly, we have [ˆHs,ˆ aq] = ¯ hωqˆ aq, where

i,ˆS−

i

?

= 2ˆSz

i≈ 2m(T)

k= (1/N)?

k≡ˆS∓

iˆS±

ie−ik·Ri,

k,ˆS−

k/?2m(T), one has

q] = 2m(T)δkq

q′] = δq,q′.

¯ hωq= gµBHex+ 2m(T)

?

q

[J0− J(q)](10)

where J(q) = (1/N)?

consistently determine the magnetization m(T) and

other macroscopic variables such as the spin energy and

specific heat. A particular simple case is for the spin-half

S = 1/2 where the identity

<ij>Jijexp[iq · (Ri − Rj)].

With the above bosonic approximation, one can self-

ˆSz= S −ˆS−ˆS+= 1/2 −

?

q

2m(T)a+

qaq

(11)

immediately leads to the self-consistent determination for

m(T)

m(T) = 1/2 −1

N

?

q

2m(T)

eβ¯ hωq(T)− 1. (12)

At low temperature, one can approximately replace m(T)

by 1/2 in the right-hand side of the equation and one

immediately sees that the above solution produces the

well-known Bloch relation, i.e., 1/2−m(T) ∝ T3/2. Near

the Curie temperature, one expands eβ¯ hωq= 1+β¯ hωq+

(1/2)(β¯ hωq)2and notice that ωqis proportional to m(T)

at zero magnetic field, see Eq. (10).

expansion into Eq. (12), the zero order term in m(T)

determines the Curie temperature and the second order

term gives the scaling m2(T) ∝ (Tc− T), i.e. the mean

field result is recovered, m(T) ∝ (1 − T/Tc)1/2. Thus

the self-consistent approach captures both low and high

By placing this

Page 5

5

temperature limiting cases. In fact, the Green’s function

technique26has been developed to justify this approxi-

mation.

For the cases other than S = 1/2, the relation between

ˆSz

iand the number of magnons is more complicated due

to non-constant (ˆSz

ately lead to a self-consistent equation for m(T). Instead,

one needs to relate ?(ˆSz

sity. Tyablikov33introduces a decoupling method to ap-

proximate ?(ˆSz

of magnons

i)2and thus Eq. (8) cannot immedi-

i)2? to m(T) and the magnon den-

i)2? with m(T) and the normalized number

n0≡

1

N

?

q

?ˆ a+

qˆ aq? =

1

N

?

q

1

exp(βωq) − 1. (13)

Here finds that, for arbitraryS, the self-consistent equa-

tion for determining m(T) is

m(T) =(S − n0)(1 + n0)2S+1(1 + S + n0)n2S+1

(1 + n0)2S+1− n2S+1

By replacing S = 1/2, the above equation reduces

to Eq. (12). The magnetic energy can be similarly

obtained34

0

0

. (14)

E = E0+S − m(T)

2n0

?

q

¯ hωq(0) + ¯ hωq

exp(βωq) − 1

(15)

where E0 is the ground state energy and ¯ hωq(0) is the

spin wave energy at T = 0. Once the internal energy is

obtained, the specific heat, Cp= ∂E/∂T, may be numer-

ically calculated.

m(Ts) is uniquely determined from Eq. (14) or Eq. (12)

for s = 1/2, if the spin temperature is known. Thus, the

laser-induced demagnetization is solely dependent on the

the time-dependent spin temperature Ts. Before we pro-

ceed to calculate Ts(t) or m(t), we show the solutions of

Eq. (14) or Eq. (12). In Figure 1, the reduced magneti-

zation m(T)/S and the specific heat as a function of the

normalized temperature T/Tcwith [Figs. 1(a) and (b)]

and without [Figs. 1(c) and (d)] the magnetic field are

shown. A few general features can be readily identified.

First, the shapes of the magnetization curves for differ-

ent spins are very similar. Second, the magnetic field

removes the divergence of the specific heat at the Curie

temperature. As expected, the magnetization reduces to

that of the mean field result near the Curie temperature

and to that of the spin wave approximation at low tem-

peratures.

IV.DYNAMIC EQUATIONS

The energy or heat transfer among electrons, spins and

lattice may be captured by the general rate equations

given below,

dEe

dt

= −Γes− Γel

(16)

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

FIG. 1:

magnetization and (b) specific heat (arbitrary unit) for

spin=1/2,1,2,8 in the absence of the external field; Temper-

ature dependence of (c) magnetization and (d) specific heat

for spin=1/2,1,2,8 in an external field H/Tc = 0.001.

(Color online) Temperature dependence of (a)

dEl

dt

dEs

dt

= Γsl+ Γel

(17)

= Γes− Γsl

(18)

where Ei are the energy densities (i = e,s,l) and the

rate of the energy transfer Γij should be determined by

Eq. (1). Since we have neglected the weaker interaction

between spins and lattice, we set Γsl = 0 in the above

equations. In the following, we explicitly derive the re-

laxation rates of Γeland Γesfrom Eqs. (3) and (5).

A.Electron-lattice relaxation Γel

The energy transfer rate between electrons and

phonons does not involve the spin. The Fermi golden

rule applied to Eq. (3) immediately leads to

Γel =

4π

¯ h

?

k,q

¯ hωp

q|Bq|2δ(ǫk− ǫk+q+ ¯ hωp

q) ×(19)

(nk+q(1 − nk)(1 + np

where the first (second) term represents the energy trans-

fer from (to) the electrons to (from) lattice by emit-

ting (absorbing) a phonon. Note that the electrons and

phonons have different temperatures; otherwise the de-

tailed balance will make the net energy transfer zero. The

electron and phonon densities are given by their respec-

tive equilibrium temperatures at Ts and Tl, i.e., nk =

[exp((ǫk−ǫF)/kBTe)+1]−1and np

1]−1.We consider polarization-independent acoustic

phonons, i.e., ωp

q= vsq where vs is the phonon veloc-

ity. By replacing Bpgiven in Eq. (4) into Eq. (19), we

have

q) − nk(1 − nk+q)np

q),

q= [exp(¯ hωp

q/kBTl)−