Theory of laserinduced demagnetization at high temperatures
ABSTRACT Laserinduced 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
subsystems, the temperature dynamics is given by the energy transfer between
the thermalized interacting baths. These energy transfers are accounted for
explicitly through electronmagnons and electronphonons interaction, which
govern the demagnetization time scale. By properly treating the spin system in
a selfconsistent 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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 SourceAvailable from: Paweł Adam BuczekA B Schmidt, M Pickel, M Donath, P Buczek, A Ernst, V P Zhukov, P M Echenique, L M Sandratskii, E V Chulkov, M Weinelt[Show abstract] [Hide abstract]
ABSTRACT: We report on a combined experimental and theoretical study of the spindependent relaxation processes in the electron system of an iron film on Cu(100). Spin, time, energy and angleresolved twophoton photoemission shows a strong characteristic dependence of the lifetime of photoexcited electrons on their spin and energy. Ab initio calculations as well as a manybody treatment corroborate that the observed properties are determined by relaxation processes involving magnon emission. Thereby we demonstrate that magnon emission by hot electrons occurs on the femtosecond time scale and thus provides a significant source of ultrafast spinflip processes. Furthermore, engineering of the magnon spectrum paves the way for tuning the dynamic properties of magnetic materials.Physical Review Letters 11/2010; 105(19):197401. · 7.73 Impact Factor
Page 1
arXiv:1112.2428v1 [condmat.meshall] 12 Dec 2011
Theory of laserinduced 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)
Laserinduced 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 subsystems, the temperature dynamics is given by the energy
transfer between the thermalized interacting baths. These energy transfers are accounted for explic
itly through electronmagnons and electronphonons interaction, which govern the demagnetization
time scale. By properly treating the spin system in a selfconsistent 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
Laserinduced 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 laserinduced 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 roomtemperature 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
subnanosecond, three orders of magnitude larger com
pared to LID.
LIDobservations have been carried outina
number of magnetic materials including transition
metals1,4–6, insulators7, halfmetals8–10and dilute mag
netic semiconductors11. A general consensus of the laser
induced demagnetization process is that the high energy
nonthermal 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 threetemperature
model1,5,9where the three interacting subsystems (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 electronspin
and electronlattice relaxations.
proposed that the laser field can directly excite the spin
polarized ground states to spinunpolarized excited states
in the presence of spinorbit coupling, i.e., the spinflip
transition leads to the demagnetization during the laser
pulse. In this picture, the demagnetization is instan
taneous (≈50150fs).
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
spinorbit coupling and impurities or phonons, through
an ”ElliotYafet”type (EY) spinflip 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 LandauLifshitz
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 threetemperature 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
laserinduced 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 selfconsistent 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 spinorbit coupled bands may lose
their spin under spinindependent momentum scatter
ing events (such as electronelectron or electronimpurity
interaction). This mechanism was later extended to
electronphonon scattering by Yafet and Overhauser22.
Consequently, the spin relaxation time τsis directly pro
portional to the momentum relaxation time τp. Whereas
the electronelectron relaxation time is on the order of
a few femtoseconds23(fs), the electronimpurity and
electronphonon relaxation time is on the picoseconds
(ps) scale. In semiconductors, bulk and structural inver
sion symmetry breaking as well as electronhole interac
tions lead to supplementary spin relaxation mechanisms
such as D’yakonovPerel24and BirAronovPikus25that
are beyond the scope of the present study.
Relaxation processes also apply to collective spin exci
tations such as magnons. Whereas the electronmagnon
interaction conserves the angular momentum, magnon
magnon interactions and magnonlattice interactions in
the presence of spinorbit 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 spinorbit coupling and is con
sidered to occur on the 100ps time scale. Therefore, in
a laserinduced 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 nonthermal 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 photoinduced electron transition is
considered spin conserving and thus does not significantly
contribute to the demagnetization although the spinflip
electron transition could occur in the presence of the spin
orbit coupling13.
In step (ii), the strong Coulomb interaction among
electrons relaxes these nonthermal highenergy electrons
to form a hot electron bath which may be described by
a thermalized hot electron temperature Te. During this
electron thermalization process, strong electronelectron
interactioninduced momentum scattering in the pres
ence of spinorbit 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) spinorbit
coupling.
Following the definition of the threetemperature
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 subsystems. Un
der the electronmagnon 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 subsystem (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 electronelectron relaxation time is τee≈ 10fs,
which is about 100 times faster than the electronspin and
electronphonon interactions23. 2) The latticelattice in
teraction is about one order of magnitude smaller than
the electronelectron relaxation time, τll ≈ 100fs29. 3)
Multiple spinwaves processes are known to take place
in the ferromagnetic relaxation leading to socalled 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
subpicoseconds.
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 subsystems. 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 zdirection. 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 electronlattice 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 electronphonon 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 electronspin 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 spinsplitting
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 spinlattice interactionˆHsl has been at
tributed to spinorbit coupling30. The energy and the
angular momentum conservations requireˆHslcontaining
twomagnon (ˆ a+
Since the spinorbit 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 twophonon operators (ˆb+
kˆbk′).
Page 4
4
D. Materials considerations
As stated in the introduction, laserinduced 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) laserinduced hot elec
trons, ground state spin excitations and phonons can be
treated as separate interacting subsystems; (ii) there ex
ists a direct interaction between hot electrons and collec
tive spin excitations; (iii) the excited spin subsystem can
be described in terms of spinwaves.
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 4frare earth or carriermediated 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 halfmetals (electron
magnon interaction is quenched by the 100% spin polar
ization) or magnetic insulators. Nevertheless, in metallic
materials such as transition metals and rareearth, 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 selfconsistent 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 selfconsistent random phase approximation treats
the resulting commutator,
?ˆS+
as a cnumber, where m(T) is the thermal average of
ˆSz
ito be determined selfconsistently. 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 spinhalf
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 selfconsistent 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 righthand side of the equation and one
immediately sees that the above solution produces the
wellknown 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 selfconsistent 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 nonconstant (ˆSz
ately lead to a selfconsistent 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 selfconsistent 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
laserinduced demagnetization is solely dependent on the
the timedependent 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.Electronlattice 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
qBq2δ(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 polarizationindependent 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)−