Nonlocal effects of partial measurements and quantum erasure
Nonlocal heat transport with phonons and electrons: Application to metallic nanowires
21
Nonlocal heat transport with phonons and electrons: application to metallic nanowires D. Jou * Departament de F´ ısica, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Catalonia, Spain and Institut d’Estudis Catalans, Carme 47, Barcelona 08001, Catalonia, Spain V. A. Cimmelli † and A. Sellitto ‡§ Department of Mathematics and Computer Science, University of Basilicata, Campus Macchia Romana, 85100 Potenza, Italy Abstract In the framework of Extended Irreversible Thermodynamics we develop a model for coupled heat conduction by phonons and electrons. Particular emphasis is given to nonlocal effects, which may arise when the mean-free paths of phonons and/or electrons are comparable to the size of the system. As particular cases, we recover two parabolic equations of the Guyer-Krumhansl type which model the concurrent presence of the diffusion of heat superposed to the propagation of heat waves, and two hyperbolic equations of the Maxwell-Cattaneo type. In the latter case, the phase speed of temperature waves is calculated. The size dependence of the Wiedemann-Franz law is briefly analyzed for metallic nanowires. PACS numbers: 44.10.+i,66.70.-f,05.70.Ln Keywords: heat transport in metallic nanowires, nonlocal constitutive equations, size dependence of the Wiedemann-Franz, electron-phonon coupling ‡ Corresponding author. * Electronic address: [email protected] † Electronic address: [email protected]. § Electronic address: [email protected] 1
Transcript of Nonlocal heat transport with phonons and electrons: Application to metallic nanowires
Nonlocal heat transport with phonons and electrons: application
to metallic nanowires
D. Jou∗
Departament de Fısica, Universitat Autonoma de Barcelona,
08193 Bellaterra, Catalonia, Spain and
Institut d’Estudis Catalans, Carme 47, Barcelona 08001, Catalonia, Spain
V. A. Cimmelli† and A. Sellitto‡§
Department of Mathematics and Computer Science,
University of Basilicata, Campus Macchia Romana, 85100 Potenza, Italy
Abstract
In the framework of Extended Irreversible Thermodynamics we develop a model for coupled
heat conduction by phonons and electrons. Particular emphasis is given to nonlocal effects, which
may arise when the mean-free paths of phonons and/or electrons are comparable to the size of
the system. As particular cases, we recover two parabolic equations of the Guyer-Krumhansl type
which model the concurrent presence of the diffusion of heat superposed to the propagation of heat
waves, and two hyperbolic equations of the Maxwell-Cattaneo type. In the latter case, the phase
speed of temperature waves is calculated. The size dependence of the Wiedemann-Franz law is
briefly analyzed for metallic nanowires.
PACS numbers: 44.10.+i,66.70.-f,05.70.Ln
Keywords: heat transport in metallic nanowires, nonlocal constitutive equations, size dependence of the
Wiedemann-Franz, electron-phonon coupling
‡ Corresponding author.∗Electronic address: [email protected]†Electronic address: [email protected].§Electronic address: [email protected]
1
I. INTRODUCTION
The analysis of nonlocal effects is an attracting topics of nonequilibrium thermodynamics,
besides being of outstanding interest in practical applications because of the recent develop-
ments in materials sciences, high-power lasers and optimization of energy generation [1–4].
Here we consider heat transport in metals. Although their thermal conductivity is domi-
nated by that of electrons, in several cases the lattice heat conductivity, due to phonons, has
to be added [5]. For instance, in small systems (nanowires and nanoribbons with charac-
teristic sizes of the order of 100 nm, or less) the electronic part of the thermal conductivity
drops faster than the phononic contribution [6]. Other examples are some epitaxial su-
perconducting films with high critical temperature, where the phonons are responsible for
50− 70 percent of thermal conduction near the critical temperature [7, 8].
The aim of the present paper is to model heat transport due to phonons and electrons
taking into account relaxation and nonlocal effects in the framework of Extended Irreversible
Thermodynamics (EIT) [3, 4]. The basic idea underlying EIT is to upgrade the physical
fluxes of energy, electric charge and others, to the status of independent variables at the
same level as the classical variables like energy, mass, and momentum [3, 4]. Particular
emphasis will be given to nonlocal effects, which are ruled by the ratio of the mean-free
path to the characteristic length of the system, the so called Knudsen number. Since the
phonon mean-free path and the electron mean-free path may be considerably different from
each other (for instance, in Au, Ag and Cu the former is several times shorter than the
latter, whereas in Pt, Ni, W both are comparable to each other [6]), the size of the system
may have different effects on both constituents and, therefore, it may allow a high degree of
control of the transport properties of systems, at spatial scales comparable to the mean-free
path of some of the species. Thus, heat transport equations specifying both mean-free paths
are useful for a more detailed description and exploration of new applications.
We assume that the heat flux q(u)i is formed by two different contributions: q
(e)i due to
the electrons, and q(p)i due to phonons. Thus, in the framework of EIT we assume that the
state-space Z is spanned by the specific internal energy u, the two vectorial variables q(e)i
and q(p)i , and their first-order spatial derivatives, namely,
Z = Z{u;u,i ; q
(e)i ; q
(e)j,i
; q(p)i ; q
(p)j,i
}. (1)
It is important to note that we included in Z not only the fluxes, but also their first-order
2
spatial derivatives in view of a weakly nonlocal description [9]. In order to elucidate the
need of the gradients in the state space in some situations, let us consider, for the sake of
simplicity, a one-dimensional crystal, and let F (q) a constitutive functional for it, which
depends on the local heat flux q. As it is well-known, in crystals the heat is carried by the
phonons [10–13]. The heat flux and, consequently, the value of F in a point x, is influenced
by the phonons’ movement after their last scattering which, of course, has taken place at the
position x′ = ±`, ` being the phonons mean-free path. This is tantamount to admit all the
points in the interval [−`; `] contribute to F (q (x)), which may be represented as follows:
F (q (x)) =1
`
∫ x
x−`F (q (x− x′)) dx′ + 1
`
∫ x+`
x
F (q (x′ − x)) dx′. (2)
Here we have assumed that each point in the interval contributes to the value of F in
x through an amount of heat flux which depends on its distance from x and which can
be represented by the function q (x− x′) if the point is placed to the left of x, and by the
function q (x′ − x) if the point is placed to the right of x. Thus, we can see that F (q (x)) is
a genuinely nonlocal functional of the local heat flux, in the sense that its value in x depends
on the state of the system in all the points of a neighborhood of x of length 2`.
On the other hand, in situations in which such an accurate description of F (q (x)) is
not necessary, we may eliminate the integral in its representation by developing F as a
power series around the point q = 0, and the quantities q (x− x′) and q (x′ − x) as power
series around the point x′ ≡ x. In this case we face with a weakly nonlocal constitutive
equation. Note that the representation of F (q (x)) is weakly nonlocal irrespective of the
order of approximation of the integrands. If we stop at the first-order approximation, we
may write
F (q) = F (0) +
(dF
dq
∣∣∣∣q≡0
)q, (3)
and
q (x− x′) = q (x)−
(dq
d (x− x′)
∣∣∣∣x′≡0
)x′, (4a)
q (x′ − x) = q (x) +
(dq
d (x′ − x)
∣∣∣∣x′≡x
)(x′ − x) . (4b)
3
Finally, substituting Eqs. (3) and (4) in Eq. (2), we have
F (q (x)) = 2F (0) + 2
(dF
dq
∣∣∣∣q≡0
)q (x)
+
(dF
dq
∣∣∣∣q≡0
)[(dq
d (x− x′)
∣∣∣∣x′≡0
)(1
2`− x
)+
(dq
d (x′ − x)
∣∣∣∣x′≡x
)1
2`
]. (5)
For nanosystems (that is, for systems the characteristic length of which is of the same
order of `, or less) the quantity `− x has order of magnitude of `. Moreover, the derivatives
dq/d (x− x′) and dq/d (x− x′) take the same order of magnitude of q/`. Therefore the last
two terms in the right-hand side of Eq. (5) are not negligible.
It is worth observing that this procedure may be pursued up to an arbitrary order of
approximation, allowing to represent F (q (x)) as a linear function of q and its higher-order
gradients evaluated in x. In some more general situations the expansion in Eq. (3) is not
allowed, so that Eqs. (4) must be substituted directly in Eq. (2). In such a case F (q (x))
becomes an arbitrary function of q and its gradients, still evaluated in x. Note further that
this method may be applied similarly if F depends on additional state functions, too. The
previous consideration constitutes one of the physical motivations of the nonlocality of the
state space postulated in EIT [3, 4].
It seems also important to observe further that the independent character of the spatial
derivative of the heat flux plays a relevant role in going beyond the usual Maxwell-Cattaneo
equation [14], which does take into account time-relaxation effects, but neglects nonlocal
effects. In the present paper, this description of nonlocal effects will be also applied to
nonlocal effects in electrical transport.
To summarize, the paper is structured as follows. In Sec. II we write down the model
transport equations and investigate their compatibility with the second law of thermody-
namics by a generalized Liu procedure [15–18]. In Sec. III, we derive the heat transport
equations for phonons and electrons. In Sec. IV we point out how this formalism could be
enlarged by incorporating the electrical current to analyze the influence of nonlocal effects on
the Wiedemann-Franz law in metallic nanowires [6, 19]. Final remarks are given in Sec. V.
4
II. MODEL EQUATIONS AND COMPATIBILITY WITH SECOND LAW
A. Balance equations and entropy inequality
In the absence of heat supply, the first law of thermodynamics, which expresses the local
balance of the total energy of the system, takes the form
u = −q(u)i,i, (6)
where the superposed dot stands for the time derivative, and the subscript ,i means the
derivative with respect to the i-th spatial coordinate. In Eq. (6) the total heat-flux q(u)i is
given by
q(u)i = q
(e)i + q
(p)i . (7)
On the other hand, q(e)i and q
(p)i have their own evolution equation which, according to
the general principles of EIT [3, 4], may be postulated in the form
q(e)i = −Φ
(e)ij,j
+ σ(e)i , (8a)
q(p)i = −Φ
(p)ij,j
+ σ(p)i . (8b)
In the equations above, the first terms in the right-hand side mean the divergence of the
flux of the corresponding state variable, and the second terms mean its production. All
these quantities have to be given as constitutive equations, that is, they have to be assigned
as suitable functions of the state variables in Z.
A possible way of modeling the fluxes Φ(e)ij and Φ
(p)ij is:
Φ(e)ij = −
[Γ
(e)1 (u) + Γ
(e)4 (u) q
(e)k,k
]δij − Γ
(e)2 (u) q
(e)i q
(p)j − Γ
(e)3 (u) q
(e)i,j− Γ
(ep)4 (u) q
(p)i,j, (9a)
Φ(p)ij = −
[Γ
(p)1 (u) + Γ
(p)4 (u) q
(p)k,k
]δij − Γ
(p)2 (u) q
(p)i q
(e)j − Γ
(p)3 (u) q
(p)i,j− Γ
(pe)4 (u) q
(e)i,j, (9b)
whereas, for the productions σ(e)i and σ
(p)i we assume:
σ(e)i = Γ
(e)0 (u) q
(e)i + γ(e) (u)u,i + Γ
(ep)0 (u) q
(p)i , (10a)
σ(p)i = Γ
(p)0 (u) q
(p)i + γ(p) (u)u,i + Γ
(pe)0 (u) q
(e)i , (10b)
with Γ(e)i , Γ
(p)i (i = 0, ..., 4), γ(e) and γ(p) phenomenological coefficients. Moreover, the phe-
nomenological coefficients Γ(ep)0 , Γ
(pe)0 , Γ
(ep)4 and Γ
(pe)4 account for mutual couplings between
5
the fluxes, whereas the terms in Γ(e)2 and Γ
(p)2 establish a coupling between q
(e)i and q
(p)i
in their respective fluxes. However, since here we are especially interested in the nonlocal
effects, in the next we will neglect the former.
Although more refined models could be considered, the constitutive equations above have
the advantage of preserving the essential physics of the problem still retaining a sufficient
simplicity. Incorporation of additional terms in the constitutive equations would be straight-
forward, but cumbersome.
Due to the form of the state-space Z and the assumptions in Eq. (7), Eqs. (9) and (10),
the time rates of u, q(e)i and q
(p)i can be rearranged as:
u =− q(e)i,i− q(p)
i,i, (11a)
q(e)i =
[∂
∂u
(Γ
(e)1 δij + Γ
(e)2 q
(e)i q
(p)j + Γ
(e)3 q
(e)i,j
+ Γ(e)4 q
(e)k,kδij
)+ γ(e)δij
]u,j
+ Γ(e)2
(q
(e)i,jq
(p)j + q
(e)i q
(p)j,j
)+ Γ
(e)3 q
(e)i,jj
+ Γ(e)4 q
(e)j,ji
+ Γ(e)0 q
(e)i , (11b)
q(p)i =
[∂
∂u
(Γ
(p)1 δij + Γ
(p)2 q
(p)i q
(e)j + Γ
(p)3 q
(p)i,j
+ Γ(p)4 q
(p)k,kδij
)+ γ(p)δij
]u,j
+ Γ(p)2
(q
(p)i,jq
(e)j + q
(p)i q
(e)j,j
)+ Γ
(p)3 q
(p)i,jj
+ Γ(p)4 q
(p)j,ji
+ Γ(p)0 q
(p)i . (11c)
Finally, the time rates of u,i , q(e)j,i
, and q(p)j,i
, instead, directly follow by taking the gradient
of Eqs. (11). These equations, also called extended balances [16–18, 20], read
u,i =− q(e)j,ji− q(p)
j,ji, (12a)
q(e)j,i
=Γ(e)3 q
(e)j,kki
+ Γ(e)4 q
(e)k,kji
+ F (e)ji
(u;u,i ;u,ji ; q
(e)i ; q
(e)j,i
; q(e)k,ji
; q(p)i ; q
(p)j,i
; q(p)k,ji
), (12b)
q(p)j,i
=Γ(p)3 q
(p)j,kki
+ Γ(p)4 q
(p)k,kji
+ F (p)ji
(u;u,i ;u,ji ; q
(e)i ; q
(e)j,i
; q(e)k,ji
; q(p)i ; q
(p)j,i
; q(p)k,ji
), (12c)
wherein F (e)ji and F (p)
ji are regular tensorial functions of the indicated arguments.
At this step, let us observe that Eq. (6) expresses the principle of energy conservation, but
it does not care about the direction of thermodynamic processes. It is well-known that each
thermodynamic process has its own privileged direction. Second law of thermodynamics,
restricting the form of constitutive equations [21], accounts for the natural evolution of a
system in any possible thermodynamic process. In the present case, its local form reads
σ(s) = s+ Φ(s)i,i≥ 0, (13)
wherein s is the specific entropy, Φ(s)i its flux, and σ(s) its production, which must be non-
negative in any admissible thermodynamic process.
6
Let us observe that, in order to derive the consequences of Eq. (13), both the entropy and
the entropy flux have to be assigned by constitutive equations, too. We do not postulate
any particular form for these functions, letting the second law to give more information on
them.
B. Exploiting the second law of thermodynamics
In order to investigate whether our model is compatible with second law of thermody-
namics, we must determine a set of conditions (restricting the constitutive equations [21])
which are necessary and sufficient to guarantee that the unilateral differential inequality
(13) is satisfied along arbitrary thermodynamic processes. Such an inequality, on the state
space takes the form
σ(s) =∂s
∂uu+
∂s
∂u,iu,i +
∂s
∂q(e)i
q(e)i +
∂s
∂q(e)j,i
q(e)j,i
+∂s
∂q(p)i
q(p)i +
∂s
∂q(p)j,i
q(p)j,i
+∂Φ
(s)i
∂uu,i +
∂Φ(s)i
∂u,ju,ji +
∂Φ(s)i
∂q(e)j
q(e)j,i
+∂Φ
(s)i
∂q(e)k,j
q(e)k,ji
+∂Φ
(s)i
∂q(p)j
q(p)j,i
+∂Φ
(s)i
∂q(p)k,j
q(p)k,ji≥ 0. (14)
To achieve that task, we apply an extension of the classical Liu procedure [15], according
to which the thermodynamic restrictions on the constitutive functions may be obtained by
checking the positiveness of the linear combination of σ(s) and the evolution equations of the
state variables for all thermodynamic processes [16–18]. This linear combination is obtained
by means of Lagrange multipliers, which depend on the state variables themselves [17, 18].
Thus, we add to σ(s) Eqs. (11a)–(11c) and Eqs. (12a)–(12c), multiplied by the respective
Lagrange multipliers −λ(u), −λ(e)i , −λ(p)
i , −Λ(u)i , −Λ
(e)ji and −Λ
(p)ji . That way, after rear-
rangement, inequality (14) takes the form:(∂s
∂u− λ(u)
)u+
(∂s
∂u,i− Λ
(u)i
)u,i +
(∂s
∂q(e)i
− λ(e)i
)q
(e)i +
(∂s
∂q(e)j,i
− Λ(e)ji
)q
(e)j,i
+
(∂s
∂q(p)i
− λ(e)i
)q
(p)i +
(∂s
∂q(p)j,i
− Λ(p)ji
)q
(p)j,i
+ Λ(e)ji
[Γ
(e)3 q
(e)j,kki
+ Γ(e)4 q
(e)k,kji
]+ Λ
(p)ji
[Γ
(p)3 q
(p)j,kki
+ Γ(p)4 q
(p)k,kji
]+ I
(u;u,i ;u,ji ; q
(e)i ; q
(e)j,i
; q(e)k,ji
; q(p)i ; q
(p)j,i
; q(p)k,ji
)≥ 0, (15)
with I as a scalar regular function of indicated arguments. This inequality, which is also
called extended Liu inequality [16–18], is linear both in the time derivatives u, ui, q(e)i ,
7
q(e)j,i
, q(p)i , q
(p)j,i
, and in the spatial derivatives q(e)z,kji and q
(p)z,kji . It is therefore inferred that its
positiveness demands that:
∂s
∂u= λ(u), (16a)
∂s
∂u,i= Λ
(u)i , (16b)
∂s
∂q(e)i
= λ(e)i , (16c)
∂s
∂q(e)j,i
= Λ(e)ji , (16d)
∂s
∂q(p)i
= λ(p)i , (16e)
∂s
∂q(p)j,i
= Λ(p)ji , (16f)
as well as
Λ(e)ji
[Γ
(e)3 q
(e)j,kki
+ Γ(e)4 q
(e)k,kji
]= 0, (17a)
Λ(p)ji
[Γ
(p)3 q
(p)j,kki
+ Γ(p)4 q
(p)k,kji
]= 0, (17b)
and
I(u;u,i ;u,ji ; q
(e)i ; q
(e)j,i
; q(e)k,ji
; q(p)i ; q
(p)j,i
; q(p)k,ji
)≥ 0. (18)
The left-hand side in the last inequality represents, locally, the net entropy production
along the thermodynamic process.
Note that Eqs. (17) can be fulfilled in different ways, which will imply different ther-
modynamic restrictions. A possible way is to request Λ(e)ji = 0 and Λ
(p)ji = 0. These two
assumptions cut out the extended balances in Eqs. (12b) and (12c) as constraints for the
entropy inequality. In particular, in view of both these assumptions, and of Eqs. (16d)
and (16f), we have that the entropy cannot depend on q(e)j,i
and q(p)j,i
. Since we restrict our-
selves to isotropic bodies, a possible representation of s therefore may be [22]
s = s0 (u)− 1
2s(u) (u)u,iu,i −
1
2s(e) (u) q
(e)i q
(e)i −
1
2s(p) (u) q
(p)i q
(p)i , (19)
where the function s0 may be interpreted as the equilibrium entropy defined for homoge-
neous states, whereas the remaining terms in the right-hand side of Eq. (19) are genuine
8
nonequilibrium contributions. It is worth noticing that Eq. (19) ensures that the principle
of maximum entropy at the equilibrium is fulfilled [23].
Furthermore, it is easy matter to verify that I contains both quadratic terms and linear
ones in the second-order spatial derivatives of the unknown fields [18]. Therefore, once the
Lagrange multipliers λ(e)i and λ
(p)i have been expressed through the combination of Eqs. (16)
and (19), one has that the condition I ≥ 0 holds whatever the thermodynamic process is,
if, and only if, the following further thermodynamic restrictions are satisfied:
∂Φ(s)i
∂u,j= 0, (20a)
∂Φ(s)i
∂q(e)k,j
=(s(u)u,i + s(e)Γ
(e)4 q
(e)i
)δjk + s(e)Γ
(e)3 q
(e)k δij, (20b)
∂Φ(s)i
∂q(p)k,j
=(s(u)u,i + s(p)Γ
(p)4 q
(p)i
)δjk + s(p)Γ
(p)3 q
(p)k δij, (20c)
whereas inequality (18) reduces to G(u;u,i ; q
(e)i ; q
(e)j,i
; q(p)i ; q
(p)j,i
;)≥ 0, G being a suitable
function of the state variables, only.
The thermodynamic restriction in Eq. (20a) points out that the entropy flux does not
depend on u,i . Since the only way of removing the dependence of u,i from Eqs. (20b)
and (20c) is to set s(u) = 0 therein, we may also conclude that the entropy cannot depend
on u,i . This means that neither Eq. (12a) represents a constraint for the entropy inequality,
and Eq. (19) has to be changed as
s = s0 (u)− 1
2s(e) (u) q
(e)i q
(e)i −
1
2s(p) (u) q
(p)i q
(p)i . (21)
Consequently, Eqs. (20b) and (20c) become
∂Φ(s)i
∂q(e)k,j
= s(e)Γ(e)4 q
(e)i δjk + s(e)Γ
(e)3 q
(e)k δij, (22a)
∂Φ(s)i
∂q(p)k,j
= s(p)Γ(p)4 q
(p)i δjk + s(p)Γ
(p)3 q
(p)k δij. (22b)
Then, integrating Eq. (22a) with respect to q(e)k,j
, we may obtain the following form of the
entropy flux:
Φ(s)i = s(e)
[Γ
(e)3 q
(e)j q
(e)j,i
+ Γ(e)4 q
(e)i q
(e)j,j
]+ Ai
(u; q
(e)i ; q
(p)i ; q
(p)j,i
), (23)
9
with Ai as the components of a vectorial function, whose admissible expression may be [24]:
Ai = a1 (u) q(e)i + a2 (u) q
(p)i + a3 (u) q
(p)j q
(p)j,i
+ a4 (u) q(p)i q
(p)j,j, (24)
With this assumption, if we further differentiate Eq. (23) with respect to q(p)k,j
and compare
with Eq. (22b), by direct calculations we infer
Φ(s)i = a1 (u) q
(e)i +s(e)
[Γ
(e)3 q
(e)j q
(e)j,i
+ Γ(e)4 q
(e)i q
(e)j,j
]+a2 (u) q
(p)i +s(p)
[Γ
(p)3 q
(p)j q
(p)j,i
+ Γ(p)4 q
(p)i q
(p)j,j
].
(25)
This equation clearly shows that the entropy flux has a nonlinear form. In the next section
it will be compared with the classical form of the entropy flux in Rational Thermodynam-
ics [25]. In particular, it will be shown that the phenomenological coefficients Γ(e)3 , Γ
(p)3 , Γ
(e)4
and Γ(p)4 are related to nonlocal effects in the heat-transport equations. Furthermore, the
terms a1 (u) and a2 (u) could be interpreted, respectively, as the reciprocal of electron and
phonon absolute temperatures, as defined from the entropy flux.
III. HEAT TRANSPORT EQUATIONS WITH PHONONS AND ELECTRONS
A. Evolution equations of the heat fluxes
In order to obtain practical results, in this section we suppose that the different material
functions in Eqs. (9) and (10) are constant. In this way, from Eqs. (8a) and (8b), respectively,
we have:
q(e)i = γ(e)cvθ,i + Γ
(e)0 q
(e)i + Γ
(e)2
(q
(p)j q
(e)i,j
+ q(e)i q
(p)j,j
)+ Γ
(e)3 q
(e)i,jj
+ Γ(e)4 q
(e)j,ji, (26a)
q(p)i = γ(p)cvθ,i + Γ
(p)0 q
(p)i + Γ
(p)2
(q
(e)j q
(p)i,j
+ q(p)i q
(e)j,j
)+ Γ
(p)3 q
(p)i,jj
+ Γ(p)4 q
(p)j,ji, (26b)
where cv = ∂u/∂θ is the specific heat at constant volume, θ being the temperature.
Equations (26) point out that q(e)i and q
(p)i are ruled by nonlinear equations which are
coupled through the functions Γ(e)2 and Γ
(p)2 . Let us observe that, although with a differ-
ent meaning, an electron-phonon coupling factor, which is temperature dependent, plays
a very important role in the kinetics of energy distribution in femtosecond laser applica-
tion [26–28]. Moreover, it also strongly influences the effective thermal conductivity and the
total interface thermal resistance of an alternate metal-nonmetal multilayered (superlattice)
10
system, where the phonon-phonon transport across interfaces are modeled by the Kapitza
interfacial thermal resistance [29]. Indeed, the electron lattice nonequilibrium, which drives
electron-phonon coupling, has been the focus of several studies and recently this interest
has been extended to nanowires [30]. In bulk materials at room temperature, the resistance
to electron transport is dominated by phonon scattering. However, when the characteristic
length of a material is of the order of the mean-free path of the electrons, the resistance
becomes affected by scattering of electrons at surfaces. For instance, in nanoporous gold,
the electron-surface scattering has been attributed to electronic coupling of adsorbates to
the conduction electrons which gives rise to the change in resistance exploited in several
applications [31].
If we disregard this coupling, that is, if we suppose Γ(e)2 = Γ
(p)2 = 0, from Eqs. (26) the
following equations arise:
τeq(e)i + q
(e)i = −κeθ,i + `2
e
(q
(e)i,jj
+ 2q(e)j,ji
), (27a)
τpq(p)i + q
(p)i = −κpθ,i + `2
p
(q
(p)i,jj
+ 2q(p)j,ji
), (27b)
where we have identified
Γ(e)0 = −τ−1
e ; Γ(p)0 = −τ−1
p ;
γ(e)cv = −κeτ−1e ; γ(p)cv = −κpτ−1
p ;
Γ(e)3 = `2
eτ−1e ; Γ
(p)3 = `2
pτ−1p ;
Γ(e)4 = 2Γ
(e)3 ; Γ
(p)4 = 2Γ
(p)3 .
(28)
These equations are of the Guyer-Krumhansl (G-K) type [32]. Therein τe and τp may be
identified as the relaxation times due to electrons’ interactions and phonons’ interactions,
respectively. In more detail, τe may be defined though the electron-electron scattering-time
τe−e and the electron-phonon scattering-time τe−p by the Matthiessen rule [26, 33], i.e.,
τ−1e = τ−1
e−e + τ−1e−p. Along with the original proposal of Guyer and Krumhansl [32], instead,
τp may be related both to the resistive mechanisms between the different particles and to the
scattering with the boundaries, in such a way that τ−1p = τ−1
u + τ−1i + τ−1
d + τ−1p−w, where τu is
the relaxation time of umklapp-phonon collisions, τi the relaxation time of phonon-impurity
collisions, τd the relaxation time of phonon-defect collisions, and τp−w the relaxation time of
phonon-wall collisions. Indeed, the relaxation time τe−p may be included in the definition of
τp, too [34].
11
Moreover, we suppose that `e is the mean-free path of electrons in the bulk [30], which
is dominated by electron and phonon scattering in such a way that `e = νF τe, with νF as
the Fermi velocity. Indeed, the effects of electron-boundary scattering should be also taken
into account when calculating `e in nanosystems. They yield another relaxation time (τe−w)
which should be added to τe via Matthiessen rule. Let us further observe that the electron
mean-free path can be also expressed by the Drude formula [35]. Analogously, in Eq. (27b)
`p means the mean-free path of phonons, which is related both to resistive scattering and to
the normal scattering of phonons [32, 36] as `2p = c2τnτp/5, τn being the relaxation time of
normal collisions, and c the modulus of the average phonons speed. Finally, in Eqs. (27) we
identify κe as the thermal conductivity due to electrons, and κp as the thermal conductivity
due to phonons. Both of them are size dependent [26, 30, 32, 36–41].
Let us observe that the last line in Eqs. (28) refers to the coefficients of the last terms
in Eqs. (27). Since those terms are related to the divergence of the heat flux (related to
the time derivative of temperature) they will be zero in steady-state situations, but they
may be relevant in processes with fast time-temperature variations. It is seen in the last
line of Eqs. (28) that, since we look for a structure of the equations of the G-K type, the
physical coefficients Γ3 and Γ4 must be proportional. This fact follows by the the solution
of the linearized Boltzman equation for phonons [32, 36], from which one obtains that the
coefficient of the gradient of the divergence of the heat flux is two times that of the divergence
of the gradient. Thus, the proportionality of Γ3 and Γ4 is dictated by phonon hydrodynamics.
Recalling the constitutive assumption in Eq. (7), we also conclude that in our model the
time rate of the local heat flux is given by a linear combination of two G-K type equations.
Thus, from Eq. (25) one may infer that an equation of G-K type for the heat flux leads to
an expression of the entropy flux which is more general than the classical expression θ−1q(u)i
postulated in Rational Thermodynamics [25]. The same conclusion, but with a different
approach, has been obtained by Lebon et al. [24].
When the terms in the round brackets in the right-hand side of Eqs. (27) can be neglected
with respect to others, we simply have
τeq(e)i + q
(e)i = −κeθ,i , (29a)
τpq(p)i + q
(p)i = −κpθ,i . (29b)
These equations are of the Maxwell-Cattaneo (M-C) type [14], and predict a hyperbolic
12
propagation of the heat flux. Along with the observations above, in such a case the time
rate of q(u)i is given by a linear combination of two M-C type equations, and the entropy flux
is more general than Φ(s)i = θ−1q
(u)i .
B. Heat waves
Here we study the propagation of heat waves in rigid solids on the basis of Eqs. (12a)
and (29). To this end, let us suppose that in our system a smooth surface, whose equation
is φ (x; t) = 0, is propagating. We suppose that across this surface each state variable is
continuous, whereas their first-order spatial derivatives, instead, suffer jump discontinuities
defined by
δ =
(∂
∂φ
)φ=0+
−(∂
∂φ
)φ=0−
. (30)
For the sake of simplicity, we put ourselves in the one-dimensional case (i.e.,
q(e) ≡ q(e) (x; t) and q(p) ≡ q(p) (x; t), where x means the direction of propagation). Making
use of the standard transformations ξ → −Uδξ and ξ,x → δξ, U being the speed of the wave
(which we aim to calculate) and ξ a generic function of the state variables, from the balance
law of internal energy (namely, Eq. (12a)) it follows
cvUδθ = δq(e) + δq(p), (31)
once the internal energy has been expressed as a function of temperature through the relation
du = cvdθ. Equations (29), instead, reduce to
τeUδq(e) = κeδθ, (32a)
τpUδq(p) = κpδθ. (32b)
By straightforward calculations from the combination of Eqs. (31) and (32) we have
U =
√κecvτe
+κpcvτp
. (33)
The result above allows us to infer that the phonons’ contribution to the heat transport
in metal may contribute significantly to the phase speed. This is particularly important in
metals as Pt, Ni and W, which are characterized by comparable values of κe and κp [6].
Other metals, as Au, Ag, Cu and Al, instead, have κe � κp, and the phase speed reduces
to U ≈ Ue =√κe/ (cvτe).
13
IV. HEAT TRANSPORT AND ELECTRIC TRANSPORT: WIEDEMANN-
FRANZ LAW IN METALLIC NANOWIRES
The Wiedemann-Franz law states that κtot/σ = L0θ, being κtot the sum of the electronic
thermal conductivity and the phonon thermal conductivity, σ the electron conductivity, and
L0 the so-called Lorenz factor. When the phonon contribution is negligible, L0 is suggested
to be constant for all metals in the Sommerfeld theory, that is, L0 = (π2/3) (kB/e) =
2.44×10−8 V2/K2, where e is the elementary charge and kB the Boltzmann constant. Indeed,
a topic of current interest in metallic nanowires is whether L0 depends on the size of the
system, whenever the characteristic size becomes comparable to (or smaller than) the mean-
free paths [6, 19].
To discuss this topic in the framework of our analysis one should also include the electric
current Ji and its spatial first-order derivatives Jj,i as further independent state variables, in
analogy with the heat flux and its first-order spatial derivatives. In the case of qi a relevant
quantity was `qj,i , as we already said. In the electrical case, instead, the analogous relevant
quantity would be `EJj,i , with `E as the mean-free path of the carriers related to electrical
current transport.
In analogy with the evolution equations for q(e)i and q
(p)i (that is, Eqs. (27)), one may
assume
τEJi + Ji = σEi + `2E
(Ji,jj + 2Jj,ji
), (34)
wherein Ei means the electric field, and τE the relaxation time. It seems worth noticing that
the mean-free path of the electric flux `E, in principle, could be different from the electron
mean-free path `e under the effects of a temperature gradient.
In Refs. [38–41], in order to obtain a size-dependent behavior of the thermal conductivity,
we have proposed to complement the time rate of the phonon heat flux with a suitable
boundary condition accounting for the nonvanishing value of the phonon heat flux on the
walls. This approach may be also extended both to the electron heat flux and to the electric
current. Therefore, limiting ourselves only to the specular and diffusive interactions between
heat carriers and walls, in order to have the wall contributions q(e)w , q
(p)w and Jw, we propose
14
to couple Eqs. (27) and Eq. (34) with the following boundary conditions:
q(e)w = Ce`e
(∂q
(e)b
∂r
∣∣∣∣∣r≡R
), (35a)
q(p)w = Cp`p
(∂q
(p)b
∂r
∣∣∣∣∣r≡R
), (35b)
Jw = CE`E
(∂Jb
∂r
∣∣∣∣r≡R
), (35c)
being q(e)b , q
(p)b and Jb the solutions of Eqs. (27) and Eq. (34) for the bulk, r the radial
distance from the longitudinal axis of the nanostructure, and R the radius of the nanowire.
Moreover, in Eqs. (35) Ce, Cp and CE are numerical constants which depend on the relative
portion of specular and diffusive collisions of the corresponding heat carriers with the walls.
As we have included the possibility that `e is different than `E, we also assume that Ce differs
from CE, namely, the electron-wall collisions under a temperature gradient could be different
that under an electric field. Extending the results of Refs. [38–41], when R is smaller than
each mean-free path `e, `p and `E, we have
κeffe = Ce
κe2
R
`e, (36a)
κeffp = Cp
κp2
R
`p, (36b)
σeff = CEσ
2
R
`E. (36c)
Note that these effective transport coefficients depend on the size and are reduced when
the radius R is reduced. The question now is to examine in which situations such a size
dependence of the transport coefficients implies a size dependence on the Wiedemann-Franz
law. From Eqs. (36), we conclude that the effective Lorenz factor Leff is given by
Leff =1
T
(κeffe + κeff
p
σeff
)=
1
T
Ceκe`e
+Cpκp`p
CEσ
`E
. (37)
As it is possible to observe from the results above, when R is smaller than `e, `p and `E,
Leff does not depend on the characteristic size of the system, but it is no longer the value
for the bulk system, nor is the same for all metals, since the coefficients Ce, Cp and CE may
be different, as well as the mean-free paths `e, `p and `E. The relation in Eq. (37) would be
valid for thin nanowires of metals as Pt, Ni and W, for which `e and `p are comparable [6].
15
In contrast, in metals as Au, Ag and Cu (which have `e several times longer than `p, and
therefore `p � R � `e) it is more realistic to have κeffp ≈ κp and the effective Lorenz factor
becomes
Leff =1
T
Ceκe2`e
+κpR
CEσ
2`E
, (38)
Thus, in this case decreasing R would slightly rise the effective value of the Lorenz factor,
as it has been observed in Ref. [6] (see Fig. 6 therein for the mentioned materials). In that
paper, the breakdown of the Wiedemann-Franz law is imputed to the importance of phonon
thermal conductivity, and to the difference between `e and `E. However, in Eq. (38) it
appears still a new factor which could in principle contribute to the discrepancy between L
and L0, that is, the difference between Ce and CE, related to collisions with the walls.
Let us finally observe that if we suppose `e ≈ `E and Ce ≈ CE, in metals for which
the phonon thermal conductivity may be neglected with respect to the electrons thermal
conductivity, Eq. (38) reduces to the classical value L0 above for the Lorenz factor.
V. CONCLUSIONS
In the framework of EIT we have developed a model for heat conduction by phonons
and electrons. Particular emphasis has been given to nonlocal effects, which may arise at
the nanometric length-scale, when the mean-free paths of phonons and/or electrons are
larger than the physical dimensions of the system. In situations in which the coupling
can be neglected, we recovered two parabolic equations of the G-K type, modeling the
concurrent presence of the diffusion of heat superposed to the propagation of heat waves.
These equations may be applied, for instance, to heat transport in metallic nanowires, in
which both electrons and phonons contribute. Moreover, they can be easily extended to
encompass heat transport in semiconductor nanowires, in which phonons, electrons and
holes contribute. From the purely thermodynamic point of view, we have stressed that
nonlocal terms in the heat transport equations imply nonclassical contributions to the heat
flux.
Under the hypothesis that in those equations the nonlocal terms are negligible, we have
recovered two equations of M-C type. In such a case, the phase speed of one-dimensional
16
temperature waves has been calculated, once each material function has been supposed to
be constant. It turns out that the electrons’ contribution to the heat transport may change
significantly the value of the phase speed.
A more explanatory comment has to be made about the use of the nonequilibrium tem-
perature and the constitutive assumption cv = ∂u/∂θ we made in Eqs. (26). Indeed, in a
more refined approach, heat conduction due to phonons and electrons should be described in
the framework of the two-temperature model [2, 42–44], namely, we should consider a set of
two coupled differential equations describing the evolution of electron and lattice tempera-
tures. Along this way, we should also consider two different heat capacities for the electrons
and the phonons [26, 45].
Incorporation of electrical current is cumbersome in the procedure, but simple in the
final issue, and it allows us to discuss the size-dependence of the Wiedemann-Franz law. We
have considered it both when the characteristic size of the system is negligible with respect
to `e and `p, and when `p � R � `e. Both cases show different behaviors in the Lorenz
factor. In particular, in the latter case, Eq. (38) shows that besides the two usual reasons
for the discrepancy in nanosystems between Leff and L0 (namely, κp no longer negligible
with respect to κe and `e 6= `E), a third factor, related to the boundary effects, appears, i.e.,
Ce 6= CE.
Here we have considered the effects of the several mean-free paths on the equations
for heat and electrical transport. A further step along the same line would be to include
thermoelectric effects (namely, Seebeck, Peltier and Thomson effects), which will be also
modified by nonlocal contributions. Thermoelectric effects are relevant from the point of
view of nanorefrigeration (Peltier effect) and thermoelectric generation (Seebeck effect),
and therefore they deserve a detailed attention. In future research it would be interesting
to incorporate those effects and to study, for instance, how the different mean-free paths
and device sizes contribute to the figure of merit Zθ of different thermoelectric devices,
Z being the square of Seebeck coefficient times electrical conductivity divided by thermal
conductivity. Since the product Zθ has a direct role on the efficiency of thermoelectric
conversion, such an analysis is expected to be useful for practical purposes.
17
Acknowledgements
D. J. acknowledges the financial support from the Direccion General de Investigacion of
the Spanish Ministry of Science and Innovation under grant FIS No. 2009-13370-C02-01,
the Consolider Project NanoTherm (grant CSD-2010-00044), and the Direccio General de
Recerca of the Generalitat of Catalonia under grant No. 2009-SGR-00164.
V. A. C. acknowledges the financial support from the University of Basilicata and the
Italian Gruppo Nazionale per la Fisica Matematica.
A. S. acknowledges the financial support from the Departament de Fısica of Universitat
Autonoma de Barcelona for his stay in the Autonomous University of Barcelona, and the
Italian Gruppo Nazionale per la Fisica Matematica under grant Progetto Giovani 2010.
[1] A. Majumdar, K. Fushinobu, and K. Hijikata, “Effect of gate voltage on hot-electron and
hot-phonon interaction and transport in a submicrometer transistor,” J. Appl. Phys., vol. 77,
pp. 6686–6694, 1995.
[2] D. Y. Tzou, Macro to micro-scale heat transfer. The lagging behaviour. New York: Taylor and
Francis, 1997.
[3] G. Lebon, D. Jou, and J. Casas-Vazquez, Understanding nonequilibrium thermodynamics.
Berlin: Springer-Verlag, 2008.
[4] D. Jou, J. Casas-Vazquez, and G. Lebon, Extended Irreversible Thermodynamics. Berlin:
Springer-Verlag, Third ed., 2001.
[5] F. Koechlin and B. Bonin, “Parametrisation of the Niobium thermal conductivity in the
superconducting state,” in Proceedings of the 1995 Workshop on RF Superconductivity, Gif-
sur-Yvette, France (B. Bonin, ed.), (New York), pp. 665–669, Gordon and Breach, 1996.
[6] N. Stojanovic, D. H. S. Maithripala, J. M. Berg, and M. Holtz, “Thermal conductivity in
metallic nanostructures at high temperature: electrons, phonons, and the Wiedemann-Franz
law,” Phys. Rev. B, vol. 82, p. 075418 (9 pages), 2010.
[7] M. F. Crommie and A. Zettl, “Thermal conductivity of single-crystal Bi-Sr-Ca-Cu-O,” Phys.
Rev. B, vol. 41, pp. 10978–10982, 1990.
[8] K. E. Goodson and M. I. Flik, “Electron and phonon thermal conduction in epitaxial high-Tc
18
superconducting films,” ASME Journal of Heat Transfer, vol. 115, pp. 17–25, 1990.
[9] V. A. Cimmelli, A. Sellitto, and D. Jou, “Nonlocal effects and second sound in a nonequilibrium
steady state,” Phys. Rev. B, vol. 79, p. 014303 (13 pages), 2009.
[10] C. C. Ackerman and R. A. Guyer, “Temperature pulses in dielectric solids,” Annals of Physics,
vol. 50, pp. 128–185, 1968.
[11] H. E. Jackson, C. T. Walker, and T. F. McNelly, “Second sound in NaF,” Phys. Rev. Lett.,
vol. 25, pp. 26–28, 1970.
[12] H. E. Jackson and C. T. Walker, “Thermal conductivity, second sound and phonon-phonon
interactions in NaF,” Phys. Rev. Lett., vol. 3, pp. 1428–1439, 1971.
[13] V. Narayanamurti and R. D. Dynes, “Observation of second sound in bismuth,” Phys. Rev.
Lett., vol. 26, pp. 1461–1465, 1972.
[14] C. Cattaneo, “Sulla conduzione del calore,” Atti Sem. Mat. Fis. Univ. Modena, vol. 3, pp. 83–
101, 1948.
[15] I.-S. Liu, “Method of Lagrange multipliers for exploitation of entropy principle,” Arch. Ration.
Mat. Anal., vol. 46, pp. 131–138, 1972.
[16] V. A. Cimmelli, “An extension of Liu procedure in weakly nonlocal thermodynamics,” J.
Math. Phys., vol. 48, p. 113510 (13 pages), 2007.
[17] V. A. Cimmelli, A. Sellitto, and V. Triani, “A new thermodynamic framework for second-grade
Korteweg-type viscous fluids,” J. Math. Phys., vol. 50, p. 053101 (15 pages), 2009.
[18] V. A. Cimmelli, F. Oliveri, and V. Triani, “Exploitation of the entropy principle: Proof of Liu
theorem if the gradients of the governing equatios are considered as constraints,” J. Math.
Phys., vol. 52, p. 023511 (16 pages), 2011.
[19] F. Volklein, H. Reith, T. W. Cornelius, M. Rauber, and R. Neumann, “The experimen-
tal investigation of thermal conductivity and the Wiedemann-Franz law for single metallic
nanowires,” Nanotechnology, vol. 20, p. 325706 (9 pages), 2009.
[20] V. A. Cimmelli, A. Sellitto, and V. Triani, “A generalized Coleman-Noll procedure for the
exploitation of the entropy principle,” Proc. R. Soc. A, vol. 466, pp. 911–925, 2010.
[21] W. Muschik and H. Ehrentraut, “An amendment to the Second Law,” J. Non-Equilib. Ther-
modyn., vol. 21, pp. 175–192, 1996.
[22] G. F. Smith, “On isotropic functions of symmetric tensors, skew-symmetric tensors and vec-
tors,” Int. J. Engng. Sci., vol. 9, pp. 899–916, 1971.
19
[23] J. Verhas, Thermodynamics and Rheology. Dordrecht: Kluwer Academic Publisher, 1997.
[24] G. Lebon, D. Jou, J. Casas-Vazquez, and W. Muschik, “Weakly nonlocal and nonlinear heat
transport in rigid solids,” J. Non-Equilib. Thermodyn., vol. 23, pp. 176–191, 1998.
[25] C. Truesdell, Rational Thermodynamics. Berlin: Springer Verlag, second ed., 1984.
[26] Z. Lin and L. V. Zhigilei, “Temperature dependences of the electron-phonon coupling, electron
heat capacity and thermal conductivity in Ni under femtosecond laser irradiation,” Appl. Surf.
Sci., vol. 253, pp. 6295–6300, 2007.
[27] P. B. Allen, “Theory of thermal relaxation of electrons in metals,” Phys. Rev. Lett., vol. 59,
pp. 1460–1463, 1987.
[28] X. Y. Wang, D. M. Riffe, Y.-S. Lee, and M. C. Downer, “Time-resolved electron-temperature
measurement in a highly excited gold target using femtosecond thermionic emission,” Phys.
Rev. B, vol. 50, pp. 8016–8019, 1994.
[29] J. Ordonez-Miranda, J. J. Alvarado-Gil, and R. Yang, “The effect of the electron-phonon
coupling on the effective thermal conductivity of metal-nonmetal multilayers,” J. App. Phys.,
vol. 109, p. 094310 (7 pages), 2011.
[30] P. E. Hopkins, P. M. Norris, L. M. Phinney, S. A. Policastro, and R. G. Kelly, “Thermal
Conductivity in Nanoporous Gold Films during Electron-Phonon Nonequilibrium,” Journal
of Nanomaterials, vol. 2008, p. 418050 (7 pages), 2008.
[31] Z. Liu and P. C. Searson, “Single nanoporous gold nanowire sensors,” J. Phys. Chem. B,
vol. 110, pp. 4318–4322, 2006.
[32] R. A. Guyer and J. A. Krumhansl, “Thermal conductivity, second sound and phonon hydro-
dynamic phenomena in nonmetallic crystals,” Phys. Rev., vol. 148, pp. 778–788, 1966.
[33] N. W. Ashcroft and N. D. Mermin, Solid state physics. New York: Saunder College, 1976.
[34] J. Zou and A. Baladin, “Phonon heat conduction in a semiconductor nanowire,” J. Appl.
Phys., vol. 89, pp. 2933–2938, 2001.
[35] G. Grimwall, The Elecron-Phonon Intercation in Metals. North-Holland Pub. Co., 1981.
[36] R. A. Guyer and J. A. Krumhansl, “Solution of the linearized phonon Boltzmann equation,”
Phys. Rev., vol. 148, pp. 766–778, 1966.
[37] F. X. Alvarez and D. Jou, “Size and frequency dependence of effective thermal conductivity
in nanosystems,” J. Appl. Phys., vol. 103, p. 094321 (8 pages), 2008.
[38] F. X. Alvarez, D. Jou, and A. Sellitto, “Phonon hydrodynamics and phonon-boundary scat-
20
tering in nanosystems,” J. Appl. Phys., vol. 105, p. 014317 (5 pages), 2009.
[39] A. Sellitto, F. X. Alvarez, and D. Jou, “Second law of thermodynamics and phonon-boundary
conditions in nanowires,” J. Appl. Phys., vol. 107, p. 064302 (7 pages), 2010.
[40] A. Sellitto, F. X. Alvarez, and D. Jou, “Temperature dependence of boundary conditions in
phonon hydrodynamics of smooth and rough nanowires,” J. Appl. Phys., vol. 107, p. 114312
(7 pages), 2010.
[41] F. X. Alvarez, D. Jou, and A. Sellitto, “Phonon boundary effects and thermal conductivity of
rough concentric nanowires,” J. Heat Transfer, vol. 133, p. 022402 (7 pages), 2011.
[42] M. I. Kaganov, I. M. Lifshitz, and M. V. Tanatarov, “Relaxation between electrons and
crystalline lattices,” Sov. Phys. JETP., vol. 4, p. 173, 1957.
[43] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces
exposed to ultrashort laser pulses,” Sov. Phys. JETP, vol. 39, p. 375, 1974.
[44] T. Hertel and G. Moos, “Electron-Phonon Interaction in Single-Wall Carbon Nanotubes: A
Time-Domain Study,” Phys. Rev. Lett., vol. 84, pp. 5002–5005, 2000.
[45] Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity
of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B, vol. 77,
p. 075133 (17 pages), 2008.
21
