Propagation of temperature waves

along core-shell nanowires∗

Vito Antonio Cimmelli (a)†, David Jou (b,c)‡, Antonio Sellitto (a)§

(a) Department of Mathematics and Computer Science,

University of Basilicata,

Campus Macchia Romana, 85100, Potenza, Italy

(b) Departament de Fısica, Universitat Autonoma de Barcelona,

08193 Bellaterra, Catalonia, Spain

(c) Institut d’Estudis Catalans, Carme 47, Barcelona 08001, Catalonia, Spain

Abstract

In a recent paper the authors proved that the dispersion relation of heat waves along

nanowires could allow to illustrate the difference between different definitions of nonequi-

librium temperature. It turns out that, from the practical perspective, one is led to the

same conclusion both with using the absolute nonequilibrium temperature and a dynam-

ical nonequilibrium temperature. Starting from these results, in the present paper the

propagation of heat waves in core-shell nanowires is analyzed by using the concept of

absolute nonequilibrium temperature. It is shown how the wave speed depends on the

properties of both media and on their mutual interaction. Some useful information about

the system is carried out.

Keywords. Dynamical nonequilibrium temperature; thermal waves; core-shell nanowires.

1 Introduction

In classical transport theory, the fluxes J and the conjugated thermodynamic forces X are

related by linear transport laws of the form J = L ·X, with L as suitable phenomenological

∗Work presented at the 10th Joint European Thermodynamics Conference, Copenhagen, June 22-24, 2009.†Electronic address: vito.cimmelli@unibas.it‡Electronic address: david.jou@uab.es§Corresponding author. Electronic address: ant.sellitto@gmail.com

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coefficients. For instance, one may identify J with the heat flux q, and X with ∇(T−1), being

T the absolute nonequilibrium temperature. Then, if L = λT 2I, where λ denotes the heat

conductivity, and I the identity tensor, for isotropic systems the well-know Fourier law

q = −λ∇T, (1.1)

ensues. This equation, which implies an instantaneous response of the system, breaks down

when the perturbation is very fast or when the response time is very long. This can happen,

for instance, in heat conduction in miniaturized systems, such as nanowires and thin layers

submitted to high-frequency perturbation. Heat transport in these devices differs significa-

tively from the predictions of Fourier law [1–4], because their behavior is strongly influenced

by memory and nonlocal effects [5, 6], which lack in the Fourier law. Furthermore, in mi-

crodevices working at high frequencies, one has also to take into account the relaxation time

of the heat flux [7], because heat flux will not have time enough to accommodate to the

value given by the Fourier law. As a consequence, a more general heat transport equation

must be introduced [8–10]. On the other hand, in situations as those mentioned above, the

relaxational constitutive equation for J

τ J = −(J− L ·X), (1.2)

seems to be more appropriate. In Eq. (1.2) τ denotes a suitable relaxation time, while a

superposed dot denotes the material time derivative which, in the case of rigid bodies at rest

considered in the present paper, reduces to the usual partial time derivative. In the case of

heat transport, Eq. (1.2) takes the explicit form

τ q+ q = −λ∇T, (1.3)

which is generally referred to as the Cattaneo or Maxwell-Cattaneo equation [11]. The term

containing the time τ represents the heat flux relaxation. When τ is negligible, or when

the time variation of the heat flux is slow, Eq. (1.3) will reduce to Fourier law. It is worth

noticing that the equation above is not the most general one to describe heat conduction

in miniaturized systems. For instance, Chen [4] regarded the transport of heat in nanosize

systems as a combination of the heat conduction described by a kinetic equation (ballistic

2

transport) superposed to the heat conduction governed by Eq. (1.3). Thus, the state variables

are the energy density and the heat flux, describing the overall Cattaneo heat conduction,

together with the one-phonon distribution function describing the superposed on it ballistic

component. The thermodynamic compatibility of Chen’s model has been investigated by

Grmela et al. [12], in the framework of the GENERIC approach to nonequilibrium thermo-

dynamics [13]. They found that both the Cattaneo and the kinetic equations are modified

by the appearance of new terms in which their coupling is expressed. Here we limit ourselves

to those nonequilibrium situations which are well described by Eq. (1.3) only.

If the heat transport is modeled through the phonon gas hydrodynamics [14, 15], τ is

identified with the relaxation time τR due to the resistive scattering processes of phonons. In

fact, in solid crystals the phonons form a rarefied gas whose kinetic equation can be derived

similarly to that of an ordinary gas. Moving through the crystal lattice they undergo two

different types of scattering:

i) Normal-(N) scattering, conserving the phonon momentum;

ii) Resistive-(R) scattering, in which the phonon momentum is not conserved.

The frequencies νN and νR of normal and resistive scattering determine the characteristic

relaxation times τN = 1/νN and τR = 1/νR. Whenever τN = 1/νN tends to zero, then a

wave like energy transport may occur. In such a case the evolution of the heat flux is well

described by Eq. (1.3).

Recently, core-shell nanowires have become the focus of intensive research owing to their

very promising technological performance. For instance, indium-gallium-arsenide (InGaAs)

nanowires have applications in high-mobility electronics and optoelectronics in a wavelength

region of interest for telecommunications, as they exhibit photoluminiscence, but with little

optical efficiency. However, it has been seen that surrounding this nanowire with a thin

shell of gallium-arsenide (GaAs) – or, in more general terms, a smaller band gap material

core surrounded by a large gap material shell – highly improves the photoluminiscent perfor-

mance [16]. Another example are gallium-nitride (GaN) nanowires surrounded by a thin shell

of amorphous silicon (a-Si), which provide miniaturized diodes with reliable useful proper-

ties. Other devices based on core-shell nanowires of the group III-nitride (Al/Ga/In-N) have

outstanding properties for developing electronic and optical components (diodes, transistors,

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detectors, and light emitters), or Si-based core-shell nanowires may improve some batteries

[17]. Many of the miniaturized devices are usually submitted to high-frequency perturba-

tions, because electronic and optical components are often used to transmit and process big

amounts of information, which must be done very fast. Thus, it is logical to ask for the

response of these systems to fast perturbations.

Here, we specialize our attention to thermal phenomena, i.e., to the response to high-

frequency temperature perturbations. The behavior at high frequency is useful to describe

the propagation of pulses which may arise, for instance, when one boundary of the nanowires

receives a short laser pulse giving to it optical information; this optical information will be

accompanied by some thermal effects, which will also propagate.

The aim of the present paper is to explore the behavior of high-frequency thermal per-

turbation in core-shell nanowires, by incorporating relaxational effects in the heat flux.

In Sec. 2 we briefly review our previous work [18] on heat pulses along simple nanowires,

where we showed that, although the speed of the pulses may depend on the definition of

nonequilibrium temperature being used in the description, from the practical point of view

one is led to the same conclusion both with using the absolute nonequilibrium temperature

and a dynamical nonequilibrium temperature.

In Sec. 3 we explore the details of heat propagation in core-shell nanowires.

2 Relaxational and surface effects in nanowires

In the present section we will focalize our attention on small cylinders of radius r. If they

are not laterally isolated, the local balance of energy reads

cT = −∇ · q− 2σ

r(T − Tenv) , (2.1)

with c as the specific heat per unit volume, and the second term in the right-hand side as the

heat exchanged by the system with the surrounding environment across the lateral walls [18].

It expresses the Newton cooling law, being σ a suitable heat exchange coefficient, and Tenv the

temperature of the environment, which will be assumed to be constant and homogeneous.

Furthermore, in Eq. (2.1) q stands for the longitudinal heat flux along the length of the

cylinder. Note that in the Newton cooling law we assume that the heat flux across the lateral

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surface is qs = σ (T − Tenv)n, being n the outward unit normal to the closest wall. In analogy

with Eq. (2.1) we could also have assumed the presence of relaxation terms in the surface

heat transfer, namely, τsqs + qs = σ (T − Tenv)n. However, the relaxation time τs of the

transversal heat flux will be of the order of the radius of the nanowire divided by the average

phonon speed, whereas the relaxation time τ of the longitudinal heat flux will be of the order

of the mean-free path of heat carriers divided by the mentioned speed. For nanowires the

radius may be smaller than the mean free-path, and the transversal relaxational effects could

be negligible as compared with longitudinal relaxational effects.

From Eqs. (1.3) and (2.1), one obtains

τcT + cT = λ∇2T − 2σ

r

[τ T + (T − Tenv)

]. (2.2)

We aim to study the propagation of plane temperature-waves

T (x; t) = T 0 exp [i (ωt− κx)] , (2.3)

which may be experimentally realized by imposing at one end of the system a sinusoidally

time-dependent temperature, and detecting the consequent temperature perturbation at dif-

ferent points along the system [10, 19]. Then, from Eq. (2.2) the following dispersion relation

holds

κ2 =1

χ

(τω2 − 2σ

rc

)− i

ω

χ

(1 +

2στ

rc

), (2.4)

with χ ≡ λ/c as the thermal diffusivity. Thus, the phase velocity Up ≡ |ω/Re (κ)| and the

attenuation distance α ≡ |1/ Im (κ)| are, respectively

Up =ω

Γ

√2

1 + ϕ, (2.5a)

α =1

Γ

√2

1− ϕ, (2.5b)

where Γ =

√1

λr4

√(τω2rc− 2σ)2 + ω2 (rc+ 2στ)2,

ϕ =

(τω2rc− 2σ

)√(1 + τ2ω2) (4σ2 + ω2r2c2)

.

(2.6)

5

At low frequencies (lf), i.e., when ωτ ≪ 1, we get

U (lf)p =

√2σλr

[1− ω2

(r2c2 + 4σ2τ2

16σ2

)](2

rc+ 2στ

), (2.7a)

α(lf) =

√λr

2σ

[1− ω2

8

(rc− 2στ

2σ

)2], (2.7b)

which are the results predicted by Fourier law, plus contributions in τ , which vanish in the

Fourier diffusive regime.

In the high-frequency (hf) limit (ωτ → ∞), instead, the phase velocity and the attenuation

distance become

U (hf)p =

√χ

τ, (2.8a)

α(hf) =

√χ

τ

(2τrc

rc− 2στ

). (2.8b)

The result (2.8a) for the phase speed is the same as in the bulk of isolated cylinders, and

it does not depend on the radius. In contrast, the radius influences the attenuation length

(2.8b) and in the limit of high radius (r → ∞), or of isolated nanowires (σ ≡ 0), one recovers

the usual result for thermal waves in hyperbolic heat propagation [8, 10, 18–20].

3 Temperature waves along core-shell nanowires

In the present section we study the propagation of temperature waves along core-shell

nanowires, i.e., nanowires composed of a cylindrical inner core of one material and a concentric

cylindrical outer shell of a different material. In this case, since we face with two different

materials, the Maxwell-Cattaneo equation [Eq (1.3)] takes the form

τicqic + qic = −λic∇Tic, (3.1a)

τosqos + qos = −λos∇Tos, (3.1b)

where the subscript (ic) means the contribution of the inner core, and the subscript (os) means

the contribution of outer shell. If the system at hand is externally isolated (i.e., the heat may

flow only from the inner core to outer shell, or conversely, but not to the environment), the

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evolution equations of both temperatures Tic and Tos read

cicTic = −∇ · qic −2σ

ric(Tic − Tos) , (3.2a)

cosTos = −∇ · qos +2σ

ric(Tic − Tos) . (3.2b)

Coupling Eqs. (3.1) with Eqs. (3.2) one obtains

τiccicTic + cicTic = λic∇2Tic −2σ

ric

[Tic − Tos + τic

(Tic − Tos

)], (3.3a)

τoscosTos + cosTos = λos∇2Tos +2σ

ric

[Tic − Tos + τos

(Tic − Tos

)]. (3.3b)

We suppose now the propagation of plane temperature-waves

Tic (x; t) = T 0,ic exp [i (ωt− κx)], (3.4a)

Tos (x; t) = T 0,os exp [i (ωt− κx)], (3.4b)

namely, we are assuming that, after an initial moment, the temperature wave will propagate

with the same phase speed Up both in the inner core and the outer shell, due to their mutual

heat exchange. If they were insulated from each other (σ ≡ 0), heat waves would propagate

in each material with different speed. It will be discussed below.

Then, taking into account Eqs. (3.4), from Eqs. (3.3) it follows

[κ2λic +

2σ

ric− ω2τiccic + iω

(cic +

2σ

ricτic

)]T 0,ic −

2σ

ric(1 + iωτic)T 0,os = 0,[

κ2λos +2σ

ric− ω2τoscos + iω

(cos +

2σ

ricτos

)]T 0,os −

2σ

ric(1 + iωτos)T 0,ic = 0,

(3.5)

which has a nonvanishing solution (i.e., T 0,ic = 0 and T 0,os = 0) if, and only if, the determi-

nant of the system (3.5) is equal to zero. This yields the dispersion relation

κ4 − ω2 (Γ1 − iΓ2)κ2 + ω4 (Γ3 + iΓ4) = 0, (3.6)

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where

Γ1 =1

λic

(τiccic −

1

ω2

2σ

ric

)+

1

λos

(τoscos −

1

ω2

2σ

ric

),

Γ2 =1

ω

[1

λic

(cic +

2σ

ricτic

)+

1

λos

(cos +

2σ

ricτos

)],

Γ3 =τicχic

τosχos

− 1

ω2

(1

χic

1

χos+

2σ

ric

τiccic + τoscos + τiccos + τoscicλicλos

),

Γ4 =1

ω3

2σ

ric

(cic + cosλicλos

)− 1

ω

[1

χic

1

χos(τic + τos) +

2σ

ric

τicλic

τosλos

(cic + cos)

],

(3.7)

with χic = λic/cic as the thermal diffusivity of the inner core, and χos = λos/cos as the

thermal diffusivity of the outer shell.

With straightforward calculations, from Eqs. (3.6)-(3.7), one may obtain the following

phase speeds

Up/± =√2

4

√A2

(±) +B2(±)

√√√√1

2+

A(±)

2√

A2(±) +B2

(±)

−1

, (3.8)

being

A(±) = Γ1 ±√∆

√1

2+

Γ21 − Γ2

2 − 4Γ3

2∆,

B(±) = Γ2 ∓√∆

√1

2− Γ2

1 − Γ22 − 4Γ3

2∆,

∆ =

√(Γ21 − Γ2

2 − 4Γ3

)2+ 4 (Γ1Γ2 + 2Γ4)

2.

(3.9)

The results above point out that there will be two kinds of waves, each of them propa-

gating with its own speed. As it is possible to observe, these speeds are influenced by the

characteristic of both inner core and outer shell, and this result is only due to the possibility

of heat exchange between the inner core and the outer shell.

When there is no heat exchange (σ ≡ 0), the set of Eqs. (3.5) splits into two independent

equations, yielding the following dispersion relations

κ2λic + iωcic − ω2τiccic = 0, (3.10a)

κ2λos + iωcos − ω2τoscos = 0, (3.10b)

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which correspond again to two different waves with their own phase speeds, namely,

Up/ic =√χic

4

√τ2ic +

1

ω2

√√√√1

2+

1

2

ωτic√1 + ω2τ2ic

−1

, (3.11a)

Up/os =√χos

(4

√τ2os +

1

ω2

√1

2+

1

2

ωτos√1 + ω2τ2os

)−1

, (3.11b)

but, in contrast with previous situation [Eqs. (3.8)], these speeds are independent of each

other, i.e., each material propagates its own wave at the speed determined by its own material

properties.

In the high-frequency limit (ωτic → ∞ and ωτos → ∞), from Eqs. (3.10) one may re-

cover that the two phase speeds have the form of Eq. (2.8a), i.e., U(hf)p/ic =

√χic/τic and

U(hf)p/os =

√χos/τos. The interaction between the inner core and the outer shell does not

modify this result, since Eqs. (3.7) become

Γ1,∞ =τicχic

+τosχos

,

Γ2,∞ = 0,

Γ3,∞ =τicχic

τosχos

,

Γ4,∞ = 0,

(3.12)

and the aforementioned phase speeds (U(hf)p/ic ≡ U

(hf)p/+, and U

(hf)p/os ≡ U

(hf)p/−) are easily recovered

from Eqs. (3.8)-(3.9). Indeed, in the high-frequency limit (in this case κ is real) the dispersion

relation (3.6) becomes

Γ3,∞U4p − Γ1,∞U2

p + 1 = 0, (3.13)

which leads to U(hf)p/+ =

√χic/τic and U

(hf)p/− =

√χos/τos.

In the low-frequency limit (ωτic ≪ 1 and ωτos ≪ 1), the dispersion relation (3.6) reduces

to

κ2 = ω2Λ1 − iωΛ2, (3.14)

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with Λ1 =

1

λic + λos

(ric2σ

ciccos + τiccic + τoscos + τiccos + cicτos

),

Λ2 =cic + cosλic + λos

.(3.15)

In this case, we will recover only one wave traveling through the core-shell nanowire with

the phase speed

U(lf)p/+ ≡ U

(lf)p/− =

√2ω

Λ2

[1− ω

2

Λ1

Λ2

(1− ω

4

Λ1

Λ2

)]. (3.16)

It is interesting to note that the influence of the radius of the inner core appears explicitly

in Λ1. Thus, in principle, if the properties of the materials of the inner core (τic, cic, λic),

of the outer shell (τos, cos, λos) and the heat exchange parameter σ are known, a detailed

measurement of the phase speed Up as a function of the wave-frequency ω would allow to

obtain the value of the radius ric of the inner core, if it is not known. That way, Eq. (3.16)

could be viewed in the light of exploring the inner structure of the wire. However, in realistic

situations the problem is more complicated, because for very small systems the heat capacity

λ and the relaxation time τ depend on the size of the system itself [21].

It is evident that, since in Λ1 and Λ2 the properties of the internal core and the external

shell enter in a symmetric way, the velocity of propagation would be the same if the materials

of the core and of the shell would be exchanged each other. This is not directly evident without

a detailed calculation. Note that the ratio Λ1/Λ2 has dimensions of time, whose value does

not depend on the thermal conductivities, but only on the specific heats and on the relaxation

times of each material.

One could also wonder which kind of material for the inner-core/outer-shell could be more

appropriate if the material of the outer-shell/inner-core is chosen in order to get predeter-

mined values of the two phase speeds. In other words, one may determine the best couple

of materials the nanowire has to be made in order to have a relatively high (or low) value of

the heat-exchange coefficient σ in Eqs. (3.2). Note that the exchange coefficient σ between

two media 1 and 2 is related to the rate of phonons crossing the interface between 1 and

2; on the other side, the reflection coefficient of the phonons at the surface is related to the

acoustic impedance Z of such media as (Z1 − Z2)2 / (Z1 + Z2)

2. Thus, σ will be higher for

lower reflection coefficient, i.e., for materials having close values of Z [22]. This impedance

is related to the product of the sound speed times the density.

10

Finally, for low frequencies and insulated systems, both materials behave independently

with their own speed of propagation U(lf)p/ic and U

(lf)p/os, namely,

U(lf)p/ic =

√2χicω

[1− τicω

2

(1− τicω

4

)], (3.17a)

U(lf)p/os =

√2χosω

[1− τosω

2

(1− τosω

4

)]. (3.17b)

4 Conclusions

Starting from the results obtained by Jou et al. [18], giving the phase speed and the

attenuation distance for temperature waves along nanowires, in this paper we have studied

the propagation of thermal waves in core-shell nanowires, namely, in those systems composed

of a cylindrical inner core and a concentric cylindrical outer shell. Inner core and outer shall

are made by different materials. In the case that the heat may flow only from the inner

core to outer shell, or conversely, we have obtained the wave speed along the system, which

is a complicated function of the properties of both materials. Moreover, we have seen that

measurement of the speed of heat waves on the external layer could allow, in principle, to

obtain some useful information about the geometrical characteristics of the system as, for

instance, the determination of the radius of the inner core, or the best combination of the

materials the nanowire has to be made in order to have specific properties.

Acknowledgements

D. J. acknowledges the financial support from the Direccion General de Investigacion of

the Spanish Ministry of Education and Science under grant FIS No. 2006-12296-CO2-01,

and the Direccion General de Recerca of the Generalitat of Catalonia under grant No. 2009-

SGR-00164. V. A. C. and A. S. acknowledge the financial support from the University of

Basilicata.

References

[1] Wang, J., Wang, J.-S., Carbon nanotube thermal transport: ballistic to diffusive, Appl.

Phys. Lett., 88 (2006), 111909.

11

[2] Fujii, M., Zhang, X., Xie, H., Ago, H., Takahashi, K., Ikuta, T., Abe H., Shimizu, T.,

Measuring the thermal conductivity of a single carbon nanotube, Phys. Rev. Lett., 95

(2005), 065502.

[3] Cahill, D. C., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J.,

Merlin, R., Phillpot, S. R., Nanoscale thermal transport, J. Appl. Phys., 93 (2003),

793-818.

[4] Chen, G., Ballistic-diffusive heat-conduction equations, Phys. Rev. Lett., 86 (2001),

2297-2300.

[5] Alvarez, F. X., Jou, D., Memory and nonlocal effects in heat transport, Appl. Phys.

Lett., 90 (2007), 083109.

[6] Jou, D., Casas-Vazquez, J., Lebon, G., Grmela, M., A phenomenological scaling ap-

proach for heat transport in nano-systems, Appl. Math. Lett., 18 (2005), 963-967.

[7] Alvarez, F. X., Jou, D., Size and frequency dependence of effective thermal conductivity

in nanosystems, J. Appl. Phys., 103 (2008), 094321.

[8] Jou, D., Casas-Vazquez, J., Lebon, G., Extended Irreversible Thermodynamics, 3rd rev.

ed., Springer Verlag, Berlin, 2001.

[9] Muller, I., Ruggeri, T., Rational Extended Thermodynamics, Springer-Verlag, Berlin,

1998.

[10] Dreyer, W., Struchtrup, H., Heat pulse experiments revisited, Continuum Mech. Ther-

modyn., 5 (1993), 3-50.

[11] Cattaneo, C., Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1948),

83-101.

[12] Grmela, M., Lebon, G., Dauby, P. C., Bousmina, M., Ballistic-diffusive heat conduction

at nanoscale: GENERIC approach, Phys. Lett. A, 339 (2005), 237-245.

[13] Ottinger, C. H., Beyond Equilibrium Thermodynamics, John Wiley and Sons, Hoboken

(New Jersey), 2005.

12

[14] Liboff, R. L., Kinetic Theory (Classical, Quantum, and Relativistic Descriptions), Pren-

tice Hall, Englewood Cliffs, New Jersey, 1990.

[15] Reissland, J. A., The physics of phonons, John Wiley and Sons, London, 1973.

[16] Jabeen, F., Rubini, S., Grillo, V., Felisari, L., Martelli, F., Room temperature lumines-

cent InGaAs/GaAs core-shell nanowires, Appl. Phys Lett., 93, (2008), 083117.

[17] Motayed, A., Davydov, A. V., GaN-nanowire/amorphous-Si core-shell heterojunction

diodes, Appl. Phys. Lett., 93, (2008) 193102.

[18] Jou, D., Cimmelli, V. A., Sellitto, A., Nonequilibrium temperatures and

second-sound propagation along nanowires and thin layers, Phys. Lett. A,

doi:10.1016/j.physleta.2009.09.060.

[19] Joseph, D. D., Preziosi, L., Heat Waves, Rev. Mod. Phys., 61 (1989), 41-73, Addendum,

Rev. Mod. Phys., 62 (1990), 375-391.

[20] Cimmelli, V. A., Frischmuth, K., Nonlinear effects in thermal wave propagation near

zero absolute temperature, Physica B, 355, (2005), 147-157.

[21] Alvarez, F. X., Jou, D., Sellitto, A., Phonon hydrodynamics and phonon-boundary scat-

tering in nanosystems, J. Appl. Phys., 105, (2009) 014317.

[22] Zhang, Z. M., Nano/Microscale heat transfer, McGraw-Hill, New York, 2007.

13