GENERATION OF HIGH FREQUENCY PHONONSPHONON GENERATION BY JOULE HEATING IN METAL FILMS

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GENERATION OF HIGH FREQUENCY

PHONONSPHONON GENERATION BY JOULE HEATING IN METAL FILMS

N. Perrin, H. Budd

To cite this version:

N. Perrin, H. Budd. GENERATION OF HIGH FREQUENCY PHONONSPHONON GENERATION

BY JOULE HEATING IN METAL FILMS. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-33-

C4-39. �10.1051/jphyscol:1972408�. �jpa-00215085�

PHONON GENERATION BY JOULE HEATING IN METAL FILMS

N. PERRIN and H. BUDD

Groupe de Physique des Solides de 1'Ecole Normale SupCrieure Tour 23, 11, quai Saint-Bernard, Paris 5e- France

Rbume. -

Nous Btudions la dependance temporelle et 1'6tat stationnaire du systbme couple dY6lectrons et de phonons dans un

mCtallique mince qui genere, par effet Joule, des phonons de haute frkquence. Nous tenons compte du problbme de I'interface en introduisant un temps de relaxation phenomCnologique comprenant un parametre de desadaptation acoustique v. Nous montrons que la forme detaillee de la distribution de phonons n'est pas une distribution de Bose.

En dkfinissant une temperature de phonon dans chaque mode Tp(q), nous voyons que cette derniere differe de f a ~ o n appreciable de la tempbature electronique Te caracterisant la distribution de Fermi-Dirac. Nous montrons que les electrons voient pratiquement une distribution de phonons

l'equilibre

la temperature ambiante

OK). Nous comparons notre modkle au modele

une tempkrature unique, dans la limite de I'adaptation parfaite.

Abstract.

- We study both the time dependence and steady state of the coupled electron- phonon system

a thin metallic film, which provides a source of high energy phonons generated by Joule heating. The interface problem is included through a phenomenological relaxation time involving an acoustic mismatch parameter

The detailed form of the phonon distribution is shown to be a non Bose like distribution.

phonon temperature TP(q) defined in each mode is seen to deviate significantly from the electron temperature Te characterizing the Ferrni-Dirac distribution. We show that the electrons see an essentially equilibrium phonon distribution at the ambient temperature

OK). We compare our model with the

one temperature model

in the black body limit.

I. Introduction. - One of the most commonly usual configuration of the emitter (Fig.

Here a employed technique for the generation of high fre- metallic film, generally in the 500-1 000 A thickness quency phonons is that of Joule heating [I]. The range, is evaporated onto a substrate. The characte- electrical power dissipated in passing a current through

some conducting body is converted into heat, thus providing a source of high frequency phonons. The detailed nature of this conversion process has received relatively little attention since it requires a complicated analysis of the electron-phonon system as well as a model for the reflection and transmission of phonons across the interfaces of the emitter, i. e. the phonon source. The interfacial problem has been studied in considerable detail [2]-[6], particularly in connection with the Kapitza resistance arising at a liquid helium- solid interface [7], [8].

Most studies of phonon generation by Joule heating

are based on the simplifying assumption that the

-

configuration

phonon emitter.

phonons are characterized by a Bose distribution a t the same temperature as the electron distribution [I], [4]-[6], [9], [SO], the latter being a Fermi-Dirac distri- bution. The common temperature which characterizes the electron-phonon system is then the only parameter which one needs to determine, and this is accomplished by a simple energy balance equation. This sort of analysis is essentially the strong electron phonon coupling case, since the electrons and phonons are assumed to be in almost perfect internal equilibrium.

A qualitative estimate of the validity of such a model can be readily deduced by considering the

,,I

ristic time for the relaxation of the metallic film phonons towards the substrate temperature is of the order of z,

where

u an appropriately averaged sound velocity, and where

is a numerical factor which measures the acoustic mis- match between the metal film and substrate. It is typically of the order of one for a solid-solid contact, but can be quite large for a solid-helium interface [IS].

This time is to be compared with z,,, the phonon- electron relaxation time, i. e. the characteristic time for the phonon distribution to relax towards equili-

. ₌ 0

- _E E .=

* L

,d,

Substrate

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972408

C4-34 N. PERRIN AND H. BUDD

brium with the electrons. One would expect a quasi-

equilibrium between electrons and phonons, only if the latter have a zep

On the other hand those phonons for which zep >> zb would be expected to be essentially pegged at the substrate (ambient) tempe- rature. Since zeil cc

[12],

the phonon frequency, one would expect the low frequency modes to remain essentially at the ambient temperature and the high frequency modes to be essentially in equilibrium with the electrons.

The model calculation which we present

in sec- tions I1 and I11 confirms this picture and in section 1V we compare our results with those obtained using a Bose distribution to describe the non-equilibrium phonon distribution.

11. Model calculation.

-

phonon emission by Joule heating in the structure shown in figure 1 requires the solution of the coupled electron-phonon transport equations with the appro- priate boundary conditions a t the interface. In order to avoid solving this complicated system, we take the electron and phonon distributions spatially uniform and replace the boundary condition by a phenomeno- logical description of the energy exchange between the phonons and the substrate. We therefore write

where &(To) is a Bose distribution at the ambient temperature To and zb is the relaxation time discussed in the introduction (7,

We take the electron distribution to be a Fermi- Dirac distribution at temperature T,, and neglect the slight anisotropy induced by the current flow. In this case, it is straightforward to calculate the rate of change of the phonon distribution due to the electrons

1

-

- -

-- rep 2 ti3

where E, is the deformation potential constant, m is the electron mass,

is the mass density of the metal, and

is the frequency of phonons of wave vector q.

In deriving this result, spherical constant energy surfaces have been assumed as well as a simple defor- mation potential constant. Umklapp processes as well as interactions with transverse phonons have been neglected. These are beyond the scope of the model calculation presented here. A Debye model is assumed for the phonons.

The evolution of the phonon distribution is therefore given by

The electron temperature Te is determined from the energy balance equation

where

is the electronic specific heat,

the resistivity and E the electric field. This expression simply expresses the rate of increase of the electron energy as the diffe- rence between the electrical power input

rate of energy transfer to the phonons PN.

The resistivity

is given by a residual contri- bution

and a phonon contribution

contribution which we calculate as follows. The pre- sence of an electric field induces a slight anisotropy in the electron distribution which becomes a Fermi- Dirac distribution displaced by a small drift velocity vd

The phonon generation rate corresponding to this distribution is given by

where the starred quantities are the same as in eq. (2) and

frequency o -

For small

we therefore have

and the rate of momentum transfer to the phonon system is

Equating this to the electrical force acting on the electrons we obtain the phonon contribution to the resistivity

aN, - zq(K) - N,

@,(TO)

Nq

- -

The first term is identical to that obtained by Koehler's

at z e ~

zb variational method

while the second re~resents

(*I A preliminary version of this work has been published

the contribution arising-from the lack of equilibrium

in the Phys. Rev. Lett., 1972, 28, 1701.

between the electrons and phonons.

111. Results.

-

TIME

We have solved eq. (4) and (5) on an IBM 370.165 in order to obtain the evolution of both the electron tempera- ture T, and the phonon distribution Nq for a typical monovalent metal film. The following constants were used

n

6 x 10" m-3,P,

5

lo-' ohm.m,

5 x

m.s-', film thickness

A

(2,

2 x 10-l1 s), mass density

19.32 g . ~ m - ~ .

small time interval At is considered, during which Nq and Te undergo small variations given by eq. (4) and (5). Having thus determined the values of 6Nq and 6Te and therefore the values of N, and Te at the time t + At, we continue to the next time interval, etc ...

Some of the results are shown in figures 2-4. In figure 2 is shown the time dependence of the electron

T I M E t ( I O - ~ S )

FIG. 2. - Electron temperature Te(t) vs time for different energy fluxes P. The dashed Iine represents the electron temperature calculated under the assun~ption of an equilibrium phonon

distribution at the substrate temperature To.

temperature for different input powers P,, along with the steady state value of the electron temperature T: . It is seen that in the first time intervals following the application of the electric field to the film, dTe/dt increases with the input power P, and that the time required to reach the steady state is a decreasing function of Pe. It is readily seen from eq. (5) that the initial rate of energy transfer to the phonons P, is zero. It remains quite small for short times, so that at least initially, the values of dT,/dt calculated from the curves in figure 2 are in good agreement with the values obtained from

which yields

,This expression gives a linear variation of T, with

time and with Pe for sufficiently small t. Figure 3 shows the ratio of the rate of energy transfer P, to the phonons to the input power P, as a function of

FIG. 3. - Ratio of the rate of energy transfer PN to the phonons to the input power Pe = aE2 vs time for different energy fluxes

P

-

time. The necessary time for this ratio to reach the steady state value of 1 is larger than that required for Te to reach T:. However neglecting P, in the equation for the evolution of T, is a rough approximation which is no longer valid for long times particularly in the case of large input power. By solving eq. (4) exactly and looking at the integro-differential equation for Te(t) obtained from eq. (5)

with

We see that the term proportional to z*/T,,, which is small for most values of q involved in the sum, can be neglected in a first approximation. The evolution of T, is then approximately given by

The electrons see an essentially equilibrium phonon

distribution at the substrate temperature

of T, calculated from eq. (15) have been compared

with the exact values deduced from the coupled

C4-36 N. PERRIN AND H. BUDD

eq. (4) and (5) for 3 of the input powers presented in

figure 2 (P,= 176.7 x lo7 W . ~ m - ~ , 512 x lo5 W . ~ m - ~ and lo6 W . C ~ - ~ ) . They do not differ significantly.

The largest differences occur in the region where T, approaches T: for large input power. Even in this region, the error is only about a few per cent.

In figure 4 is shown the evolution of the energy

PHONON WAVE NUMBER q (10'm-')

FIG. 4. - Energy density spectrum q 3 N g vs phonon wavc number at different times. The lower curve is _- he thermal equi-

librium energy density spectrum q 3 N q (To 4.2 OK).

density spectrum of the phonons for an electric field E

16 V.cm-'

512 x lo5 W . c n ~ - ~ ) . It is seen that the low frequency part of the spectrum remains unchanged by the application of the electric field while the energy localized in the high frequency part of the spectrum increases significantly with increas- ing time and therefore with increasing

It is essen- tially the high frequency modes that are strongly perturbed by the Joule heating.

The electron temperature

following energy balance equation

It is readily seen from eq. (3) and (17) that T: is an increasing function of the input power density.

Eq. (17) also shows that the approximations made above in the evolution of T, give better agreement in the steady state for Iow input power than for high one, as we found before. Indeed, this approximation yields the following energy balance equation in the steady state,

which differs from eq. (17) through the term

+ ^(rb/zep)

in the denominator of the sum, withrb/2,,

When

increases, the term on the right hand side of eq. (IS), P,, involves larger q for which zb/zeP is larger.

Then, neglecting this term in the steady state remains a very good approximation for not too high input power. The exact values of T, as deduced from eq. (17) are shown in figure 5 for different input power densities.

PHONON WAVE N U M B E R q (10' m-' )

FIG 5. - Phonon temperature TD(q) vs wave number q for dinerent energy fluxes and the corresponding electron rempe- iature. The dashed lines represent the corresponding electron and phonon tcmperature in the one temperature model

(PI = 10-1oRm).

B. THE STATIONARY STATE. -

The steady state

phonon distribution is given directly by eq. (4)

This may be compared to the phonon tempera- ture Tp(q) defined in each mode by

Zb

+ %(TO)

q Tep

+

Nq ⁼

( ^exp (

^{1 ) - I}

For sufficiently low frequencies reP/rb > N,(T,)/~(T,)

and Nq

%(TO) while for sufficiently high frequencies and shown in figure 5. T,(q) is close to T, for high

z, >

and

%(T,). The low frequency modes frequency modes but is far below T, for the low fre-

are relatively weakly perturbed by the Joule heating, quency ones. This can be understood by noting that

while the high frequency modes tend to be in quasi- the electron phonon interaction is very weak in this

equilibrium with the electrons. frequency range (r,,

q -

and therefore the phonons

interact essentially with the substrate through

In any case, a frequency independent phonon tempera- ture

(i. e. a Bose approximation) cannot realistically be defined to represent the non-equilibrium phonon distribution.

We present in figure 6 the phonon density spec- trum

Hq for different input powers up to

P H O N O N WAVE N U M B E R q ( l o 8 m-'1

FIG. 6. - Steady state energy density spectrum ^{q 3} Nq vs phonon wave number q for different energy fluxes. The lower curve is the energy density spectrum of the equilibrium Bose distribution

(at 4.2 O K ) .

(which corresponds to a power density per unit of area P

200 W .cm-'). These curves clearly show the non Bose like form of the phonon energy distri- butions, as well as the preferential heating of the high frequency modes.

IV.

In most studies of phonon generation by Joule heating and of thermal contact resistance, a common temperature T : is assumed to characterize the electron phonon system in the metal film, the phonon distribution being a Bose distribution at T!.

Under these assumptions along with the assumption of a perfect match between the heater and the substrate (blackbody limit) referred to as the one temperature (OT) model, the results obtained are compared with ours. In figures 7 and 8 we present the energy density spectrum

N, for two different input powers PC (which correspond to a power density per unit of area of 50 and 400 W .cm-' respectively).

In both cases, our results are drastically different from the Bose distribution in the OT model. The latter grossly overestimates the energy density in the low frequency modes. In figure 9, we compare the power dependence of the electron temperature as calculated in the one temperature model with our results. The lower curve refers to the OT model while the two upper curves refer to our results for

P H O N O N W A V E N U M B E R cl ( l o a m - ' )

FIG. 7. -Steady state energy density spectrum q 3 Nq in our model and in the one temperature model for a n energy flux

P = 50 W. cm-2 (pr = 10-1 0 Rm).

P H O N O N W A V E N U M B E R q ( 1 0 8 m - ' )

FIG. 8. - Steady state energy density spectrum q 3 N, in our model and in the one temperature model for an energy flux

P = 400 W.cm-2 fpr = 10-10 Om).

90 -

-

L 8 0 -

Tp =Te {X black body limit

-

W

5

t

(r 50 - a

4 0 - Z 0 3 0 -

LT k-

Y

2 W

I

,

10 l o 2 lo3 lo4 lo5

FIG. 9. - Electron temperature vs power density P in the one temperature model and in our model with two acoustic mismatch

parameters, q = 1 and q

-

C4-38 N. PERRIN A N D H. BUDD

two different values of z, (z,

2 x

s and

7, =

8 x lo-" s).

The ratio of the input powers required to obtain the same electron temperature in the OT model and in our model is of the order of five and tends t o unity at high power densities. This can be understood from eq. (17) and from the similar equation in the OT model

The right hand side of eq. (20) is the limit of the right hand side of eq. (17) with z,,

0. It is the case of a strong electron phonon coupling. The ratio of the denominators in the sum of eq. (17) and (20) is a decreasing function of q

Therefore the contribution of the term ze,/z, to the sum in eq. (17) is large for low frequency modes and the differences between eq. (17) and (20) are large when the maximum of the integrand occurs for small q, which is the case of small T,.

In comparing our results for the two different values of z, (see Fig. 9 and lo), we note that the curves are

POWER DENSITY P ( w c f 2 )

FIG. 10. - Ratio of the electrical conductivity G to the residual resistivity GI in the one temperature model and in our model with two acoustic mismatch parameters, ^q = 1 and = 4

(PI = 10-10 Qm).

essentially the same at low input power and begin to show non-negligible differences only at the highest power levels. This is to be expected since it is only for the high frequency modes that z,, =

whereas for most of the modes z,,

It is therefore only at high temperatures or power levels that the details of zb are important.

We see in figure 9 that the electron temperature obtained for a given input power are not strongly different in the OT model and in our model, even with different acoustic mismatches. This can also be seen in figure 5 where the electron temperature in the OT model is represented by the dashed lines along with the phonon and electron temperatures in our model. Similarly the ratio of the electrical conducti- vity a to the residual conductivity

is not signifi- cantly different in the OT model and in our calcu- lations as is shown in figure 10. The largest differences are of the order of 15 %. We see that the resistivity depends weakly on the form of the phonon distribu- tions which are very different as can be seen in figures 7 and 8. These distributions appear in the expression for z, through a n integral which averages them.

V.

In summary, a single temperature cannot be used to characterize both the electron and phonon systems. This result agrees with some studies of heat exchangers in dilution refrigerators [14].

In the evolution of the coupled electron-phonon system, the electrons can be considered to interact with an almost equilibrium phonon distribution a t the substrate temperature To. The characteristic time of the evolution of the electron distribution depends strongly on the input power. I t is about 10-lo seconds for relatively moderate input powers and could possibly be measured.

The phonon distribution is a distinctly non Bose type distribution function. Its form depends strongly on the electron distribution through Te as well as on the film thickness and acoustic mismatch parameter

Since the resistivity is only weakly dependent on the form of the phonon distribution, it does not allow one to easily distinguish between the different theories.

Bose distributions as well as our distributions which are very different, also give approximately the same values of the resistivity.

The weak heating of the low frequency modes emerging from our calculations probably explains the power independence of some results of heat pulses experiments in semiconductors [15],

where low frequency phonons are detected after ballistic propa- gation. The high frequency modes are usually strongly attenuated by isotope scattering and are not detected.

In thermal contact resistance studies, the Bose approximation and our model would result in diffe- rences certainly larger than those arising from diffe- rent interface models.

Our model is a simple one. A more realistic calcu-

lation of the coupled electron phonon system requires

taking account of the spatial dependence of the distri-

bution functions along with the dispersion in the

high frequency modes. The same qualitative results

would probably be obtained. Still further, the electron-

electron interaction might be introduced instead of

the assumption of the electron temperature.

References

[I

1

GUTFELD

Physical Acoustics (Academic

NARAYANAMURTI (V.),

Press, New York,

Vol.

GUREVICH

and GASYMOV

[2]

LITTLE (W. A.),

37,

(English transl.

[3]

HERTH (P.) and Wers (O.),

9,

29,

ZIMAN

1,

and corri-

[4]

WEIS

gendum.

[5]

WIGMORE (J.

ZIMAN

Electrons and Phonons (Clarcndon Press, Oxford, England,

f6] ANDERsoN

(R.

(A.

ROUBEAU (P.), LE FUR

and VAROQUAUX

J.

in Third Intern. Cryogenic Engineering Conf.

[7] KHALATNIKOV (J. M.), Zh. Eksp. i Teov. Fiz., 1952,

(Iliffe Science and Technology Publishers, Guil-

22, 687.

ford,

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CHEEKE

Colloque, SUPPI.

MANEVAL

P.), ZYLBERSZTEJN

and HUET

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31,

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KWOK

C.),

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ISHIGURO

DISCUSSION

R. J. VON GUTFELD. - T O what extent, if any,

have you considered the importance of Umklapp processes which would appear to be very important in obtaining equilibrium between electrons and phonons

H. F. BUDD. -

Umklapp processes as well as other band structure related effects are not within the scope of our model calculation. They would obviously increase the electron phonon coupling but would never be sufficiently strong so as to compete with the relaxation of the low frequency phonons towards the substrate temperature.

R. ORBACH. - HOW important are phonon-phonon

action itself becomes increasingly effective in esta- blishing internal equilibrium between electrons and phonons since the relaxation time varies inversely with frequency.

J. CHEEKE.

Experimentally we find that the radiation temperature of thin film metal phonon radiators can be quite happily accounted for using a Bose-Einstein distribution. In view of your calcula- tions do you think that this could be due to

a high defect concentration in the films which therma- lise the phonons or (b) the relative insensitivity of the results to the exact form of the phonon distribution function

interactions in restoring equilibrium between low and

high energy phonons

H.

- Our results show quite clearly that

the resistivity, which you measure. is auite insensitive

F.

- At low power input levels the to the form-of the phonon distributioi. This kind of

phonon-phonon interaction is unimportant compared experiment doesn't therefore seem very useful in

t o the electron-phonon interaction or to the phonon- distinguishing the factors determining the latter. In

substrate interaction. At sufficiently high power levels any event one would require enormously high defect

these effects presumably becomeimportant and should densities to prevent the low frequency modes from

be included. Here however the electron-phonon inter- escaping ballistically from the film.