Change of Electronic Properties Due to Ultrashort Laser Pulses

Change of electronic properties due to ultrashort laser pulses

Bernd Hüttner 1 CPhys FInstP DLR-Institute of Technical Physics, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany

Abstract

We begin with a short discussion of existing approaches to the description of nonequilibrium processes and the possibility of how to overcome the problem of the definition of temperature in this case. Next, we explain why the equations of fluxes and conductivities as derived in standard solid state theory have to be reconsidered. Starting from the Boltzmann equation then we derive expressions for the thermal and electrical conductivity with new qualitative and quantitative properties. These results are supported by comparison with experiments.

Keywords: nonequilibrium, temperature definition, thermal and electrical conductivity, diffusivity, metals, reflectivity

1. Introduction – temperature concept

The interaction of short laser pulses with matter is obviously a nonequilibrium process and, therefore, the standard thermodynamics, one should better say thermostatics, is not suitable for any description. Nonequilibrium thermodynamics, however, inherently has two quite complicated conceptual problems: the definition of entropy and temperature beyond equilibrium. Consequently, we are faced with the difficulty that up to this very day no generally accepted definition of temperature exists for situations of nonequilibrium. Several approaches have been developed in the past to deal with this matter. There is no space here to go into details because the literature is abundant about this topic. We will focus here only on one method, the projection operator formalism1,2,3 together with a hierarchy of relaxation times4, because we can take this as a more sophisticated but also more complicated basis for tackling the problem. Such a scheme is usually called the nonequilibrium statistical operator method5. The main point is that the system is divided into a “relevant” part containing mechanisms with very short relaxation times and in a “perturbed” part related to mechanisms with longer relaxation times. This way, a quasitemperature can be introduced for the nonequilibrium system after a short initial period of time. Besides of nonequilibrium, short pulse laser interaction is characterized by a second important effect: non- stationarity. Both peculiarities are in strong contrast to the basic assumptions of standard solid state physics 6, namely, local thermal equilibrium, i.e., electrons and phonons have one common temperature, and a steady state. For this reason, we have to reconsider the equations for the electrical and thermal currents and the corresponding conductivities. In the following we derive the change of the equations for some few properties and compare the results with experimental data. A more detailed treatment is given in7.

2. Thermal conductivity

Although there are extensive compilations of data for the thermal conductivity8, they are of limited value for the description of short laser pulse experiments because the experiments were carried out under steady state conditions where the electrons and phonons are described by a single common temperature. In metals, the contribution of the electrons to the thermal conductivity is about two orders higher than that of the phonon part. Therefore, we restrict us in the following on the former one. In the semi classical theory of conduction in metals the electronic thermal conductivity is defined by

   f FD l =轾E(k) - m (T) v(k)2 t 轾 E k k  臌臌 ( ) k T

1 [email protected]; phone +49 711 6862 375; fax +49 711 6862 348 where we have assumed without loss of generality that the heat is only carried by the electrons in a single band. Four points are remarkable in equation : 1. The Fermi-Dirac-distribution indicates that we are very close to equilibrium. 2. The temperature T is the same for electrons and phonons at all places and at all times.

3.: The chemical potential µ (T) is usually taken as the Fermi energy µ0 at T=0K because the first correction is propor- 2 tional to (kBT/µ0) . 4. In pure systems the scattering time  is dominated by the electron-phonon scattering since the electron-electron scattering is hindered by the exclusion principle.

For the description of short pulse laser interaction with metals we have to revise all four points: 1. We need a nonequilibrium distribution function in view of the fact that the system may be far from equilibrium. 2. The electrons and phonons must be characterized by different temperatures where the electronic one can even exceed the critical temperature of the system. 3. Due to this possibility the temperature correction to the Fermi energy can be not insignificant. 4. The electron-electron scattering must be taken into account while its phase space is enhanced. Therefore, it can even become the primary process. The first task, the determination of the nonequilibrium distribution function, can be solved by a perturbation expansion of the Boltzmann equation    抖f( k, t)  f( k, t)   f( k, t)       +v - e v E = P(k, k ', t)轾 f k ', t - f k, t d k ' - n G f k, t 抖t r E 臌( ) ( ) ( ( )) up to the second order where all terms, except the last one, keep their usual meaning. This additional term represents the phonon-assisted absorption or emission of photons and is discussed in more details inError: Reference source not found. The expansion reads formally

m n 2 f= p fn = f o + p f 1 + p f 2 + ... n= 0

where f0 is the Fermi-Dirac distribution and the small parameter, p = · , is defined by

I( t) t p( t) = u( t) t = n hwd where I(t) is the laser intensity,  the scattering time, n the electron density,  the laser frequency, and  the optical absorption depth. We skip the evaluation; this was done in Error: Reference source not found; and we give here only the first and second order results

 t2 t 2 骣  -i w t - - et v E2 讯 T骣 f 2 p f= -琪 o e2 tL +( E -m) v t e琪 - o + p e t L G( f ) 1琪1- i wt T E o o 桫 e 桫

t '2 t t ' 骣 - - t 娑(p f1) 鲦   饿( p f 1 ) 2 2 t t 琪 tL p f2= - e d t 'e v� T e琪 - e v E 琪 n o G e( p f 1 ) . - 琪琪抖T 琪 E 桫桫桫

where Te is the electron (quasi)temperature. Consequently, the short initial time is identified by the time needed for the establishment of the electron temperature. This time may be quite short due to the strong electron-electron interaction as discussed below. The temperature dependence of the chemical potential is found with a Sommerfeld expansion up to the 2nd order to 22 2 4 4 4 骣 pkB T e7 p k B T e m(Te ) =m o 琪 1 -2 - 4 . 桫 12mo 360 m o

By means of the Fermi liquid theory one obtains for the electron-electron scattering time

-1 - 1 - 1 2 2 2 2 te- e = t T + t E =4 p b k B T e +b( E - m) where  can be taken from the experiment9 or calculated by means of an appropriate theoretical expression10,11. After insertion of equations to into equation and integration over the k-space we get for the thermal conductivity in first order

T骣 1 骣p2 T 2 7 p T 4 l = le琪 G(T ) -琪 e + e 1 LTE琪 e G(T ) 24m2 240 m 4 Tph( 1+ z( T e ,T ph )) 桫 e 桫 0 0 and in second order

骣 hw 琪 G( Te ) + m 骣 z w 7p4 z w k 2 T 2 琪 0 N T ,T ,w +h +h B e + 琪 3琪 ( e ph ) 3 N T ,T ,w 桫 12m0 120m0 琪 ( e ph ) 琪 w 琪 G( T ) - h 琪 e 4 2 2 m0 骣 z hw 7p zh w kB T e 琪 3琪N( Te ,T ph ,w) - - 3 N T ,T ,w 桫 12m0 120m0 Te 琪 ( e ph ) l2 = l LTE2p 0( T ph , w ,t)琪 Tph 琪骣琪琪 p2 1 1 骣k2 T 2 7p2 k 4 T 4 琪-琪 + B e + B e 2琪琪 2 4 琪 N T ,T ,w hw h w 桫24m0 120m0 ( e ph ) 琪 G( T) + G( T ) - 琪琪 em e m 琪桫 0 0 琪 1 骣 w p2k 2 T 2 7 p 4 k 4 T 4 琪 h B e B e -2琪 + 2 + 4 琪 G T 1+ z T ,T 桫2m0 48m0 480 m 0 桫 ( e)( ( e ph ))

where the function z(Te, Tph) is defined by the ratio of the electron-phonon scattering time over the temperature dependent part of the electron-electron scattering time

t T ph( ph ) 2 2 z( Te ,T ph) = = 4 p b T e (eV) t ph( T ph ) . tT(T e )

The function G(Te) is specified in equation by the expression in the parentheses. The abbreviations N(Te, Tph, ) and p0(Tph, , t) have the following meaning. 2 2 z( Te ,T ph )h w N( Te ,T ph ,w) = 1 + z( T e ,T ph ) + 2 2 2 . 4p kB T e (t - t )2 - L It (T ) 2 o ph ph tL p0( T ph ,w ,t) = e nhwd

The first term in equations and , respectively, corresponds to the standard expression for the electronic thermal conductivity as predicted by the Wiedemann-Franz law. The subscript “LTE” indicates that the conductivity LTE is related to the case of local thermal equilibrium, i.e., Te=Tph=T. It is worthwhile noting that the second order term possesses some peculiarities: It is explicitly depending on the laser properties (I0, , L) and on time. For this reason, a universal plot of the thermal conductivity as a simple function of temperature is not possible anymore. As an example, the values of 2 in figure 1 are calculated for the fluence 2 F=50mJ/cm , L=500fs, and =1eV. The abscissa in figure 1 has a double meaning: For the case of local thermal equilibrium, the three shorter curves runs up to the melting point and T represents the common temperature, Te=Tph=T, of the system. For the remaining curves T corresponds to the electron temperature, T=T e, because we have chosen a fixed Tph=300K for a better presentation. Additionally, we have incorporated the often used linear approximation of the electronic thermal conductivity.

14  =  ·T /T 0 e 0 ]

K 12 m c / 10  + W 1 2 [

y t i v

i 8 t c u

d 6 n o c

l Wiedemann-Franz

a 4 m r  + but with T =T e  1 2 e ph 2

0 1000 2000 3000 4000 5000 6000 7000 T (K)

Figure 1: Thermal conductivity of gold for the case of nonlocal thermal equilibrium at fixed Tph=300K upper solid

curve: 1+2, dotted curve: linear approximation, dashed-dotted curve: 2, and for the local thermal equilibrium

Te=Tph=T: lower solid curve 1, dotted curve: LTE, o experimental data taken from Error: Reference source not found; for the laser parameter used cf. text Furthermore, it is remarkable that the incorporation of the electron-electron scattering improves the agreement between theory and experiment even in the case of local thermal equilibrium. But, we have to add that due to the explicit time dependence of 2 the upper full curve and the dash-dotted curve in figure 1 are only a snapshots, however representative ones, taken at t=L. This aspect is more thoroughly explored in Error: Reference source not found.

Although the linear approximation results from in the limit z<<1 and kB·Te<<0 it has to be used with caution since, at least for gold, the true behavior is completely different for temperatures above 2000K. Instead of linearly increasing the thermal conductivity decreases roughly inversely proportional to Te. Closely related to this point is the behavior of the electronic thermal diffusivity, the ratio of the thermal conductivity over the electronic specific heat, a e. In the linear approximation ae is almost constant because both the electronic specific heat and the thermal conductivity are proportional to Te. In contrast to this it follows from equation

le 1 lLTE anl = @ c轾1+ b·T2 g T e臌 e e ph

2 with b=4 ph. As long as b is much smaller than unity the nonlinear diffusivity corresponds to the standard expression but if the electron temperature becomes high enough anl starts to decrease. Such a changed behavior must certainly have consequences for the calculation of the electron and phonon temperature distribution in metals. Since the thermal diffusivity and the transport of heat inside the metal are reduced at higher temperatures one anticipates a stronger increase of the temperatures near the surface. For example, b is approximately 8.4 for gold at Te=6000K. This way, the energy transport into the material is hindered leading to a further rise of the temperature in the surface region. A direct measurement of this effect is not possible but we can make an indirect check utilizing an experiment performed by Wang et al. (●) 12. The authors have measured the maximum of the electron temperature at the surface.

Electron temperature

Au 500 nm nonlinear, I + I 1 2 6000 linear, I + I 1 2 nonlinear, I 1 Wang et al. PRB 50 (1994) 8016  t = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1 ps 5000 2 I = 23 GW/cm 1 I = 21 GW/cm2 2 )  = 120 fs

a 4000 m T

2000 0,0 0,5 1,0 1,5 2,0 t (ps)

Figure 2: Maximum electron temperature of gold as a function of the delay time, experimental (●) and calculated behavior; for details see text. The gold sample has been irradiated with two almost identical pulses at different delays Δt as indicated in figure 2. The curves were calculated by means of the extended two temperature model (ETTM) 13, which is based on the hyperbolic differential equation, with  in the linear approximation (dashed curve) and with the above nonlinear expression (dotted and full curves). The horizontal curve corresponds to the maximal electron temperature available with the single laser pulse I1. For two delayed pulses this means that the temperature produced by the first pulse is completely relaxed before the second pulse arrives. The agreement with the theory (dotted curve) is perfect. The same is true for the zero delay (full curve). Taking, however, the linear relation (dashed curve), the electron temperature stays below the maximum because the transport into the metal is to fast. It should be mentioned that the two temperature model, which is based on the parabolic differential equation, gives too low values even if we take the nonlinear conductivity. The physical origin is lying in the prediction that a temperature gradient is build up instantaneously. In the ETTM, however, there is a delay, which is governed by Allen’s14 temperature relaxation time. In the next section we give a brief overview about the change of the electrical and optical properties.

3. Electrical conductivity – optical properties

Up to second order the electrical current is represented by    j= s E = - e v f + pf + p2 f e臌轾 0 1 2

Restricting ourselves to the first order term and taking into account that odd powers of the velocity vanish we obtain from equation for the electrical conductivity

t2 - 2 2 v t 骣 f s w,T ,T ,t = e2 e 2 tL - 0 ( e ph )  琪 k (1- i wt) 桫 E where we have neglected the thermoelectric contribution because it contributes only small correction as shown in 15. A much more detailed treatment including the second order contributions was published in16 . After integration we obtain for the complex electrical conductivity

骣骣 z p2k 2 T 2 z Gw2 t 2 琪 B e D 琪(1+ z) - -2 ( 1 + z) + t2 琪桫 1224m0 6 - 2 tL 琪 s( w,Te ,T ph , t) = s D G e 琪琪骣 z p2k 2 T 2 z Gw2 t 2 琪+i wt琪 1 - -B e + D 琪 D 琪 2 桫桫 6( 1+ z) 24m0 6( 1 + z) with the new abbreviation

1 G = G( w,Te , T ph ) = 2 2 2 . (1+z( Te , T ph)) + w t D

The interested reader can find details again inError: Reference source not found orError: Reference source not found. For a cw-laser the Drude expression follows from equation in the limit z<<1. Although in the static case the corrections to the electrical conductivity are comparable to the thermal one their quantitative contribution is reduced for >0 2 2 through the additional term  D . Nevertheless, we shall see that the optical properties react quite sensible even to small corrections. They can be determined using the familiar relation between the frequency dependent conductivity and the dielectric function 4p e( w,Te ,T ph) = e 0 + i轾 s 1( w ,T e ,T ph) + s 2( w ,T e ,T ph ) w 臌 from where the reflectivity, e.g., follows by

e( w,Te ,T ph ) - 1 R(w ,Te ,T ph ) = . e( w,Te ,T ph ) + 1

Figure 3 shows a comparison between the measured relative change of the reflectivity17 and the theoretical calculation. The measurements of reflectivity were carried out in a pump probe experiment at incident angles of 43° for the pump 2 and 48° for the probe pulse on a 20 nm thick sample of gold. The laser was p-polarized with I=10GW/cm ,  L=120fs, and =2 eV. Taking these experimental inputs, we have calculated the electron and phonon temperatures with the

ETTM. These values were inserted into equation with (, Te, Tph) given by expression . Additionally, the normalized electron temperature is plotted in figure 3 to illustrate the physical origin of the damped wavy structure of the reflectivity. In the ETTM an electron temperature wave is running through the film and is reflected back and forth from the rear side. This gives rise to the undulatory appearance.

Relative change of reflectivity

1,0 Au d=20nm I =12GW/cm2 0,8  =100fs L d e z i l

a 0,6 m r

o n ( 0,4 experiment R / theory R

 electron temperatue 0,2

0,0 0 1 2 3 4 5 t (ps)

Figure 3: Normalized transient thermal reflectivity of gold as a function of time. The experiment is taken from Error: Refer- ence source not found. See text for the details of the theoretical calculations.

The slower decrease of the reflectivity signal compared to the temperature results from the growth of the phonon temperature with elapsed time leading to a complicated balance of contributions which are either more dominated by the electron or more by the phonon temperature. This point is discussed in greater detail inError: Reference source not found (see table 1 there).

4. Conclusions

In this paper, methods for the description of nonequilibrium processes are briefly sketched. Furthermore, we have given specific reasons for the necessity of reconsidering the equations for the thermal and electrical fluxes and the conductivities related to them. It is shown that two points are mainly responsible for the appearance of new phenomena: the local thermal nonequilibrium expressed by different temperatures in the electron and phonon system and the excitation of the electrons far above the Fermi energy. The latter opens up a large phase space for electron-electron scattering and makes this process, therefore, much more important and sometimes even dominant compared to its contribution in the case of local thermal equilibrium. Based on a perturbation expansion of the Boltzmann equation, we derived new expressions for the thermal and electrical conductivity. For example, the thermal conductivity becomes a function of the laser properties (frequency, pulse duration, intensity) in second order. In first order it is a strong nonlinear function of the electron temperature. Accordingly, the thermal diffusivity and with it the energy transport become modified.

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