Ultrashort pulse damage of semiconductors
Bernd Hüttner* CPhys FInstP
DLR-Institute of Technical Physics, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany
Abstract
First, we give a briefly critically discuss the existing definitions of melting and damage thresholds and the different
kinds of experimental determinations of the thresholds.
Then we investigate the thermal and athermal melting of oxides (wide-band gap semiconductors) and of silicon by solving a rate equation for the excited electrons and a by complete self-consistent solution of a coupled system of differential
equations for the electron density and for the electron and phonon temperatures. In particular, we direct our attention to
the still open question about the value for the critical electron density in the case of athermal melting.
Keywords: melting threshold, semiconductor, laser pulse duration, fs-range
1. Introduction
The fundamental processes which are related to the kinetics of high-density plasmas generated in semiconductors by
ultrashort laser pulses have been investigated by many authors. Despite this numerous work, to this very day, we cannot
calculate the melting or damage thresholds results with certainty. There is still a large scatter in the theoretical and as
well as in the experimental values. This is partly related to a lack of the knowledge of accurate input parameters for the
models but also, as we believe, to the more fundamental question of the true criterion for melting induced by ultrashort
laser pulses. For longer pulses there is a clear thermal criterion saying melting appears if the phonon temperature becomes equal to the melting one. Even if the melting temperature may be increased due to superheating this remains a
thermal process. In the case of fs pulses the situation can drastically be changed because during the pulse duration an
extremely high electron density of up to and above the order of the critical density can be excited into the conduction
band. As a result of this highly nonequilibrium process an electronically induced solid-to-liquid phase transition takes
place leading to an ultrafast disordering of the crystal. The crucial and open question is, however, at which electron
density this happens. Some authors prefer the critical plasma density related to the laser wavelength1 others favor smaller ones2. By contrast, molecular dynamics calculation proposes a much higher value3. Below we will show that this
choice is decisive for the calculation of threshold values.
2. Definition and determination of thresholds
First, one has to distinguish between single pulse and multi pulse laser irradiation. For the latter case, the measured
thresholds are in general lower than for the former one due to the effect of incubations. Some incubation effects are
known as, e.g., the color centers or the self-trapped excitons but in many cases their nature is obscure. For this reason we
will regard in the following only single pulse experiments.
Second, there may be a difference between thresholds determined on the surface and in the bulk. For example, the surface damage threshold of CaF2 is by a factor two lower than the bulk value while for SiO2 both values coincide4. It must
be by careful preparation, however, excluded that this is not affected by surface imperfections, such as scratches, cracks,
grooves, and chemical contamination. Since the overwhelming majority of experiments are done on surfaces we shall
restrict ourselves in the following on the consideration of experiments and theoretical calculations concerning the front
surfaces. Due to the nonlinear effects of self-focusing and self-phase modulation the rear surface or the bulk is often
damaged before the front surface. Therefore, it must be careful controlled in the experiment that the damage does not
propagate from the inside to the front surface.
*
Bernd.huettner@dlr.de; phone +49 711 6862 375; fax +49 711 6862 348
Third, in a simplified manner we can discriminate between melting and ablation thresholds. Unfortunately, there is no
general agreement on the definition of thresholds and, therefore, a lot of further definitions exist in the literature as
phase transition and optical breakdown related to melting or damage, evaporation and residual damage, as introduced in
this conference by Efimov, related to ablation. In addition, to this conceptual confusion a broad number of methods of
threshold detection with different sensitivities are used for monitoring. There are ex-situ investigations of the morphology by AFM, SEM, optical5 and Nomarski microscope6 or of the shape and depth of craters by profilometry7. On the
other hand, a multitude of in-situ procedures are applied like TOF8, monitoring the plasma formation9, x-ray diffraction10, light scattering4, plasma radiation11, time-resolved microscopy12, time-resolved interference13, and transient reflection14, only to mention the most applied ones. The various methods and their different detection limits may lead to a
scattering of the determined threshold values. A molten surface, for example, can recrystalize and it may be hard if not
impossible to see a change by ex-situ microscope investigation. The temporal phase transition, however, is seen by an
in-situ optical measurement.
3. Melting threshold
In the standard approach melting appears if the phonon temperature becomes equal or larger than the melting temperature. This purely thermal process is characteristic of long laser pulse duration. An athermal melting, however, 15 is possible for very short pulse durations. In this case, a large number of electrons are excited into the conduction band during
the laser pulse. If the electron density becomes equal or larger than a critical density a phase transition takes place. There
is still a controversial discussion about what is the critical density. In molecular dynamics calculations16 it is shown that
the lattice becomes destabilized if about 10% (~ 1022cm-3) of the valence band electrons are excited. In the modeling of
thresholds is often used, however, the critical plasma density17 belonging to the laser frequency. This density is for the
often applied Ti-sapphire laser (=800nm) about one order of magnitude smaller (~ 1021cm-3). Measurements of the
excited density by Quere et al.2 suggest a still lower value (~ 1020cm-3). Consequently, the authors come to the conclusion “that the breakdown threshold should not be defined as the achievement of a critical excitation density.” The present author, however, deems that this conclusion is misleading because in the experiment an average value is determined
over about half of the diameter of the pump laser beam. Due to the Gaussian shape of the pump laser beam profile the
intensity is much higher in the centre where the athermal melting appears at first. In the following we present calculations with a critical density close to the plasma value.
3.1 Rate equation
A rate equation model is often used to describe the production of the conduction electron density, as given below:
n e  t 
t
 I  t    k I  t  
k
ne  t 
rel
(1)
where  is the avalanche coefficient, k is the k-photon absorption cross section with k as the smallest number satisfying
k·≥Egap, where  and Egap is the laser frequency and the band gap energy, respectively. The last term on the rhs of
(1) summarizes the loss due to recombination and diffusion.
The distinction between the several models depends on the choice of the input quantities, the addition of extra terms and
on the interpretation of the respective roles of multi-photon ionization and impact ionization.
For example, the loss term is totally neglected by Stuart et al.17, Lenzner et al.18, and Jasapara et al.19. Whereas a very
short time (rel=60fs) was used by20 in contrast to the fairly long rel=100ps in21. A comparable scatter may be found for
the respective roles in going from multiphoton ionization alone is capable of producing high electron densities17 over the
statement that damage is still done by avalanche but with the assistance of photoionization22 to avalanche is dominant
down to L=10fs18.
Knowing the experimental values of  and k, equation (1) can be integrated for a Gaussian time-dependent intensity
and the solution is given by
t



 y
1
n e  t    k I 0k  L  L exp   ky 2  I 0  L erf  y   L  dy  n e0 
0
2
rel 



1
 t
 exp  I 0 erf 
 L
2
 t

 rel
(2)

.

10
Gaussian temporal pulse
L = 1.55 eV
rel = 0.1 ps
rel = 1.0 ps
rel = 10.0 ps
Lenzner et al. L = 1.59eV
Tien et al.
2
Fthr (J/cm )
The threshold values are found from (2) for a given pulse duration L by setting the lhs to ne=ncrit=1021cm-3 and then
seeking the appropriated fluence fulfilling the equation. The calculated results for SiO2 together with the experimental
results of 18 and 22 are plotted in figure 1 for the laser frequency =1.55eV.
solution of
-1
k
dne/dt = ( I(t) - rel ) ne + k I(t)
21
3
for ne = ncr = 10 /cm
1
0,01
0,1
1
10
Fig. 1: Damage threshold of SiO2 as a function of L in ps
The value used for the avalanche (=4·cm2/J) and the multiphoton coefficient (6=6·108cm-3s-1) were determined in18 by
fitting to experimental data. Taking the exact value (ncrit=1.84·1021cm-3) for the plasma frequency belonging to
=1.55eV, the points would be shifted slightly to higher values. The largest change at L=100fs is, however, smaller
than 6% indicating that the rate equation is not very sensitive to the choice of the critical electron density. A good
agreement with the experiment can be found by taking relaxation times in the order of a few picoseconds. This, however, is not possible for the fs-value or for much larger times. Lenzner et al.18 noticed that the observed multiphoton ionization rate is substantially lower than predicted by Keldysh23. This is not unexpected because in Keldysh’s theory the band
gap is fixed. In reality, however, it is a function of the phonon temperature and electron density in the conduction band.
Because the energy gap decreases with increasing temperature and increasing electron density the multiphoton ionization rate effects a strong modification during the laser pulse. Consequently, the fitted quantity is an average over the
whole pulse duration. We encounter here the drawback of any rate equation approach because the restriction to only one
quantity, the conduction electron density, cannot reflect the complex processes occurring during laser matter interaction.
We shall deal with this point in more details in the following section.
3.2 Coupled system of partial differential equations
Next, we shall describe a fully self-consistent calculation of the electron temperature Te, the phonon temperature Tph and
electron density ne including the dependence of the band gap Eg on Tph and ne. In this model three coupled differential
equations are solved numerically for the space and time-dependent density and temperatures.
The question of validity and applicability of the temperature concept on short time scales is discussed in a second paper
given in this conference24.
Basically, the model deals with coupled Boltzmann’s equations in the relaxation time approximation to describe the
development of the electron density and the electron and phonon temperature generated by short laser pulses.
The equations, which govern the dynamics of the macroscopic variables, ne, Te, and Tph are given by:
(1  R)  Tph  I(t) (1  R) 2 I(t) 2
n e
 j 

 (Te , Tph , n e )n e  n 3e
t

2 


 (1  R)   Tph    n e ·I(t)  (1  R) 2  I(t) 2 



 Te
1  ce  Te 
3



Te  Tph   n e  E g  k B Te   n e E g 


 t 3n e k B 
T
2





   E g  4k B Te  j     e  Te , n e  Te





 Tph
t

(3)
 ph  Tph 
h ex
Te  Tph  
Tph .

cph
cph
(4)
(5)
The quantities have the following meaning: j is the particle current, R is the reflectivity,  and  are the linear and
nonlinear absorption coefficient, respectively,  is the impact ionization,  is the Auger recombination coefficient,  is
the free carrier absorption, ce and cph are the specific heat of the conduction electrons and phonons, respectively, ph is
the phonon thermal conductivity, hex is the heat exchange coefficient, and T is the temperature relaxation time.
The particle current is defined by


2 ne
n
j  D0  Tph   n e 
Eg  e Te 
k B Te
2Te


(6)
where D0 is the diffusion coefficient. In lack of experimental data, the free electron expression is used for the electronic
thermal conductivity as listed below. The different dependencies of the remaining input quantities on the electron density and the electron and phonon temperature together with the numerical values for silicon are given by
 ph  1585  Tph1.23 W / cmK
e 
2 k B2   n e  Te
3 m eff
c ph  1.978  3.54·104 K 1 ·Tph  3.68K 2 ·T 2 J / cm3
  3.8  1031 cm 6 / s
  3.6  1010 exp(3.2  E g / Te ) s 1
 300 K  cm 2
D0  18 
 Tph  s


(7)

E g  1.117  4.1  104  Tph  T0   1.5  108 n
 Tph  1
  5.02  103 exp 
 cm
 430 K 
2
cm
W
1
3
 eV
 Tph  2
  5  1018 
 cm .
 300 K 
In comparison with the rate equation model such an approach is much more sophisticated. It is able to describe both the
thermal and athermal melting by monitoring if the phonon temperature becomes equal to or larger than the melting temperature or if the electron density becomes equal or larger than the critical density, respectively. Clearly, at long pulse
durations we await the melting threshold to be governed by the thermal melting because due to diffusion and relaxation
the electron density can not rise high enough.
0,50
0,45
 = 1.55eV
0,40
2
Fth (J/cm )
0,35
0,30
0,25
21
-3
n = ncr= 1.84·10 cm
n=0.5·ncr
experiment-literature
experiment-DLR
rate equation
0,20
0,15
0,10
0,1
1
10
100
Figure 2: Melting threshold of Si as a function of L in ps
Figure 2 shows the calculated melting threshold of silicon together with experimental data (◊,○) of 6,25 for two different critical densities n=ncr () and n=0.5·ncr, (■) respectively. Additionally, we have included the results of calculations based on the rate equation ().
Although, the absolute values obtained from the coupled system of differential equations and the rate equation are comparable the latter shows below L = 1ps an increase of the melting threshold with decreasing pulse durations. Such behaviour is not seen for the former one and also seems not to be supported by the experiments.
As expected, for pulse durations L  20ps the curves for n=ncr and n=0.5·ncr coincide because the melting is caused by
the phonons, Tph=Tm. At shorter pulse durations, however, two features are remarkable: First, the melting threshold
possesses a strong dependence on the value of the critical electron density especially in comparison with the small effect
for the rate equation model. Second, the crossover from the thermal to the athermal melting, indicated by the arrows, is
shifted by one order of magnitude to longer pulse durations if the critical density is merely altered by a factor 0.5. Furthermore, the good agreement between theory and experiment supports the assumption that the critical electron density
is close to the value of the plasma frequency. The final clarification of this point would be very important because in this
case the athermal melting depends on the laser frequency and not on the material properties. The best way would be to
repeat the measurements at longer wavelengths. Taking 1.6µm, the critical density is reduced by a factor of four. Unfortunately, we have not the equipment and literature data are not known to the author.
4. Conclusions
We have discussed two different approaches, the rate equation and a system of coupled differential equations, respectively, for the determination of the melting threshold of semiconductors. It was shown that the simple rate equation
model works only well for wide band gap semiconductors. This is mainly caused by the fact that the band gap shrinkage
is not taken into account. A reduction of the pulse duration is accompanied by a rise of the production rate of conduction
band electrons leading to a decrease of the band gap. Obviously, such an effect is much more important to semiconductors like silicon than for wide band gap semiconductors as silica.
The system of coupled differential equations is able to describe both the thermal and athermal melting in good agreement with the experiments. Moreover, it predicts a critical electron density close to the plasma frequency related to the
laser wavelength. This conjecture could be most simply tested by choosing longer wavelengths.
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