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arXiv:cond-mat/0301002v1 [cond-mat.mtrl-sci] 31 Dec 2002
Supersonic strain front driven by a dense electron-hole plasma
M. F. DeCamp,
1
D. A. Reis,
1
A. Cavalieri,
1
P. H. Bucksbaum,
1
R. Clarke,
1
R. Merlin,
1
E. M. Dufresne,
1
D.
A. Arms ,
1
A. M. L indenber g,
2
A. G. MacPhee,
2
Z. Chang,
3
B. Lings,
4
J. S. Wark,
4
and S. Fahy
5
1
FOCUS Center and Department of Physics, University of Mi chigan
2
Department of Physics, University of Cal ifornia, Berkeley
3
Department of Physics, Kansas State University
4
Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK
5
Physics Department and NMRC, University College Cork, Ireland.
(Dated: February 1, 2008)
We study coherent strain in (001) Ge generated by an ultrafast laser-initiated high density
electron-hole plasma. The resultant coherent pulse is probed by time-resolved x-ray diffraction
through changes in the anomalous transmission. The acoustic pulse front is driven by ambipolar
diffusion of the electron-hole plasma and propagates into the crystal at supersonic speeds. Simu-
lations of the strain including electron-phonon coupling, modified by carrier diffusion and Auger
recombination, are in good agreement with the observed dynamics.
PACS numbers: 61.10.Nz, 63.20.-e, 42.65.RE
Subpicosecond laser-induced e lec tron-hole plasmas in
semiconductors can produce large amplitude lattice
strain and rapid los s of tra nslational order. These effects
have been studied ex tensively in ultrafast linear and non-
linear reflectivity exp eriments [1, 2, 3, 4, 5, 6, 7, 8, 9] and,
more recently, in time-re solved x-ray B ragg scattering
exp eriments[10, 11, 12, 13, 14, 15]. X-ray diffraction has
the advantage that it can provide quantitative structural
information.Many of the x-ray experiments[10, 11, 16]
have been analyzed using the thermoelastic model put
forward by Thomsen et al.[7] in which the strain is caused
by differential thermal expansion. Deviations from this
model are discussed in the work of Thomsen et al. and
have been seen in x-ray diffraction [11, 13, 14, 16, 17].
Cavalleri et al.[13, 14] studied c oherent strain near the
thermal melting threshold in Ge a nd concluded that the
strain is produced over a r e gion which is thick compared
to the optical penetration depth due to ambipola r dif-
fusion. However, their experiment was only sensitive to
structural changes in the near surface region.
In this letter we report on measurements of coher-
ent strain generation in Ge following ultrafast laser-
excitation using a bulk sensitive structural probe. We
use time-resolved ultrafast x-ray transmission to measure
strain propaga tion deep within the crystal, providing in-
formation about the generation process. Initially, the
strain front advances at speeds gre ater than the s ound
sp e e d. In our experiments, the laser intensity is sufficient
to impulsively generate a dense electron-hole plasma at
the crystal surface, the dynamics of which are governed
by ambipolar diffusion [18] and Auger recombination.
The plasma couples to the lattice through the deforma-
tion potential. In order to probe the resulting coherent
acoustic pulse as it travels deep within the bulk, we uti-
lize the Laue geometr y where by the x-rays traverse the
full thickness of the crystal, emerging on the other side
as two mutually coherent be ams[19]. We have recently
shown that a short acoustic pulse can coherently transfer
energy between these two beams o n a time scale inconsis -
tent with the thermo-elastic model. Following the initial
transient, the beam intensities oscillate as a function of
the pump-probe delay[17]. In the experiments reported
here, the stra in generation is studied as a function of the
incident laser fluence. The relative phase of the oscilla-
tions and the amplitude of the transient provide informa-
tion about the strain generation process at times s horter
than the x-ray probe duration.
In the Laue geometry, two linearly independent wave
solutions propagate through the crystal. Transverse to
the propagation, these two solutions are sta nding waves
whose wavelengths are twice the spacing of the diffracting
planes. The solutions are usually labelled α and β with
the convention that α has its nodes, and β its antinodes
on the diffracting planes. In the case that all atoms lie on
these planes, α is maximally transmitted and β is maxi-
mally absor bed. Becaus e the two solutions interact with
different electron densities, they propagate with different
velocities.
Outside the crystal, two diffracted beams are pro-
duced: one in the direction of the input be am (forward-
diffracted or “0” beam), and the other in the direction
determined by the vector sum
~
k
H
=
~
k
0
+
~
G
H
(deflected-
diffracted or “H” beam). Here
~
k
0
(
~
k
H
) c orresponds
to the wavevector of the forward-diffracted (deflected-
diffracted) beam and
~
G
H
is the reciprocal lattice vector
corresponding to the diffracting planes. These b e ams are
linear combinations of the two internal solutions, α and
β. The external intensities are g iven by:
I
0
= |a
~
E
α
e
i
~
k
α
·~z
+ b
~
E
β
e
i
~
k
β
·~z
|
2
(1)
I
H
= |c
~
E
α
e
i
~
k
α
·~z
− d
~
E
β
e
i
~
k
β
·~z
|
2
(2)
where I
0
(I
H
) is the diffracted intensity of the forward
(deflected) beam,
~
E
α,β
is the complex wave field inside
2
FIG. 1: Intensities of the forward- and deflected-diffracted
beams (upper) and the interior solutions α, β (lower) as a
function of depth inside a thick crystal. A lattice disturbance
near the exit couples the two solutions, regenerating β .
FIG. 2: Experimental setup.
the crystal,
~
k
α,β
is the complex wavevector of the α, β
solutions (including absorption), and a, b, c, d are deter-
mined by the crystal orientation. The two internal modes
~
E
α,β
oscillate in and out of phase as they propagate
through the crysta l. The wavelength of the interference,
Λ = |
~
k
α
−
~
k
β
|
−1
, is known as the Pendell¨osung length
which is typically a few to tens of microns and is often
shorter than the absorption length.
For a crystal that is thick compared to the β-
absorption length, only the α solution survives and there
are no interference effects. This is the anomalous trans-
mission of x-rays, known as the Borrmann Effect [19]. A
distortion of the lattice c an caus e a redistribution of the
interior wave s olutions [20]. Figure 1 shows the effect of
a thin region of distortion regenerating the β solution af-
ter it has decayed away in a thick crystal fo r the case of
zero α absorption. When this occurs c lose enough to the
crystal exit, the regener ated β wave does not dec ay away
and interference occur s at the exit face, despite the fact
that the crystal is thick.
In our experiments, a short acous tic pulse is generated
at the surface of a thick crystal. This pulse can be con-
sidered as a moving lattice disturbance. The diffracted
intensities will oscillate in time as the pulse travels into
the crystal bulk with a period that is given by the Pen-
dell¨osung length divided by the speed of sound. Devia-
tions from the impulsive strain generation will be evident
in the phase and/or amplitude of the x-ray modulation
as a function of pump-probe delay.
The exper iments were perfo rmed at the 7-ID undula-
tor beamline at the Advanced Photon Source. The x-ray
energy was set to 10keV using a cryogenically co oled Si
111 double crystal monochromator leading to a 1.4×1 0
−4
fractional energy sprea d. The x-ray beam is masked by
tantalum slits to ensure that the x-ray spot is smaller
than the lase r spot on the sample and to provide x-ray
collimation. The sample is a 280µm thick, (001) Ge sin-
gle c rystal. The crystal was oriented such that the x-rays
diffracted in the asymmetric 20
¯
2 Laue geometry. In this
geometry, and at 10 keV, the Pendell¨osung length is 6.2
µm and the β absorption length is 19µm, normal to the
surface. There fo re, in the unperturbed crystal, only α
survives at the exit. The only difference between the two
diffracted beams is in their direction and a mismatch in
their amplitudes due to details of the boundary condi-
tions on the exit face [19].
Coherent strain pulses are produced on the x-ray out-
put face of the crystal by sub-100fs, 800 nm laser pulses
at a 1kHz repetition rate. The ex c itation is fully re-
versible b etween subsequent laser pulses. The laser is
phase-locked to the x-ray beam to better than the x-ray
pulse duration. The las er is timed to the x-rays using
a combination of a digital delay generator and an elec-
tronic phase shifter in the phase-locked loop. In this
manner the pump-probe delay may be set across a range
of ±1 ms with 19 ps precision. A fast silicon avala nche
photodiode (APD) and a picosecond x-ray streak camera
[21] were used as the time-resolved detectors. The APD
sampled the deflected-diffracted beam intensity and the
streak camera sampled the forward-diffracted beam (see
Fig. 2). The x-ray bunch separation was ∼152 ns, large
enough to allow electronic gating and measurement of a
single x-ray pulse .
Following laser-excitation, high contrast o scillations
are observed in the pump-probe da ta over a la rge span of
excitation densities. Figure 3 shows these osc illations in
the deflected-diffracted beam. The period of oscilla tion
agrees with the Pendell¨osung length divided by the longi-
tudinal speed of sound. At an incident fluence of 35
mJ
cm
2
,
the behavior near t = 0 shows a large transient that is
unresolved with the 100 ps x-ray probe beam. After the
transient, the oscillations show a significant phase-shift
with respect to oscillations that occur following an ex-
citation of 2
mJ
cm
2
. The amplitude and frequency of the
oscillations are relatively insensitive to the fluence. How-
ever, as shown in Fig. 4, the phase is strongly dependent
on the fluence and is correlated with the amplitude of
the initial transient. The r e lative phase of the oscillation
was defined with respect to the 2
mJ
cm
2
excitation and was
retrieved from a least squares fit [22, 23]. The amplitude
of the transient is defined as the diffracted intensity at
a delay of 200 ps. Most of the energy transfer occurs in
∼40ps, measured with the forward diffrac ted be am us -
ing a streak camera (see the inset in fig.3 ). At relatively
high fluences (> 10
mJ
cm
2
), the intensity of the deflected
diffracted beam approximately doubles while the forward
3
FIG. 3: Time-resolved anomalous transmission. The t ime-
dependent intensity of the deflected-diffracted beam at three
different incident optical fluences: 35
mJ
cm
2
(solid line), 7
mJ
cm
2
(dashed line), 2
mJ
cm
2
(dot-dashed line). Inset: Streak cam-
era data showing the intensity of the forward-diffracted beam
with picosecond resolution at an incident optical fluence of 35
mJ
cm
2
.
FIG. 4: The relative phase of the Pendell¨osung oscillations
(squares) and the normalized deflected-diffracted intensity at
a time delay of 200ps (circles) as a function of incident optical
fluence.
diffracted beam is cut by more than 75%. At relatively
low fluences (< 2
mJ
cm
2
), there is no transient.
Inspe c tion of (1) and (2) shows that the maximum en-
ergy transfer between the forward and deflected beams
near the ex it of a thick crystal occ urs if the α and β
solutions are c oupled at a depth of Λ/4. This implies
that the transient behavior is due to a perturbation to
the lattice that reaches a depth of more than 1.5 µm into
the bulk. In the simplified picture that a moving inter-
face couples the α and β solutions, the excitation must
propagate into the bulk at gr e ater than 37,000 m/s, more
than seven times the longitudinal speed of sound[24].
The strain pulse has a finite spatial extent and is co m-
prised of a spectrum of phonons with different wavevec-
tors. We expect that the phonon component with wave-
length equal to the Pendell¨osung length will resonantly
couple the two interior wave solutions [25]. To model this
phenomenon, we solve the equations of dynamical diffrac-
tion within the crystal, taking into account the laser-
induced time-dependent stra in profiles, using the Takagi-
Taupin formalism adapted for Laue geometry [26, 27]. In
this method, the differential equations coupling the α and
β branches are solved numerically. The depth-dependent
strain profile for a given time is taken into account by
noting that local strain is e quivalent to a change in the
local Bragg angle. Details of this approach (for Brag g ge-
ometry) can be found in the orig inal work o f Takagi[26]
and Taupin [27]. The means by which the method can
be adapted for Laue geometry are implicit in the work o f
Zachariasen [28] a nd Batterman and Cole [19].
Pure thermoe lastic models of strain propagation do
not predict the observed fluence dependence of the phase
and amplitude of the Pendell¨osung oscillations. A proper
model must include the effects of the coupling of the pho-
toexcited plasma to the crystal lattice. The strain is
comprised of both diffusive and elastic components: the
diffusive strain is determined by the instantaneous tem-
perature and ca rrier density profiles, modified by Auger
recombination; the elastic strain is driven by changes in
the tempe rature and carr ie r density, and propagates into
the crystal at the speed of sound. In the absence of diffu-
sion, a bipolar pulse develops in the time given by the op-
tical penetration depth divided by the speed of sound[7].
For LA phonons with wavectors along [10 0] and a 0.2 µm
penetr ation depth, this corresponds to ∼40ps. Including
diffusion, the electron-hole pla sma extends ∼1 µm in the
same time, lea ding to a strain front that has propagated
into the bulk faster than the speed of sound.
Figure 5 shows the calculated diffraction intensity as
a function o f laser delay at an absorbed laser fluence of
3
mJ
cm
2
(corresponding to a carrier dens ity of ∼ 6 · 10
20
cm
−3
). Good qualitative agreement with the experiment
is seen (Fig. 3). The sharp initial rise in diffraction in-
tensity is reproduced, as well as the frequency and pha se
of the time-resolved Pendell¨osung oscillations. Figure 6
shows the calculated phase shift and the diffracted in-
tensity as a function of absorbed optical fluence. After
taking into account the surface reflectivity of the sample,
good agreement with the experiment is obtained (Fig.
4).
In conclusion we have demonstrated a bulk sensi-
tive probe of lattice dynamics using time-resolved x-
ray anomalous transmission. We have observed that
electron-phonon coupling modified by carrier diffusion
is a dominant mechanism for energy transport in laser-
excited Ge. This work could be extended to study how
the elastic response of the material can modify the elec-
4
FIG. 5: Simulated deflected-diffracted intensity for an ab-
sorbed laser fluence of 3
mJ
cm
2
.
FIG. 6: The calculated relative phase of the Pendell¨osung
oscillations (squares) and the calculated normalized deflected-
diffracted intensity at a time delay of 200ps (circles) as a
function of absorbed optical fl uence.
tronic transport properties of semiconductors.
We thank Bernhard Adams, Mar cus Hertlein, Don
Walko, and Jare d Wahlstrand fo r technical assistance
and stimulating discussions. We also thank Jin Wang
for use of the intensified CCD camera . This work was
conducted at the MHATT-CAT insertion device beam-
line at the Advanced Photon Source and was supported
in part by the U.S. Department of Ener gy, Grants No.
DE-FG02-99ER45743 and No. DE-FG02-00ER15 031, by
the AFOSR under contract F49620-00-1-0328 through
the MURI program and from the NSF FOCUS physics
frontier center. One of us (SF) acknowledges the financial
support of Science Foundation Ireland. Use of the Ad-
va nce d Photon Source was supported by the US Depart-
ment of Energy Basic Energy Sciences, Office of Energy
Research under Contract No. W-31-109 -Eng-38.
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