V
OLUME
85, N
UMBER
3
P H Y S I C A L R E V I E W L E T T E R S
17 J
ULY
2000
Anharmonic Lattice Dynamics in Germanium Measured with Ultrafast X-Ray Diffraction
A. Cavalleri,
1,
* C. W. Siders,
1,
†
F. L. H. Brown,
1
D. M. Leitner,
1,
‡
C. Tóth,
2
J. A. Squier,
3
C. P. J. Barty,
4
and K. R. Wilson
1
1
Department of Chemistry and Biochemistry, The University of California San Diego, La Jolla, California 92093-0339
2
Institute for Nonlinear Science, The University of California San Diego, La Jolla, California 92093-0339
3
Department of Electrical Engineering, The University of California San Diego, La Jolla, California 92093-0339
4
Department of Applied Mechanics and Engineering Sciences, The University of California San Diego,
La Jolla, California 92093-0339
K. Sokolowski-Tinten,
5
M. Horn von Hoegen,
5
and D. von der Linde
5
5
Institut f ür Laser- und Plasmaphysik, Universität Essen, D-45117 Essen, Germany
M. Kammler
6
6
Institut f ür Festkörperphysik, Universität Hannover, D-30167 Hannover, Germany
(
Received 17 December 1999
)
Damping of impulsively generated coherent acoustic oscillations in a femtosecond laser-heated thin
germanium film is measured as a function of fluence by means of ultrafast x-ray diffraction. By simul-
taneously measuring picosecond strain dynamics in the film and in the unexcited silicon substrate, we
separate anharmonic damping from acoustic transmission through the buried interface. The measured
damping rate and its dependence on the calculated temperature of the thermal bath is consistent with es-
timated four-body, elastic dephasing times
͑T
2
͒ for 7-GHz longitudinal acoustic phonons in germanium.
PACS numbers: 63.20.Ry, 42.65.Re, 61.10.Nz
In semiconductors, the response to impulsive, interband
optical excitation generally proceeds through intraband re-
laxation of hot carriers via phonon emission, radiative and
nonradiative recombination, vibrational acoustic transport
into the bulk or across interfaces, and eventual thermal-
ization of the phonon modes via lattice anharmonicity [1].
While the processes involving charge carriers and Raman-
active optical phonons have been extensively characterized
with optical methods [1], the ultimate steps of transport
and thermalization of nonequilibrium acoustic lattice vi-
brations have remained largely undetected. Moreover, an-
harmonic lattice effects are of general interest because they
are responsible for many other common processes in the
solid state, such as thermal expansion, volume-dependent
elastic constants, and temperature dependent thermal con-
ductivity in solids [2].
Yet, the measurement of non-
Raman-active coherent acoustic phonons is demanding and
has been indirectly achieved only at surfaces [3,4]. By
combining the temporal resolution of ultrafast laser spec-
troscopy [5] with the structural sensitivity of x-ray scat-
tering [6], a number of direct studies [7 – 13] of atomic
motion deep within the bulk of matter have been recently
achieved. In this paper, we report on ultrafast x-ray mea-
surements of strain oscillations in an impulsively heated
germanium film. Excitation-dependent damping times and
vibrational transport across a buried interface are simulta-
neously measured, thereby identifying individual coherent
phonon decay mechanisms.
In our experiment, multiterawatt, femtosecond laser
pulses [14] were focused onto a moving copper wire to
produce ultrafast x-ray bursts [15] at 20 Hz. The emitted
x rays, consisting of spin-orbit split 8-keV
͑1.54 Å͒
Cu-K
a1
and Cu-K
a2
lines, were diffracted by the sample,
a 400-nm-thick, crystalline (111) germanium film grown
on bulk (111) silicon by surfactant-mediated heteroepitaxy
[16]. After symmetric Bragg diffraction, the x rays were
recorded with a solid state area detector [x-ray charge-
coupled device (CCD)]. The two K
a
doublets where
diffracted at two distinct Bragg angles (germanium 13.6
±
,
silicon 14.2
±
) resulting from different lattice constants of
the two diamondlike materials (germanium 5.65 Å, silicon
5.43 Å). The area of the sample probed by the x rays was
illuminated by an 800-nm wavelength, 30-fs laser pulse
with fluence of 40 mJ
͞cm
2
. By simultaneously probing
the film and the substrate, we could measure the strain
dynamics in both components of the structure. Multishot
degradation at the very surface was avoided by translating
the sample after exposure to a few hundred pulses.
Typical x-ray probe lines, diffracted at a pump-probe
time delay of 100 ps, are shown in Fig. 1. In the Ge film
a full shift of the two nonbroadened lines indicates ho-
mogeneous expansion over its entire thickness. As the
sample was illuminated with a Gaussian intensity profile,
fluence resolved measurements of the time-dependent cen-
troid shifts (averaged over the K
a1
and K
a2
positions)
could be easily made and are displayed in Fig. 2 for four
regions of the photoexcited region (A D as indicated in
Fig. 1). The displayed experimental points represent the
average deviation from the static diffraction angle of the
K
a1
and K
a2
for four different fluences [
͑A͒ 40 mJ͞cm
2
,
͑ B͒ 32 mJ͞cm
2
,
͑ C͒ 25 mJ͞cm
2
,
͑ D͒ 15 mJ͞cm
2
]. Typi-
cal values of the peak strain of
ϳ0.05% 0.1% correspond
586
0031-9007
͞00͞85(3)͞586(4)$15.00
© 2000 The American Physical Society
V
OLUME
85, N
UMBER
3
P H Y S I C A L R E V I E W L E T T E R S
17 J
ULY
2000
FIG. 1.
Experimentally measured diffraction curves from the
photopumped germanium film and the silicon substrate for a
100 ps pump-probe delay. The horizontal axis is the diffraction
angle and the vertical is the position on the crystal. The regions
of the samples labeled as A D represent the positions where
fluence and time resolved centroid positions were measured.
to an increase in the distance between lattice planes of
about 150 – 300 fm and result in shifts of the Bragg angle
that range between 225 and 250 arcsec. The peak of
the measured compressions, representing a spatial aver-
age over 2.6-mm probed depth in silicon, correspond to
20 – 40 fm average change in lattice spacing (i.e., by a few
nuclear diameters). Because of better crystalline quality of
the bulk substrate, the silicon measurement was about 43
more sensitive than that of the germanium film. Approxi-
mately 75 ps after photoexcitation, expansion of the ger-
manium lattice reaches its peak value [Fig. 2(a)]. Damped
oscillations were observed at longer time delays, indica-
tive of periodic expansion and compression of the film.
Diffraction from the silicon substrate evidenced centroid
shifts toward higher angles, indicating compression simul-
taneous with Ge-film expansion [Fig. 2(b)].
Two fluence dependent effects are immediately appar-
ent in the Ge data. First, we observed a fluence-dependent
delay on the onset of expansion, becoming shorter at
higher excitation level. Second, fluence-dependent life-
times of coherent oscillations in germanium were clearly
observable.
Recently, a similar effect has been observed under dif-
ferent experimental conditions [10]. In Ref. [10], a re-
versible, nonthermal phase transition was evidenced from
strongly overdamped acoustic oscillations in InSb, with re-
ported strain near the Lindemann criterion
͑ϳ10%͒. In our
experiment, the equilibrium lattice is coherently strained
by as little as 0.1%. Second, unlike the case of ultrafast
melting [11], the total diffraction efficiency in the 111 di-
rection (integrated over the whole Ge-rocking curve) oscil-
lates around the static value but never decreases, indicating
that no significant loss of order is taking place. Finally, as
shown below, after the coherent oscillations are damped,
we can quantitatively model our data for all fluences by
FIG. 2.
Time dependent shift of the angular centroid position,
for (a) 400-nm germanium film, (b) silicon bulk. Thin lines:
theoretically calculated centroid shifts after heating of the ger-
manium film with the calculated strain profile. A fully harmonic
model for a perfect crystal and a fluence-independent heating
time of 10 ps is assumed. Thick lines: (a) phenomenological
fits to the germanium diffraction centroid, yielding total damp-
ing times
͑1͞G
tot
͒ of 44 6 20 ps (A), 62 6 15 ps (B), 75 6
12 ps (C), 109 6 10 ps (D). (b) Expected silicon response us-
ing heating and damping times from the germanium data.
assuming a hot crystalline germanium cooling by thermal
diffusion into the silicon substrate. Therefore our results
cannot be explained with a phase transition.
Our interpretation proceeds along the following lines.
The optical pump pulse excites carriers in germanium over
its absorption depth (200 nm), with initial peak surface
density of
ϳ10
21
cm
23
, with negligible photoexcitation of
the substrate. The absorption depth is largely independent
on the excitation fluence, due to the high density of states
587
V
OLUME
85, N
UMBER
3
P H Y S I C A L R E V I E W L E T T E R S
17 J
ULY
2000
available for interband excitation and to rapid electron-
hole thermalization, depleting the optically coupled states
already during absorption of the pump pulse. While effi-
cient ambipolar diffusion homogeneously distributes the
hot carriers over the entire film within a few picoseconds
[17], equilibration with the lattice takes place through a
combination of nonradiative Auger recombination and
intraband relaxation [18]. The 430-meV potential barrier
at the interface between germanium
͑E
g
0.67 eV͒ and
silicon
͑E
g
1.1 eV͒ is significantly higher than the
quasi-Fermi-levels of the relaxed electrons and holes,
confining the carriers at all times and leaving the silicon
substrate unexcited. Thus, the diffracted Ge lines shift
fully, with virtually no broadening and no significant
negative strain is generated in the silicon. The observed
delays in the onset of expansion are in good agreement
with delayed heating times of 50 ps
͑D͒, 15 ps ͑C͒, 5 ps
͑B͒, and 3 ps ͑A͒, calculated using available values for
germanium Auger recombination rates [19].
Homogeneous impulsive heating starts the coherent
film vibration with a period of 2d
͞c
L
ഠ 150 ps (d
400 nm 6 20 nm is the film thickness and c
L
5400 m͞
sec is the longitudinal speed of sound in germanium) with
the unexcited silicon substrate acting as vibrational energy
sink. In a harmonic approximation and assuming a perfect
crystal, decay of the coherent vibration results only from
transmission of acoustic pulses into the substrate.
In
a real crystal, however, a variety of additional effects,
ranging from defect and surface scattering to phonon-
phonon scattering, result in more rapid damping of the
coherent oscillations. Importantly, while all the defect-
mediated scattering mechanisms are independent of the
degree of excitation of the lattice, anharmonic interactions
between the normal modes of the crystal depend upon the
population of individual phonon modes and thus on the
temperature of the solid [20]. Therefore, the observed
fluence dependent damping is a direct indication of lattice
anharmonicity.
To estimate the relative contributions of the various
mechanisms to the measured damping rate
͑G
tot
͒, we first
compare the data to a fully harmonic model for a perfect
crystal with no defects and assuming complete acoustic
matching between the film and the substrate [Fig. 2]. The
initial stress distribution was numerically calculated by
solving two differential equations for carrier and lattice
temperature [21], taking into account screened Auger re-
combination [22] and density-dependent carrier diffusion
[17]. The one-dimensional elastic equation [23,24] was
then numerically solved in the two-layer system starting
from the calculated stress distribution. Thermal diffusion,
significant at the sharp interface between hot germanium
and cold silicon, was included in the model. The expected
time-dependent x-ray diffraction pattern was calculated
using dynamic diffraction theory [25]. The model predicts
fluence-independent damping
͑G
harm
͒, originating from
the transmission of the coherent vibrations into the sub-
strate. The long lived coherent oscillations measured at
the lowest fluence follow the calculated curve very closely,
demonstrating that defect and surface scattering play a
minor role in our sample, and that only acoustic trans-
mission should be taken into account among the fluence-
independent processes. As the laser fluence is increased
and anharmonicity becomes significant, the data start to
deviate from the model. Total damping rates
͑G
tot
͒ were
fitted to the germanium data using a phenomenological
functional form for damped coherent oscillations super-
imposed on a delayed thermal response. In the substrate,
while the harmonic model is in reasonable agreement with
the data for low excitation, inclusion of the fitted damping
times yields better matching in the higher fluence range.
No clear evidence for fluence-dependent delay in the
onset of compression can be observed in the data, prob-
ably because of the relatively large uncertainty in the
measurement.
The damping rate from acoustic transmission (dominant
at low fluence) was subtracted from the fitted rates,
yielding the fluence-dependent component of the damping
͑G
anh
͒, displayed in Fig. 3. Two different mechanisms
can explain the anharmonic damping of the 7-GHz
oscillations:
inelastic collisions with thermally popu-
lated phonons, causing decay of the population of the
coherent mode (T
1
processes) [25] and energy conserving
collisions, leading to mutual decoherence between the
individual phonons (T
2
processes). Both processes exhibit
a linear dependence on temperature for a classical thermal
bath [26], consistent with our observation. The former,
T
1
process, originates largely from three-body collisions
arising from cubic anharmonicity. For a 7-GHz phonon,
however, T
1
processes are expected to occur at a rate of
ϳ10
25
psec
21
, 3 orders of magnitude slower than the
measured decay of the oscillations [26].
On the other
hand, four-body elastic dephasing
͑T
2
͒ processes [27] can
be significantly faster. The origin of pure dephasing lies
in quartic anharmonic coupling, modulating the frequency
of the 7-GHz mode and causing loss of coherence. This
effect can occur on time scales that are much faster
than inelastic collisions and energy flow out of this very
mode.
By introducing reasonable four-body terms for
the temperature range of our experiment, we find [27]
T
21
2
of order 0.01 psec
21
, which is consistent with our
measurement.
In conclusion, we have measured ultrafast, acoustic
phonon dynamics in a germanium/silicon layered struc-
ture, measuring atomic vibrations with 10-fm resolution,
using ultrafast x-ray diffraction.
The fluence depen-
dence of the oscillations directly yields the temperature-
dependent decoherence rates resulting from anharmonic
lattice dynamics within the germanium film. We attribute
the observed damping to T
2
processes, which originate
largely from four-body interactions that cause decoherence
in the evolution of the mode. Ultrafast x-ray diffraction
allows direct measurement of coherent acoustic excita-
tions before their thermalization with the environment,
thereby making experiments of nonequilibrium ballistic
588
V
OLUME
85, N
UMBER
3
P H Y S I C A L R E V I E W L E T T E R S
17 J
ULY
2000
FIG. 3.
Anharmonic damping rates as a function of calculated
temperature in germanium. The anharmonic damping rates are
calculated by subtracting the low fluence, harmonic damping
rates from the total damping rates. The vertical error bars have
been determined from the uncertainty on the fitted value. The
error bars on the calculated temperature of the crystal (horizon-
tal) originate from the uncertainty on the optical constants of the
sample during irradiation, on laser fluence variations, and on the
thermalization time of the thermal bath. Dashed line: linear fit
to the data.
heat transport possible. Anharmonic effects of strongly
driven or shocked crystals and of solids close to phase
transitions and critical points could also be examined,
with both fundamental and technological ramifications.
K. S. T. gratefully acknowledges financial support by the
Deutsche Forschungsgemeinschaft. The authors are grate-
ful to H. J. Maris for critical discussion.
*To whom correspondence should be addressed.
Email address: acavalleri@ucsd.edu
†
Present address: School of Optics/CREOL, University of
Central Florida, Orlando, FL 32816.
‡
Present address: Department of Chemistry, University of
Nevada, Reno, NV 89557.
[1] J. Shah, Ultrafast Spectroscopy of Semiconductors and
Semiconductor Heterostructures (Springer, Berlin, 1996).
[2] N. W. Ashcroft and N. D. Mermin, Solid State Physics
(Saunders College Publishing, Fort Worth, 1976).
[3] J. J. Baumberg, D. A. Williams, and K. Köhler, Phys. Rev.
Lett. 78
,
3358 (1997).
[4] C. Thomsen, Phys. Rev. Lett. 53
,
989 (1985).
[5] C. V. Shank, Science 233
,
1276 (1986).
[6] M. Von Laue, Ann. Phys. (Leipzig) 41
,
989 (1913); W. L.
Bragg, P. R. Soc. London 89
,
248 (1913).
[7] C. Rischel et al., Nature (London) 390
,
490 (1997).
[8] C. Rose-Petruck et al., Nature (London) 398
,
310 (1999).
[9] A. H. Chin et al., Phys. Rev. Lett. 83
,
336 (1999).
[10] A. M. Lindenberg et al., Phys. Rev. Lett. 84
,
111 (2000).
[11] C. W. Siders et al., Science 286
,
1340 (1999).
[12] A. Rousse et al., in Technical Digest of the Quantum Elec-
tronics and Laser Conference ’99, Baltimore, 1999 (Optical
Society of America, Washington, DC, 1999), p. 152.
[13] J. Larsson et al., Appl. Phys. A 66
,
587 (1998).
[14] C. P. J. Barty et al., Opt. Lett. 21
,
668 (1996).
[15] A. Rousse et al., Phys. Rev. E 50
,
2200 (1994).
[16] M. Horn von Hoegen, Appl. Phys. A 59
,
503 (1994).
[17] By measuring the generation and propagation of short
acoustic pulses in a bulk germanium crystal, we confirmed
that efficient carrier diffusion causes homogeneous heating
of the germanium film. Although the optical penetration
depth is 200 nm, we measured heat deposition into the
bulk lattice over more than 1 mm in less than 20 ps, in-
dicating that heat is initially transferred through rapid car-
rier diffusion. The inferred ultrafast heat transfer velocities
are higher than 5 3 10
6
cm
͞sec, consistent with measured
high-density carrier diffusion rates of germanium [J. Young
and H. M. van Driel, Phys. Rev. B 26
,
2147 (1982)] and
similar to what was already reported for bulk metals [S. D.
Brorson et al., Phys. Rev. Lett. 59
,
1962 (1987)].
[18] M. C. Downer and C. V. Shank, Phys. Rev. Lett. 56
,
761
(1986).
[19] D. H. Auston, C. V. Shank, and P. LeFur, Phys. Rev. Lett.
35
,
1022 (1975).
[20] B. K. Ridley, Quantum Processes in Semiconductors
(Clarendon Press, Oxford, 1993).
[21] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman,
Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 66
,
776
(1974).
[22] D. H. Auston and C. V. Shank, Phys. Rev. Lett. 32
,
1120
(1974); E. Yoffa, Phys. Rev. B 21
,
2415 (1980).
[23] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys.
Rev. B 34
,
4129 (1986).
[24] The value of the screening parameter for Auger recombi-
nation and of the time- and density-dependent carrier dif-
fusion rate were known only from theoretical estimates
(Ref. [22]), but the results of our simulations were not
critically dependent on either parameter. The calculated
peak strain/centroid shift was estimated with 65% accu-
racy, mostly determined by the fluctuations of the laser
energy.
[25] S. Takagi, J. Phys. Soc. Jpn. 26
,
1239 (1969); D. Taupin,
Bull. Soc. Fr. Mineral. Cristallogr. 87
,
469 (1964).
[26] S. Tamura and H. J. Maris, Phys. Rev. B 51
,
2857 (1995);
H. J. Maris (private communication).
[27] T
21
2
ഠ p͞16 ¯h
2
P
i
w
2
11ii
n
i
͑n
i
1 1
͒͞g
i
(Ref. [28]), where
w
11ii
is the quartic coupling coefficient between the 7-GHz
mode (mode 1) and bath mode i, n
i
is the average number
of phonons in mode i, and g
i
is T
21
1
of mode i, which we
take to be of the order of 10
23
psec
21
[Ref. (26)]. To esti-
mate w
11ii
, we take the result for a one-dimensional chain,
where w
11 ii
ϳ p ¯h
2
v
1
v
i
͑͞ka
2
͒, where k is the spring con-
stant and a is the lattice spacing [29], for temperatures
around 500 K. We find T
21
2
to be
ϳ10
22
psec
21
. Note
that, for experimentally relevant temperatures, since n
i
and
g
i
vary linearly with temperature, so does T
21
2
.
[28] D. W. Oxtoby, Adv. Chem. Phys. 40
,
1 (1979); A. Stuche-
brukhov, S. Ionov, and V. Letokhov, J. Phys. Chem. 93
,
5357 (1989).
[29] R. E. Peierls, Quantum Theory of Solids (Clarendon Press,
Oxford, 1955).
589
Dostları ilə paylaş: |