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Temperature dependence of the raman linewidth and frequency shift in Ge and Si
TEMPERATURE DEPENDENCE OF THE RAMAN LINEWIDTH AND FREQUENCY SHIFT IN Ge AND Si S.
Francis
Bitter
National
Magnet
and B. LAX
SAFRAN
Laboratory+ and Physics Department, Cambridge, MA 02139, U.S.A.
Massachusetts
Institute
of Technology,
(Received 7 Ocroher 1974) Abstract-Information about the anharmonic properties of crystals can be obtained from a study of the temperature dependence of the linewidth and frequency of their Raman spectra. We have calculated the frequency shift of the zone-center optical phonon in silicon and germanium as a function of temperature assuming that the lifetime and self-energy of the k = 0 optical phonon is governed only by decay into two LA phonons. We find that our model not only fits the experimental data for the frequency shift, but is also highly consistent with calculations and experimental data for the linewidth. The consistency in the fining of the theoretical curves thus obtained to the experimental data, determining the same, single adjustable parameter for both the shift and the width seems to corroborate the view that there is a selection rule. aside from momentum conservation, which limits the optical phonon decay into two LA phonons only.
1. IhTRODUCTlON
Raman
where
scattering
due
the spectral
frequency linewidth
which
lo phonons
Here
lines have a finite vary
with
width
temperature.
and shift can be understood
is the unperturbed
eigenvalue
p signifies principal
part.
Recently,
and peak
The
EI
subscript
results in a spectrum
linewidth
other states of the system1 I]. Similarly,
assumption
interactions.
in systems with
such as phonons
have
in an anhar-
optical
has been calculated (0)
from
into two LA
the
which
shifts in the Raman dependence of the
by Klemmens[3]
that it arises only
phonon
H,,and
been performed
and frequency
spectra of Ge and Si. The temperature
as arising from the
finite lifetime of a state due to spontaneous emission into dissipative
experiments
measure the linewidths
natural
of
on the
the decay of the
phonons
of balanced
manic crystal, we can also expect a finite width and shift
momentum. This has been confirmed by the experimental
due to interactions
data of Hart el ol. in Si[4] and by Ray et al. in Ge[5]. In
arising from “collisions”
phonons. Using perturbation
theory,
between the
this paper, we extend
Brout [2] has calcu-
lated the width and shift due to interactions
frequency
of the form
this approach
shift, similarly
and calculate
the
assuming that the only decay
channel is 0 -+ LA - LA and compare our results with the
H = H, + HR -F H,,,
(1)
experimental
data in both Si and Ge. In addition,
demonstrate
and
the consistency
ing the normalizations H,=H,,+AVwithIhVIGHo
(2)
linewidth
H, is the Hamiltonian due lo the radiation field; H,.,represents the interaction between system and radiation; H, is the system Hamiltonian where H,,
where
describes the system without
interactions
and V couples
the modes of the system as in collision
techniques
and who evaluated
quency shift numerically allowed,
the difference repopulation
in the frequency and depopulation
a set of quantum
&
shifts or linewidths
due 10 by
the linewidth
(3a) and the O+ LA +
of temperature.
systems,
CALCL’LATIOZ
the dissipative
resented by V in equation
rep-
three phonon process[8] k
P
v = c c(k, k’,k”) WL’ II I-
is given by
&-r .,‘.
(a,, - a:,)(a,.K - a:.k)(a,-,--
=~~ll’~(~~-~,~)where s is the branch index,
I-k
a:-,-)
(4)
(3b)
number by the National
interaction
(2) is due to the anharmonic
(3a)
is the shift of level k, while the linewidth
Qupported
and fre-
mentioned above, and a linear approxima-
2. PERTL’RRATIOS In phonon
by k, where
=$Fyygf k
that of
are given by
of a state characterized
numbers represented
shift and the from
assuming all decay channels are
in that we use equation
shift as a function shift and the linewidth
differs
tion lo the LA dispersion relation to obtain the frequency
effects. The total frequency
Our work
Cowley [6] and Maradudin [7] who used Green’s functions
LA restriction
and anharmonic
for both the frequency
calculations.
we
of our approach by compar-
Science Foundanon.
of
anharmonic 753
atoms
in
coefficient,
the
k = the wavevector, crystal,
c(k,k’,k”)
N is the is the
and II,~ and u ;* are the annihila-
754
S. SAFRAN and
tion and creation operators for phonons on branch s with wavevector k, where (n + Ilu’ln) = J&
(n + 1)“’ (5)
(n - l]aln) = J&(n)“’
Here, M is the atomic mass and n is the occupation number of that mode. In our calculation, we shall only consider the difference in the terms in equation (4) corresponding to the creation or destruction of one optical phonon of frequency w0and the simultaneous destruction or creation of two acoustic phonons. The other terms in equation (4) give rise to either non-energy conserving terms in the case of the linewidth or either non-resonant denominators or cancellations in the expression for the shift given by equation (3a) in our approximation of decay to two phonbns~belonging to the same branch. Thus, we consider only V, =
2 ~(0, k, -k)aoa:~,a:--t
(6)
,‘I x
Vz= c c(0, k, -k) a,,+a,,ra,- t 7’l.L
B. LAX
where y is the Griineisen constant which is assumed to be frequency independent. In order to equate AE due to this strain to the energy of the anharmonic term, we calculate S(o*) from (9) and (II), considering terms to first order in e only; thus $3 remains the same and (1 +x)-l is approximated by I - X. The results give S(o*) proportional to w2 and S(L)=
-21rsw’
(12)
0
Using this expression in (8) and equating coefficients of the displacements due to the strain and ao, we find: ~(0, k, -k) = coo*/2
(13)
A more careful analysis, using the exact dispersion relation]lO] for unit maSS
o2 = a + /3 + d/(a + /3)’ -4a/3 sin*ka
(14)
and calculating 8(0’) again to first order in e by expanding the radical in equation (14) assuming
(7) (15)
where a0 refers to the k = 0 optical phonon and because of the delta function in equation (4), k’ = -k” = k. In order to approximate the frequency dependence of the c(k,k’k’? terms, we shall extend an argument presented by Klemmens [3,9] and estimate c(k, k’, k”) by equating equations (6) or (7) with the energy due to a static strain identical to the instantaneous strain field of the optical mode in its interaction with the acoustic phonons. Since the energy of an acoustic mode k is given by Mo’a’(k)a(k), (the a (k) may be thought of classically as referring to position), a strain causes a change in energy d AE = d(~2) [Mda+(k)a(kj]
1
6(wL)
(8)
yields o’+ 6(w2) =(aotPo)+2~~(uo-Bo)-~
(16)
which reduces to, upon the use of the expression for the optical mode frequency obtained from equation (14) at k ~0, 00 = 2(ao + 83,
Also, because the dispersion relations for Si and Ge resemble those of a three dimensional “linear” chain with atoms of the same mass with different force constants /3 and a, we write as a first approximation[3]. o2
2i:
--!!!k
sin’ko
(9)
a2
(17)
which agrees with equation (9) for small (o/w,). Therefore, from equation (13a)
(a+BW ~(0,k, -k) = coo2f(o/wo)
(18)
1 f(w/oo) = - (1 - 2~z/00’)“ 2
(19)
Imposing a strain e, where where Au ET-
a (Au is the atomic displacement and a is the lattice parameter) and where the strain is +z on every a link and --E on every /3 link, we get a = &(I +2ye) (11) /.?= P,(l -2ye)
Noting, as above, that we want the difference between the squares of matrix elements of VI calculated between (T~.~t Iv$~-~+ Ino = 01 and ]ns.t~,--tq~ = 1) and those of V, evaluated between (nl.k - lnr--k - 1 no= 11 and l~.~n,-_~n~= 0), the expression for the linewidth and the frequency shift become upon using equation (5) in
155
Temperature dependence of the raman linewidth
equation (8) above,
we shall make the approximation that
A = $ & co?f(&‘o)12 (o,_;~~w:;~“_,) x
r = $zk
(v+k
c,2Lf(w/wL#ws’kws”-k
-
+
8(
w,,
+
1)
+
-
1)
w$k
(vs”k
-
-
1)
w,“k)
(21)
where the population of the k th mode on branch s is given
by risk= [exp($)-lr’
(22)
Since we are assuming decay into two LA phonons only and are excluding all. other channels, we can use the Debye approximation which is applicable considering the linear nature of the LA branches of Si and Ge [ 11,121.We thus set o = vk and consider decay to only the LA branch, summing over all wave-vectors on that branch. We therefore drop all sums over s and s’ and set wssK= wssrK.In addition, in order to obtain an analytic expression for A and I, we let &sd3k=
T=I
I
&$w2dw
Since coth x is being integrated this approximation should not result in any serious error. In doing the integration we take care to note the relationships between x, 2x,, and 1 in order to perform correctly the principal part integration which can be integrated by standard analytic methods. In addition, in order to proceed with the integration in equation (25a), we take f2(o/wo) out of the integral and evaluate it at some average value of w/wO.Since, as we demonstrated above, the important part of c (0, k, -k) is the w2 dependence-this shows up even in the approximate dispersion relations-we should be justified in leaving the w2 dependence of c(0, k, -k) in the integral and taking out f’(w/w3. For convenience we define
6-4 Since f is determined by the ratio of two frequencies, its temperature variation can be neglected. The results of the integration of equation (25a) are, with y = 4kT/hwo and y,,, = 4kT/hw,
(23)
Thus, our expressions for the width and the shift become A=k,,@
I
--$&
Lf(&&
(26a)
coth x =
(20)
1) -;dk?,.“-k)
[($‘k
7js’k?s”k]
{(%‘k
(TIC+ 77-k
+
1)
F(Y)
do
A(T) = - Aocuoflyy,,,z 2
(27a)
A(T) = - AaflF(y) y,,,%2
(27b)
= [ y4+2y3~pgi-12Yt(y-l)lnll-yl+P,]
(284
(24a) r=2rko
CW I
0~4Lf(W/Wo)l* (qk + 7-k
+
1)
~(WO -
2w) do where (24b)
,=%=yo
where
00
c,’
~o=~z3t2z2+2r+ln/2z-1/
and is the same for both linewidth and the frequency shift with n7s8K = n7s,rKas above. Defining x = ho/2kT and x0 = hwo/4kT, we get for the shift
I
O*Xm Lf(w/o,,)]’&
coth x dx (25a)
while the trivial integration of equation (25b) yields for the width I = 27r& coth xo= Tocoth xo
(25b)
where ho again, is the same for both I and A. The integration in equation (25a) is carried out by integrating from zero up to xm = hw,/4kT which corresponds to the maximum frequency of the LA phonon-approximately the zone-edge frequency. In order to perform the integration in (25a) analytically, JPCSVol. 36No.7-I
(294
v 4rr
kc,=--7 h (2rr) v
A= -A,@-$
ym
po=(Y0t4z4
(29b) (29~)
We therefore see that the physical interpretation of AO is that A(0) = - A$F(O) = - A$j% is the zero temperature shift due to the self energy of the optical phonon, where /lo is given by equation (29~) and f by equation (26b). Similarly, in the case of the linewidth, from equation (25b) r(O) = I0 = 2rA0 is the zero temperature width. At finite temperatures, considering z > l/2, the shift is negative and linear with T for high temperatures, while for low temteratures, it is determined by F(y) which -PO for 0 < y < 1, if we expand the log term to 4th order. 3. RESULTS AND COMPARISON WITH EXPERIMENT
Equations (27)-(29) give the frequency shift as a function of temperature in terms of only one undermined positive parameter & = A$ which is proportional to the shift
756
S.
SAFRAN
and B. LAX
and the width at T = 0. To determine A& we fit our curves to the data of Ray et al. for Ge [S] and Hart et al. for Si [4] using values of z and w0 described below. The parameter was determined equation
using a least squares fit for
(27a) and equation
determining
each of
(27b) for all T < 700 K and
the final value of h; by taking a weighted
average of the values obtained from the fits. For Ge this gave Ab = 0.61 cm-‘,
while
check for consistency, data using equation
for
Si AI, = I.58 cm-‘.
As a
the fits to the high temperature
(27a) and the low temperature
using equation
(27b) were done separately-the
pancy in &was
less than IO per cent in both materials.
data
discre-
Since from equation (27b) we see that there is a finite shift even at zero temperature,
we had to determine
a
suitable value of w”, different from the values given by the data
at zero
temperature.
By
extrapolating
T = 0 K from the high temperature,
back
to
linear regime, we see
that values of w0 = 309 cm-’ for Ge and 532 cm-’ for Si are appropriate.
These “unclothed”
and turned
values of on were used
out to give the best tit to the experimental
data[l3]. The parameter
z = w,,/u,, was taken as 0.82 for Si and
0.80 for Ge. This is in line with our decay model; the LA
K/Kmo,.
Fig. 2. Phonon dispersion curves for Ge in the [It I] direction obtained from neutron scattering (after 8. Brockhouse[ 121). The value of w, lo, = 0.80 was obtained as in Fig. I caption.
branch is highly linear in both Ge and Si with a maximum w,,,, determined by the k/k,,,
frequency, straight
line fitted
to the dispersion
= 1 intercept of a curves for the LA w.
branches and averaged over the three principal directions. Since z is a ratio dependence
of two frequencies
-I
GERMANIUM = 309 cm-’
,
its temperature
can be neglected. (See Figs. I and 2).
Our plots of the shift A(T), along with the original data points of Hart et al. for Si [4] and Ray et al. for Ge [S] are presented
in Figs.
considering
3 and 4. The fits are satisfactory
the approximations
discrepancies
used. In particular,
at high temperatures
can be attributed
the
-
to
200
r-
---1
-GY
---I
04
06
08
Id
K/Km,,
Fig. 1. Phonon dispersion curves for Si in the [ 1001 direction obtained from neutron scaftering (after G. Dolling[l I]). The value of w, Iw, = 0.82 was obtained by approximating the LA branch by the dashed line and averaging values of w, for various directions.
600.
800
SIL!CON
-.200
02
1 (%I
Fig. 3. A plot of the frequency shift of the Stokes component of the first order Raman line in Si, as a function of temperature. The x signs represent the data points of Hart er a/.[41 and thesolid line is the curve obtained from equation (27) of the theory assuming only two LA phonon decay. See text for explanation of dashed line.
0
-0
400 TEWERATURE
Fig. 4. A plot of the the first order Raman x signs represent the is the curve obtained only two LA phonon
L_. 400 TEMPERlTURE
..I 600
I300
rK)
frequency shift of the Stokes component of line in Ge as a function of temperature. The data points of Ray er al. [5] and the solid line from equation (27) of the theory assuming decay. See text for explanation of dashed line.
Temperature
multiphonon around
processes.
traced
to the error
x = I leading this bump dashed
the
lack
in F(y) there.
terms
proportional
the frequency
parameters
agreement the data
much
better
linewidth
our
by a
than
the low
use equation
dependence The
of
temperature
fit
values
of the frequency
the simpler
calculation
(?Sb), indicates
for the shift,
shift
A(O) = - A$F(O) = 1 for
As
noted
equation determine
f
of
i’
x (z’)‘.
= - &of&,
average
Thus,
the
with
I
0
f =
To
for
p(z’)z’ d:’
Si in our
in equation
~ = 42
(32)
d:’
approximations,
and
flu as
to equations
(30) and (31) we see that,
both the linewidth
due to the 0+2LA consistent satisfy
decay
calculation the
and the frequency of
following
as we have both
r(T)
assumed, and
relationship
if in
shift are only
when
then a
A(T)
should
evaluated
at
T=OK: -2a
r(o)
= -0.37Ge
m=m
for Ge[ 141 is r(0)
PI(O) = -3.Ocmwho found equation
from Thus,
[r(O)/A(O)]
(33) evaluated
= 2.
calculate
A(O) = - 8.5 cm-’
ture data as above, excellent
implying
agreement agree
proper
choice
which
gave
addition adjustment
with
so well best
to satisfying of wa when
we estimated
that with
from
fits
Similarly.
[r(O)/I(O)]
fitting
= - 0.25, again in (33)
That
the
is due to the
we chose were the ones
to the
equation
we
low tempera-
(33) above.
equation
for Si,
pt al. [4] while
the actual
equation
with
we get et al.[S],
= 0.37 in agreement
by Hart
of w,)- the values the
while
by recent
while
of Ray
for germanium.
as indicated
I cm-’
as measured
= I.1 cm-‘.
the data points
= 0) = 306 cm-’
w(T
o0 = 309cm-‘.
(33)
= -0.26Si
The T = 0 value of the linewidth experiments
results
structure,
approxima-
that are in good
The values
of r(O)/b(O) data
are
and
ef al.[4]
phonon.
some
to determine
conservation
phonons,
experimental
that there
which
the the
is some that only
operates
in the
We suggest that besides work
further
in crystals theoretical
those possible
selection
of the work
be
rules.
Ackno&dgemtnts-We wish to thank Dr. R. Aggdrwal for several useful discussions, and Drs. T. Hart and P. G. Klemmens for discussions of their previous work. We are also grateful to Dr. V. J. Tekippe who provided us with recent values of the low temperature linewidth in Ge prior to publication. REFERENCES
Returning
r(0)
further
curves
momentum
of the optical
the
only into two
All this seems to corroborate
to two LA
pursuing
only.
of both
decays
experimental
by Hart
rule besides
diamond
phonon
the
calculations.
first suggested
undertaken
calculation decay
(29~) is given by 5.37 for Ge and 5.82
for Si. reality
both
decay
in the
in Ge and Si, assuming
yields
experiment.
relaxation
we get
branch,
for
selection
two LA
and using a Debye-type
with
allows
z’ = w,,,/o~ as above,
optical
agreement
the first
states
of a consistent
assumes
the
function
of
fitting
4. CONCLL’SIOSS we note that calculations
phonons
tion for the LA
view
which as the
shift and the fits in the case of the
which
that the zone-center
theoretical
evaluating
I .79 for Ge and 2.05 for Si. since z = 0.80 for
and 0.82
defined
by
of z’ = w/w~.
points
(33) for both the theory
shift and the linewidth
same
the
et al.141 and Ray et a/.[51 were not merely
In conclusion,
longitudinal
satisfied
fits as well
that our curve
but are the results
on a model
of z’, we calculate
p(z’)
yielding
calculation]
value
based
density
value
using
2’ =
Ge
(31)
is approximated
at some
the correct
moment p(z’)
above,
(19)
by Hart
frequency (30)
temperature
implies
case of the frequency
that if we
then at T = 0
w = 00/2 in the width
of our fits to the data
same values of the ratio in equation
width
have
to fit.
both good finite
of the
l-(O) = I-” = 27r&
Lf2(w,/wo)
consistency
and the experiment,
overall curves
The
we would
8% in Ge and 12% in Si where
data was more difficult yielded
(33). if not in
had we used our curves
data points,
(33) to within
fortuitous,
in equation
(27b)
&,
totally
[6].
(2Sa) with
resulting
have
equation
with equation
For example,
and not the actual
around
indicated
calculations.
of our calculation
equation
agreement.
and the low temperature
by Cowley
Comparison
are in good agreement
exact
that
and temperature
in all
is gratifying
calculated
which
We believe
We
157
Si can be
our constant,
to y’/c’.
of the raman linewidth
agreement
in the coth x approximation
to the bump
is not real and is, therefore,
neglected these
of
T = 200 K for
line in Figs. 3 and 4. In addition,
contains
from
while
T = 120 K for Ge and
dependence
experimental
(33).
However,
the data
leads
data
in
even
an
to results
I, Wigner E. and Weisskoff V.. Z. Phvsik 63. 54 (1930). !. Brout R., Phps. Rec. 107. 664 (1957). 3. Klemmens P. G.. Php. Rer. 148. 845 (1966). 4. Hart T. R.. Aggarwal R. I.. and Lax B.. Ph.vs. Rer. B. I. 638 (1970). 5. Ray R. K.. Aggarwal R. L. and Lax B.. in Proceedings of the Se&d /nte&rional Conference on Light Scat&n~ in Solids. (Edited hr Balkanski M.) n. XX. Flammarion Science. Paris (1971). . 6. Cowley R. A., 1. Ph,vs. (Paris) 26. 659 (196.0. 7. Maradudin A. A.. Fein A. E. and Vineyard G. H.. Phvs. Stat. Sol. 2. 1479 (I%?). of the Theory of Solids D. 68. 8. Ziman J. M.. in Princinles Cambridge University Press, Cambridge il9??). ’ 9. Klemmenb P. G.. in Solid Store Physics. (Edited by Seitz and Turnbull). Academic Press. h’.Y. (1958). p. 33. Let a-l/M. l/M,+@ and I/M,-+P IO. See reference[gl. lo get our expression. Symposium on rhe II. Dolling G., in Proceedings of International Inelasric Scartering of h’eurmns in Solids and Liquids. Vienna. I%?. Vol. ?. D. 37. International Atomic Aaencv. _ _ Vienna (1963). B. N.. 1. Ph.vs. Chem. So/ids 8. 402 (1959). I?. Brockhouse 13. Hart Tim. Ph.D. Thesis. Massachusetts Institute of Technology (1970 unpublished). On p. 70. Hart suggests a value of 529cm ’ for Si. 14. More recent measurements by Tekippe V. J. and Aggarwal R. L. on Ge at low temperatures down IO 4.2 K yield a value of I ‘I cm ’ for r(O) in contrast to the value of I .4 cm-’ deduced _ m reference 5 (prlvale commumcatum).
Francis
Bitter
National
Magnet
and B. LAX
SAFRAN
Laboratory+ and Physics Department, Cambridge, MA 02139, U.S.A.
Massachusetts
Institute
of Technology,
(Received 7 Ocroher 1974) Abstract-Information about the anharmonic properties of crystals can be obtained from a study of the temperature dependence of the linewidth and frequency of their Raman spectra. We have calculated the frequency shift of the zone-center optical phonon in silicon and germanium as a function of temperature assuming that the lifetime and self-energy of the k = 0 optical phonon is governed only by decay into two LA phonons. We find that our model not only fits the experimental data for the frequency shift, but is also highly consistent with calculations and experimental data for the linewidth. The consistency in the fining of the theoretical curves thus obtained to the experimental data, determining the same, single adjustable parameter for both the shift and the width seems to corroborate the view that there is a selection rule. aside from momentum conservation, which limits the optical phonon decay into two LA phonons only.
1. IhTRODUCTlON
Raman
where
scattering
due
the spectral
frequency linewidth
which
lo phonons
Here
lines have a finite vary
with
width
temperature.
and shift can be understood
is the unperturbed
eigenvalue
p signifies principal
part.
Recently,
and peak
The
EI
subscript
results in a spectrum
linewidth
other states of the system1 I]. Similarly,
assumption
interactions.
in systems with
such as phonons
have
in an anhar-
optical
has been calculated (0)
from
into two LA
the
which
shifts in the Raman dependence of the
by Klemmens[3]
that it arises only
phonon
H,,and
been performed
and frequency
spectra of Ge and Si. The temperature
as arising from the
finite lifetime of a state due to spontaneous emission into dissipative
experiments
measure the linewidths
natural
of
on the
the decay of the
phonons
of balanced
manic crystal, we can also expect a finite width and shift
momentum. This has been confirmed by the experimental
due to interactions
data of Hart el ol. in Si[4] and by Ray et al. in Ge[5]. In
arising from “collisions”
phonons. Using perturbation
theory,
between the
this paper, we extend
Brout [2] has calcu-
lated the width and shift due to interactions
frequency
of the form
this approach
shift, similarly
and calculate
the
assuming that the only decay
channel is 0 -+ LA - LA and compare our results with the
H = H, + HR -F H,,,
(1)
experimental
data in both Si and Ge. In addition,
demonstrate
and
the consistency
ing the normalizations H,=H,,+AVwithIhVIGHo
(2)
linewidth
H, is the Hamiltonian due lo the radiation field; H,.,represents the interaction between system and radiation; H, is the system Hamiltonian where H,,
where
describes the system without
interactions
and V couples
the modes of the system as in collision
techniques
and who evaluated
quency shift numerically allowed,
the difference repopulation
in the frequency and depopulation
a set of quantum
&
shifts or linewidths
due 10 by
the linewidth
(3a) and the O+ LA +
of temperature.
systems,
CALCL’LATIOZ
the dissipative
resented by V in equation
rep-
three phonon process[8] k
P
v = c c(k, k’,k”) WL’ II I-
is given by
&-r .,‘.
(a,, - a:,)(a,.K - a:.k)(a,-,--
=~~l
I-k
a:-,-)
(4)
(3b)
number by the National
interaction
(2) is due to the anharmonic
(3a)
is the shift of level k, while the linewidth
Qupported
and fre-
mentioned above, and a linear approxima-
2. PERTL’RRATIOS In phonon
by k, where
=$Fyygf k
that of
are given by
of a state characterized
numbers represented
shift and the from
assuming all decay channels are
in that we use equation
shift as a function shift and the linewidth
differs
tion lo the LA dispersion relation to obtain the frequency
effects. The total frequency
Our work
Cowley [6] and Maradudin [7] who used Green’s functions
LA restriction
and anharmonic
for both the frequency
calculations.
we
of our approach by compar-
Science Foundanon.
of
anharmonic 753
atoms
in
coefficient,
the
k = the wavevector, crystal,
c(k,k’,k”)
N is the is the
and II,~ and u ;* are the annihila-
754
S. SAFRAN and
tion and creation operators for phonons on branch s with wavevector k, where (n + Ilu’ln) = J&
(n + 1)“’ (5)
(n - l]aln) = J&(n)“’
Here, M is the atomic mass and n is the occupation number of that mode. In our calculation, we shall only consider the difference in the terms in equation (4) corresponding to the creation or destruction of one optical phonon of frequency w0and the simultaneous destruction or creation of two acoustic phonons. The other terms in equation (4) give rise to either non-energy conserving terms in the case of the linewidth or either non-resonant denominators or cancellations in the expression for the shift given by equation (3a) in our approximation of decay to two phonbns~belonging to the same branch. Thus, we consider only V, =
2 ~(0, k, -k)aoa:~,a:--t
(6)
,‘I x
Vz= c c(0, k, -k) a,,+a,,ra,- t 7’l.L
B. LAX
where y is the Griineisen constant which is assumed to be frequency independent. In order to equate AE due to this strain to the energy of the anharmonic term, we calculate S(o*) from (9) and (II), considering terms to first order in e only; thus $3 remains the same and (1 +x)-l is approximated by I - X. The results give S(o*) proportional to w2 and S(L)=
-21rsw’
(12)
0
Using this expression in (8) and equating coefficients of the displacements due to the strain and ao, we find: ~(0, k, -k) = coo*/2
(13)
A more careful analysis, using the exact dispersion relation]lO] for unit maSS
o2 = a + /3 + d/(a + /3)’ -4a/3 sin*ka
(14)
and calculating 8(0’) again to first order in e by expanding the radical in equation (14) assuming
(7) (15)
where a0 refers to the k = 0 optical phonon and because of the delta function in equation (4), k’ = -k” = k. In order to approximate the frequency dependence of the c(k,k’k’? terms, we shall extend an argument presented by Klemmens [3,9] and estimate c(k, k’, k”) by equating equations (6) or (7) with the energy due to a static strain identical to the instantaneous strain field of the optical mode in its interaction with the acoustic phonons. Since the energy of an acoustic mode k is given by Mo’a’(k)a(k), (the a (k) may be thought of classically as referring to position), a strain causes a change in energy d AE = d(~2) [Mda+(k)a(kj]
1
6(wL)
(8)
yields o’+ 6(w2) =(aotPo)+2~~(uo-Bo)-~
(16)
which reduces to, upon the use of the expression for the optical mode frequency obtained from equation (14) at k ~0, 00 = 2(ao + 83,
Also, because the dispersion relations for Si and Ge resemble those of a three dimensional “linear” chain with atoms of the same mass with different force constants /3 and a, we write as a first approximation[3]. o2
2i:
--!!!k
sin’ko
(9)
a2
(17)
which agrees with equation (9) for small (o/w,). Therefore, from equation (13a)
(a+BW ~(0,k, -k) = coo2f(o/wo)
(18)
1 f(w/oo) = - (1 - 2~z/00’)“ 2
(19)
Imposing a strain e, where where Au ET-
a (Au is the atomic displacement and a is the lattice parameter) and where the strain is +z on every a link and --E on every /3 link, we get a = &(I +2ye) (11) /.?= P,(l -2ye)
Noting, as above, that we want the difference between the squares of matrix elements of VI calculated between (T~.~t Iv$~-~+ Ino = 01 and ]ns.t~,--tq~ = 1) and those of V, evaluated between (nl.k - lnr--k - 1 no= 11 and l~.~n,-_~n~= 0), the expression for the linewidth and the frequency shift become upon using equation (5) in
155
Temperature dependence of the raman linewidth
equation (8) above,
we shall make the approximation that
A = $ & co?f(&‘o)12 (o,_;~~w:;~“_,) x
r = $zk
(v+k
c,2Lf(w/wL#ws’kws”-k
-
+
8(
w,,
+
1)
+
-
1)
w$k
(vs”k
-
-
1)
w,“k)
(21)
where the population of the k th mode on branch s is given
by risk= [exp($)-lr’
(22)
Since we are assuming decay into two LA phonons only and are excluding all. other channels, we can use the Debye approximation which is applicable considering the linear nature of the LA branches of Si and Ge [ 11,121.We thus set o = vk and consider decay to only the LA branch, summing over all wave-vectors on that branch. We therefore drop all sums over s and s’ and set wssK= wssrK.In addition, in order to obtain an analytic expression for A and I, we let &sd3k=
T=I
I
&$w2dw
Since coth x is being integrated this approximation should not result in any serious error. In doing the integration we take care to note the relationships between x, 2x,, and 1 in order to perform correctly the principal part integration which can be integrated by standard analytic methods. In addition, in order to proceed with the integration in equation (25a), we take f2(o/wo) out of the integral and evaluate it at some average value of w/wO.Since, as we demonstrated above, the important part of c (0, k, -k) is the w2 dependence-this shows up even in the approximate dispersion relations-we should be justified in leaving the w2 dependence of c(0, k, -k) in the integral and taking out f’(w/w3. For convenience we define
6-4 Since f is determined by the ratio of two frequencies, its temperature variation can be neglected. The results of the integration of equation (25a) are, with y = 4kT/hwo and y,,, = 4kT/hw,
(23)
Thus, our expressions for the width and the shift become A=k,,@
I
--$&
Lf(&&
(26a)
coth x =
(20)
1) -;dk?,.“-k)
[($‘k
7js’k?s”k]
{(%‘k
(TIC+ 77-k
+
1)
F(Y)
do
A(T) = - Aocuoflyy,,,z 2
(27a)
A(T) = - AaflF(y) y,,,%2
(27b)
= [ y4+2y3~pgi-12Yt(y-l)lnll-yl+P,]
(284
(24a) r=2rko
CW I
0~4Lf(W/Wo)l* (qk + 7-k
+
1)
~(WO -
2w) do where (24b)
,=%=yo
where
00
c,’
~o=~z3t2z2+2r+ln/2z-1/
and is the same for both linewidth and the frequency shift with n7s8K = n7s,rKas above. Defining x = ho/2kT and x0 = hwo/4kT, we get for the shift
I
O*Xm Lf(w/o,,)]’&
coth x dx (25a)
while the trivial integration of equation (25b) yields for the width I = 27r& coth xo= Tocoth xo
(25b)
where ho again, is the same for both I and A. The integration in equation (25a) is carried out by integrating from zero up to xm = hw,/4kT which corresponds to the maximum frequency of the LA phonon-approximately the zone-edge frequency. In order to perform the integration in (25a) analytically, JPCSVol. 36No.7-I
(294
v 4rr
kc,=--7 h (2rr) v
A= -A,@-$
ym
po=(Y0t4z4
(29b) (29~)
We therefore see that the physical interpretation of AO is that A(0) = - A$F(O) = - A$j% is the zero temperature shift due to the self energy of the optical phonon, where /lo is given by equation (29~) and f by equation (26b). Similarly, in the case of the linewidth, from equation (25b) r(O) = I0 = 2rA0 is the zero temperature width. At finite temperatures, considering z > l/2, the shift is negative and linear with T for high temperatures, while for low temteratures, it is determined by F(y) which -PO for 0 < y < 1, if we expand the log term to 4th order. 3. RESULTS AND COMPARISON WITH EXPERIMENT
Equations (27)-(29) give the frequency shift as a function of temperature in terms of only one undermined positive parameter & = A$ which is proportional to the shift
756
S.
SAFRAN
and B. LAX
and the width at T = 0. To determine A& we fit our curves to the data of Ray et al. for Ge [S] and Hart et al. for Si [4] using values of z and w0 described below. The parameter was determined equation
using a least squares fit for
(27a) and equation
determining
each of
(27b) for all T < 700 K and
the final value of h; by taking a weighted
average of the values obtained from the fits. For Ge this gave Ab = 0.61 cm-‘,
while
check for consistency, data using equation
for
Si AI, = I.58 cm-‘.
As a
the fits to the high temperature
(27a) and the low temperature
using equation
(27b) were done separately-the
pancy in &was
less than IO per cent in both materials.
data
discre-
Since from equation (27b) we see that there is a finite shift even at zero temperature,
we had to determine
a
suitable value of w”, different from the values given by the data
at zero
temperature.
By
extrapolating
T = 0 K from the high temperature,
back
to
linear regime, we see
that values of w0 = 309 cm-’ for Ge and 532 cm-’ for Si are appropriate.
These “unclothed”
and turned
values of on were used
out to give the best tit to the experimental
data[l3]. The parameter
z = w,,/u,, was taken as 0.82 for Si and
0.80 for Ge. This is in line with our decay model; the LA
K/Kmo,.
Fig. 2. Phonon dispersion curves for Ge in the [It I] direction obtained from neutron scattering (after 8. Brockhouse[ 121). The value of w, lo, = 0.80 was obtained as in Fig. I caption.
branch is highly linear in both Ge and Si with a maximum w,,,, determined by the k/k,,,
frequency, straight
line fitted
to the dispersion
= 1 intercept of a curves for the LA w.
branches and averaged over the three principal directions. Since z is a ratio dependence
of two frequencies
-I
GERMANIUM = 309 cm-’
,
its temperature
can be neglected. (See Figs. I and 2).
Our plots of the shift A(T), along with the original data points of Hart et al. for Si [4] and Ray et al. for Ge [S] are presented
in Figs.
considering
3 and 4. The fits are satisfactory
the approximations
discrepancies
used. In particular,
at high temperatures
can be attributed
the
-
to
200
r-
---1
-GY
---I
04
06
08
Id
K/Km,,
Fig. 1. Phonon dispersion curves for Si in the [ 1001 direction obtained from neutron scaftering (after G. Dolling[l I]). The value of w, Iw, = 0.82 was obtained by approximating the LA branch by the dashed line and averaging values of w, for various directions.
600.
800
SIL!CON
-.200
02
1 (%I
Fig. 3. A plot of the frequency shift of the Stokes component of the first order Raman line in Si, as a function of temperature. The x signs represent the data points of Hart er a/.[41 and thesolid line is the curve obtained from equation (27) of the theory assuming only two LA phonon decay. See text for explanation of dashed line.
0
-0
400 TEWERATURE
Fig. 4. A plot of the the first order Raman x signs represent the is the curve obtained only two LA phonon
L_. 400 TEMPERlTURE
..I 600
I300
rK)
frequency shift of the Stokes component of line in Ge as a function of temperature. The data points of Ray er al. [5] and the solid line from equation (27) of the theory assuming decay. See text for explanation of dashed line.
Temperature
multiphonon around
processes.
traced
to the error
x = I leading this bump dashed
the
lack
in F(y) there.
terms
proportional
the frequency
parameters
agreement the data
much
better
linewidth
our
by a
than
the low
use equation
dependence The
of
temperature
fit
values
of the frequency
the simpler
calculation
(?Sb), indicates
for the shift,
shift
A(O) = - A$F(O) = 1 for
As
noted
equation determine
f
of
i’
x (z’)‘.
= - &of&,
average
Thus,
the
with
I
0
f =
To
for
p(z’)z’ d:’
Si in our
in equation
~ = 42
(32)
d:’
approximations,
and
flu as
to equations
(30) and (31) we see that,
both the linewidth
due to the 0+2LA consistent satisfy
decay
calculation the
and the frequency of
following
as we have both
r(T)
assumed, and
relationship
if in
shift are only
when
then a
A(T)
should
evaluated
at
T=OK: -2a
r(o)
= -0.37Ge
m=m
for Ge[ 141 is r(0)
PI(O) = -3.Ocmwho found equation
from Thus,
[r(O)/A(O)]
(33) evaluated
= 2.
calculate
A(O) = - 8.5 cm-’
ture data as above, excellent
implying
agreement agree
proper
choice
which
gave
addition adjustment
with
so well best
to satisfying of wa when
we estimated
that with
from
fits
Similarly.
[r(O)/I(O)]
fitting
= - 0.25, again in (33)
That
the
is due to the
we chose were the ones
to the
equation
we
low tempera-
(33) above.
equation
for Si,
pt al. [4] while
the actual
equation
with
we get et al.[S],
= 0.37 in agreement
by Hart
of w,)- the values the
while
by recent
while
of Ray
for germanium.
as indicated
I cm-’
as measured
= I.1 cm-‘.
the data points
= 0) = 306 cm-’
w(T
o0 = 309cm-‘.
(33)
= -0.26Si
The T = 0 value of the linewidth experiments
results
structure,
approxima-
that are in good
The values
of r(O)/b(O) data
are
and
ef al.[4]
phonon.
some
to determine
conservation
phonons,
experimental
that there
which
the the
is some that only
operates
in the
We suggest that besides work
further
in crystals theoretical
those possible
selection
of the work
be
rules.
Ackno&dgemtnts-We wish to thank Dr. R. Aggdrwal for several useful discussions, and Drs. T. Hart and P. G. Klemmens for discussions of their previous work. We are also grateful to Dr. V. J. Tekippe who provided us with recent values of the low temperature linewidth in Ge prior to publication. REFERENCES
Returning
r(0)
further
curves
momentum
of the optical
the
only into two
All this seems to corroborate
to two LA
pursuing
only.
of both
decays
experimental
by Hart
rule besides
diamond
phonon
the
calculations.
first suggested
undertaken
calculation decay
(29~) is given by 5.37 for Ge and 5.82
for Si. reality
both
decay
in the
in Ge and Si, assuming
yields
experiment.
relaxation
we get
branch,
for
selection
two LA
and using a Debye-type
with
allows
z’ = w,,,/o~ as above,
optical
agreement
the first
states
of a consistent
assumes
the
function
of
fitting
4. CONCLL’SIOSS we note that calculations
phonons
tion for the LA
view
which as the
shift and the fits in the case of the
which
that the zone-center
theoretical
evaluating
I .79 for Ge and 2.05 for Si. since z = 0.80 for
and 0.82
defined
by
of z’ = w/w~.
points
(33) for both the theory
shift and the linewidth
same
the
et al.141 and Ray et a/.[51 were not merely
In conclusion,
longitudinal
satisfied
fits as well
that our curve
but are the results
on a model
of z’, we calculate
p(z’)
yielding
calculation]
value
based
density
value
using
2’ =
Ge
(31)
is approximated
at some
the correct
moment p(z’)
above,
(19)
by Hart
frequency (30)
temperature
implies
case of the frequency
that if we
then at T = 0
w = 00/2 in the width
of our fits to the data
same values of the ratio in equation
width
have
to fit.
both good finite
of the
l-(O) = I-” = 27r&
Lf2(w,/wo)
consistency
and the experiment,
overall curves
The
we would
8% in Ge and 12% in Si where
data was more difficult yielded
(33). if not in
had we used our curves
data points,
(33) to within
fortuitous,
in equation
(27b)
&,
totally
[6].
(2Sa) with
resulting
have
equation
with equation
For example,
and not the actual
around
indicated
calculations.
of our calculation
equation
agreement.
and the low temperature
by Cowley
Comparison
are in good agreement
exact
that
and temperature
in all
is gratifying
calculated
which
We believe
We
157
Si can be
our constant,
to y’/c’.
of the raman linewidth
agreement
in the coth x approximation
to the bump
is not real and is, therefore,
neglected these
of
T = 200 K for
line in Figs. 3 and 4. In addition,
contains
from
while
T = 120 K for Ge and
dependence
experimental
(33).
However,
the data
leads
data
in
even
an
to results
I, Wigner E. and Weisskoff V.. Z. Phvsik 63. 54 (1930). !. Brout R., Phps. Rec. 107. 664 (1957). 3. Klemmens P. G.. Php. Rer. 148. 845 (1966). 4. Hart T. R.. Aggarwal R. I.. and Lax B.. Ph.vs. Rer. B. I. 638 (1970). 5. Ray R. K.. Aggarwal R. L. and Lax B.. in Proceedings of the Se&d /nte&rional Conference on Light Scat&n~ in Solids. (Edited hr Balkanski M.) n. XX. Flammarion Science. Paris (1971). . 6. Cowley R. A., 1. Ph,vs. (Paris) 26. 659 (196.0. 7. Maradudin A. A.. Fein A. E. and Vineyard G. H.. Phvs. Stat. Sol. 2. 1479 (I%?). of the Theory of Solids D. 68. 8. Ziman J. M.. in Princinles Cambridge University Press, Cambridge il9??). ’ 9. Klemmenb P. G.. in Solid Store Physics. (Edited by Seitz and Turnbull). Academic Press. h’.Y. (1958). p. 33. Let a-l/M. l/M,+@ and I/M,-+P IO. See reference[gl. lo get our expression. Symposium on rhe II. Dolling G., in Proceedings of International Inelasric Scartering of h’eurmns in Solids and Liquids. Vienna. I%?. Vol. ?. D. 37. International Atomic Aaencv. _ _ Vienna (1963). B. N.. 1. Ph.vs. Chem. So/ids 8. 402 (1959). I?. Brockhouse 13. Hart Tim. Ph.D. Thesis. Massachusetts Institute of Technology (1970 unpublished). On p. 70. Hart suggests a value of 529cm ’ for Si. 14. More recent measurements by Tekippe V. J. and Aggarwal R. L. on Ge at low temperatures down IO 4.2 K yield a value of I ‘I cm ’ for r(O) in contrast to the value of I .4 cm-’ deduced _ m reference 5 (prlvale commumcatum).