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Anharmonic effects on two-dimensional electron lattice
432
Surface
Science
II3
North-Holland
ANHARMONIC LATTICE *
Received
IX July
Making shear Iimlt
in
(‘(7.)-C;,{ v,hcre
IYXI:
USC of
modulus and
EFFECTS
the
due the
to
tow
~ng
csperiment
an
Ko\tcrlitr~Thoul~.\s \*ith
rcccnt
function
modulus
intcrelcctron
distance, by
of the
by
two
bhear r
modulus
appraumstlcx~,
and ut, is ths
Bohr
and
Adams. electron
v.c’
lowat
order
ctwtron
~‘r found
radius.
solid
ELECTRON
lattice
correction> In
the
to
the
cla~~~al
to hc
‘)}.
the harmonic
of the
the
dimcnGonal
’ -O(T‘
437-417 Compan\
I YX I
WC calculate
the
‘,‘)+Y5] In
27 August
method.
region,
temperature)
experiment
publication
anharmomcitv
tn(0.1Y5rrj
shear
the average to
for
temperature
I --[3.5 C’,, is the
accnpted
ON TWO-DIMENSIONAL
(19x7)
Publishing
r-
obtain 7’,
c’( r;,L
Suh.\titutlng the
I
,3T)
‘, ‘;
‘i,
‘I,,.
‘il I\
‘; ~-4 X * IO3 corrc\pond-
melting
temperature
16X c2 I( r;,L
,3). which
(i c
the
is conxistcnt
et al.
A recent experiment by Grimes and Adams [l] (GA) on the two-dimensional electron lattice (Wigner solid) on the surface of helium four has revived interest in the melting or two-dimensional solid. They observed that the electron lattice is formed at the temperature T corresponding to r = 131 -m7, where I‘ is the ratio of the average Coulomb energy and the thermal energy; r = e’( ak,,T) ‘. Earlier, Platzman and Fukuyama [2] predicted that the solid would be formed at r - 3, by studying the instability in the transverse phonon in the electron solid, which overestimated grossly the stability of the solid phase. It is now believed that the transition observed by Grimes and Adams is rather triggered by the dissociation of bound dislocation pairs in the two dimensional solid [3]. Indeed Thouless [4] has already applied this idea to the two dimensional electron lattice and obtained the melting temperature r,,, = (4?+’
na,?;C,,
where n is the electron density, a,, is the lattice constant and C, is the shear modulus of the electron lattice. Making use of C, calculated by Bonsall and Maradudin [5] (BM). Thouless obtained the corresponding r = 78.71. This I was in good agreement with the results of computer simulation [6,7] and was *
Work
supported
by the
National
0039-6028/82/0000-0000/$02.75
Science
Foundation
under
0 1982 North-Holland
No
DMR
7Y- IhYO3
not far from the GA result. This already indicates that the melting of two-dimensional electron lattice is controlled the the Kosterlitz-Thouless mechanism [2]. More recently Morf [S] calculated the temperature dependence of the shear modulus by the Monte Carlo method and found C,(T) is well approximated by C,(T) = C,(I---3O.gr ~ ‘) in the low temperature region where C, is the shear modulus at T = 0 K. If C,(T) is substituted in the Thouless formula, it gives now r = 110 for the melting temperature, which farther improves the agreement between the theory and the experiment. The object of this paper is to calculate the temperature dependence of the shear modulus within a framework of the perturbation theory. The twodimensional electron lattice is strongly anharmonic. Therefore, in general. the shear modulus calculated by Bonsall and Maradudin [5t has to be corrected by anharmonic effects. It is shown that in the classical limit. the harmonic result becomes exact at T = 0 K. In the same limit the finite temperature corrections can be expanded in terms of diagrams as shown in fig. 1 [9,10]. This expansion corresponds to the expansion in powers of r ‘: each independent phonon loop contributes a factor of I - ‘. We find to the lowest order in l7 -I, C(T)
= C,[ 1 - [3.5 ln(0.195F + 951
Y’) ,
(1)
where C, is the harmonic shear modutus due to BonsaII and Maradudin f5]. It is of interest to point our the lowest order quantum correction (i.e., r, “) appears already in the logarithm. This arised from the thermal average of a In q term in the lowest order correction to the transverse phonon velocity 21~. Inserting rs = 4.8 X lo3 corresponding to the experiment by Grimes and Adams [I] and I = I, obtained by Thouless int he logarithm of eq. (1X we obtain C(T) = C,fl--9OY’f. Although she coefficient of Y ’ appears substantially larger than that formed by Morf, our expression predicts rM given by I = id& which is close but somewhat larger that the GA result. We find that the second diagram in fig. 1 contains a logarithmic divergence, if we expand the self-energy integral in powers of q. A more proper treatment of the diagram yields the correction to the transverse sound velocity depending logarithmically in q. At finite temperatures the corresponding shear modulus is given by C, = nav: , where the In q term in tiL is replaced by its thermal average resulting eq. ( 1).
a
_o _-.._
Fig. I. (a) The Feynman cubic interaction term.
diagram
due to quartic
interaction
term. (b) The Feyman
diagram
due to
We start with a model that the electrons are confined in the two-dimensional surface and interacting with each other via the Coulomb interaction. Furthermore we assume that they form a regular triangular lattice with the lattice constant a,, at low temperatures and that the electron density is sufficiently small so that the exchange effect is completely neglected. Then the Hamiltonian for an electron lattice is given by d’r / ly, -ri
’
12.)
where P, = ar,/at. II (=(&a:) ‘) is the electron density, and y, is the positionn of the ith electron. Expanding q around the equilibrium position R,. we can rewrite ey. (2) as N-
VfH,,+H,
I,': ;
+I'$ t...,
(3)
c, l I#/lR,-&I
(4) where
(5) and we have neglected the anharmonic terms arising from the last term (the positive back ground term) in eq. (2) for simplicity. The first term gives the Madelung energy, while the second term gives the harmonic phonons. In particular, BM have diagonalized H,,and found in the low monentum limit the phonons split into the transverse and the longitudinal phonon with dispersion q’(q)
= 0; q’ = 0,0362967w,;( u,,&.
w;(q) = $(q,q)
- w;(O.181483)
where C$ = (2ar’n/mr1,,), lattice. We shall here calculate
(6) (7)
(joy)‘.
the plasma
frequency
the anharmonic
inthe
corrections
case of the triangular to the above dispersion.
For this purpose
it is convenient
to introduce
the Green
function
defined
(8)
CUB(Ic,t)=i(Tu,(k,r)tcp(-k,O)). where u( k, t ) is defined u(k,r)
=N-‘/*zu,(t)
Within
the harmonic
by (9)
exp(ik.R,).
approximation
G”“( k, w) is then given by
G,;“(k.w,,)=~~“:‘i’[~~+w(k)]“+~;ellIw~+wi(k)]
(10)
‘1,
where e, = i, e, = k”X e, and k^= k/k. In the presence of H,, Hz, etc., the renormalized lated as
Green
function
is calcu-
-’ - fI(k,o,,)
r~(k,w,,)-‘=~,(k,w,,)
(11)
n where II( k, a,,) is the self-energy correction. The lowest order corrections shown in figs. la and lb and are given by [9,10] II;~(k,o,)
= -12TX
~~~(~.#~)
= l&z-C
P
I’
J
as
s
d’q --+Ypy*(k,-k.q, (24”
-----+“YS(k,q, d2g (2a)
-4)
-k-q)
G,:“tq,w,,),
@“‘“(-k,
are
(12)
-q.k$-q)
XG$tq,q,) G,s”(k+q,w,+,),
(13)
where @“fl@(k, -k,q, -4) and aay8(k,, k,, k,) are the Fourier transforms of (I)@‘@ and Qavs in eq. (5) respectively. In the classical limit eqs. (12) and (13) a;; further simplified as KI;~(k,c+,)=
ll;f(k.w,)
-E
*cD”@+(k, FE 1 (24”
=FJf&
where q’ = q + k. Substituting the phonon
-k,q,
-q)
{&w,-“(q)
WY”(k,q,-k-q)@rY(-k,-q,k+q)
spectra
given in eqs. (6) and (7). we can further
r-qy( k, 0)
P4T2
“y
2( Pru:)’
WR)(k*R’)
R.R’
aR
*
a; Y
x [ 1 - eosfq~ R’)] [I - cos(q~ R’ff .
(17)
Here we have neglected the contributions from the longitudinal mode in eqs. ( 16) and (17) which are of order of magnitude smaller than of the transverse mode. Futhermore we have dropped the contribution from the positive back ground potential. Since the shear modulus is related to 1I“‘f k 1 with k I/x, we shall limit to this particular case and we find r~~~~k~~k~l~~
- ISm u:T-‘k;,
(1st
II;i(k)/kIIx=:
[ 3.5 In
(19)
where In y = 0.577... is the Euler constant. Note that eq. (19) contains a logarithmic divergence in k. It appears that all of the higher order diagrams contains the higher power of In k in the limit k tends to zero and we cannot expand the self-energy diagrams in power of k. Recalling that the shear modulus C, is given in terms of the transverse phonon velocity as C,=mv:,
we obtain C(T)=CC,(I-[3.51n~0.195Tr,‘.“~‘“)+95]
_I +o(I?-‘)I.
where we have inserted ei corresponding to that of the thermal logarithmic divergence in eq. (20) is replaced by (ln( l/rk))
_r:ln( Du,)
= ln(0.195
(20) phonon:
the
IYrJ I/?).
We should stress that the In r5 term is the most important quantum correction to the shear modulus; the other quantum corrections are given in powers of r,- i*
Finally, if we substitute in eq. (20) the rs value corresponding to the experiment by Grimes and Adams, we shall obtain a coefficient of I ~ ’ about 3 times larger than the one obtained by Morf [8] in his Monte Carlo analysis. Furthermore it appears that Morf did not notice any logaritmic dependence on I. For the moment we do not know the origin of this discrepancy, although a part of the discrepancy may be due to inaccuracy in our numerical analysis. In any case a more accurate treatment of integral (17) is certainly desirable.
References [I] [2] [3] [4] [5] [6] [7] [8] [9] [IO]
C.C. Grimes and CJ. Adams, Phys. Rev. Letters 42 (1979) 795. P.M. Platzman and H. Fukuyama. Phys. Rev. BIO (1974) 3150 J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) I 181. D.J. Thouless, .I. Phys. Cl I ( 1978) I 189. L. Bonsall and A.A. Maradudin. Phys. Rev. B15 (1977) 1959. R.W. Hackney and T.R. Brown, J. Phys. C8 (1975) 1813. R.C. Gann, S. Chakravarty and G.V. Chester. Phys. Rev. B20 ( 1979) 326. R.H. Morf, Phys. Rev. Letters 43 (1979) 931. A.A. Maradudin and A.E. Fein. Phys. Rev. I28 (1962) 2589. R.A. Cowley, Advan. Phys. I2 (1963) 421.
Surface
Science
II3
North-Holland
ANHARMONIC LATTICE *
Received
IX July
Making shear Iimlt
in
(‘(7.)-C;,{ v,hcre
IYXI:
USC of
modulus and
EFFECTS
the
due the
to
tow
~ng
csperiment
an
Ko\tcrlitr~Thoul~.\s \*ith
rcccnt
function
modulus
intcrelcctron
distance, by
of the
by
two
bhear r
modulus
appraumstlcx~,
and ut, is ths
Bohr
and
Adams. electron
v.c’
lowat
order
ctwtron
~‘r found
radius.
solid
ELECTRON
lattice
correction> In
the
to
the
cla~~~al
to hc
‘)}.
the harmonic
of the
the
dimcnGonal
’ -O(T‘
437-417 Compan\
I YX I
WC calculate
the
‘,‘)+Y5] In
27 August
method.
region,
temperature)
experiment
publication
anharmomcitv
tn(0.1Y5rrj
shear
the average to
for
temperature
I --[3.5 C’,, is the
accnpted
ON TWO-DIMENSIONAL
(19x7)
Publishing
r-
obtain 7’,
c’( r;,L
Suh.\titutlng the
I
,3T)
‘, ‘;
‘i,
‘I,,.
‘il I\
‘; ~-4 X * IO3 corrc\pond-
melting
temperature
16X c2 I( r;,L
,3). which
(i c
the
is conxistcnt
et al.
A recent experiment by Grimes and Adams [l] (GA) on the two-dimensional electron lattice (Wigner solid) on the surface of helium four has revived interest in the melting or two-dimensional solid. They observed that the electron lattice is formed at the temperature T corresponding to r = 131 -m7, where I‘ is the ratio of the average Coulomb energy and the thermal energy; r = e’( ak,,T) ‘. Earlier, Platzman and Fukuyama [2] predicted that the solid would be formed at r - 3, by studying the instability in the transverse phonon in the electron solid, which overestimated grossly the stability of the solid phase. It is now believed that the transition observed by Grimes and Adams is rather triggered by the dissociation of bound dislocation pairs in the two dimensional solid [3]. Indeed Thouless [4] has already applied this idea to the two dimensional electron lattice and obtained the melting temperature r,,, = (4?+’
na,?;C,,
where n is the electron density, a,, is the lattice constant and C, is the shear modulus of the electron lattice. Making use of C, calculated by Bonsall and Maradudin [5] (BM). Thouless obtained the corresponding r = 78.71. This I was in good agreement with the results of computer simulation [6,7] and was *
Work
supported
by the
National
0039-6028/82/0000-0000/$02.75
Science
Foundation
under
0 1982 North-Holland
No
DMR
7Y- IhYO3
not far from the GA result. This already indicates that the melting of two-dimensional electron lattice is controlled the the Kosterlitz-Thouless mechanism [2]. More recently Morf [S] calculated the temperature dependence of the shear modulus by the Monte Carlo method and found C,(T) is well approximated by C,(T) = C,(I---3O.gr ~ ‘) in the low temperature region where C, is the shear modulus at T = 0 K. If C,(T) is substituted in the Thouless formula, it gives now r = 110 for the melting temperature, which farther improves the agreement between the theory and the experiment. The object of this paper is to calculate the temperature dependence of the shear modulus within a framework of the perturbation theory. The twodimensional electron lattice is strongly anharmonic. Therefore, in general. the shear modulus calculated by Bonsall and Maradudin [5t has to be corrected by anharmonic effects. It is shown that in the classical limit. the harmonic result becomes exact at T = 0 K. In the same limit the finite temperature corrections can be expanded in terms of diagrams as shown in fig. 1 [9,10]. This expansion corresponds to the expansion in powers of r ‘: each independent phonon loop contributes a factor of I - ‘. We find to the lowest order in l7 -I, C(T)
= C,[ 1 - [3.5 ln(0.195F
Y’) ,
(1)
where C, is the harmonic shear modutus due to BonsaII and Maradudin f5]. It is of interest to point our the lowest order quantum correction (i.e., r, “) appears already in the logarithm. This arised from the thermal average of a In q term in the lowest order correction to the transverse phonon velocity 21~. Inserting rs = 4.8 X lo3 corresponding to the experiment by Grimes and Adams [I] and I = I, obtained by Thouless int he logarithm of eq. (1X we obtain C(T) = C,fl--9OY’f. Although she coefficient of Y ’ appears substantially larger than that formed by Morf, our expression predicts rM given by I = id& which is close but somewhat larger that the GA result. We find that the second diagram in fig. 1 contains a logarithmic divergence, if we expand the self-energy integral in powers of q. A more proper treatment of the diagram yields the correction to the transverse sound velocity depending logarithmically in q. At finite temperatures the corresponding shear modulus is given by C, = nav: , where the In q term in tiL is replaced by its thermal average resulting eq. ( 1).
a
_o _-.._
Fig. I. (a) The Feynman cubic interaction term.
diagram
due to quartic
interaction
term. (b) The Feyman
diagram
due to
We start with a model that the electrons are confined in the two-dimensional surface and interacting with each other via the Coulomb interaction. Furthermore we assume that they form a regular triangular lattice with the lattice constant a,, at low temperatures and that the electron density is sufficiently small so that the exchange effect is completely neglected. Then the Hamiltonian for an electron lattice is given by d’r / ly, -ri
’
12.)
where P, = ar,/at. II (=(&a:) ‘) is the electron density, and y, is the positionn of the ith electron. Expanding q around the equilibrium position R,. we can rewrite ey. (2) as N-
VfH,,+H,
I,': ;
+I'$ t...,
(3)
c, l I#/lR,-&I
(4) where
(5) and we have neglected the anharmonic terms arising from the last term (the positive back ground term) in eq. (2) for simplicity. The first term gives the Madelung energy, while the second term gives the harmonic phonons. In particular, BM have diagonalized H,,and found in the low monentum limit the phonons split into the transverse and the longitudinal phonon with dispersion q’(q)
= 0; q’ = 0,0362967w,;( u,,&.
w;(q) = $(q,q)
- w;(O.181483)
where C$ = (2ar’n/mr1,,), lattice. We shall here calculate
(6) (7)
(joy)‘.
the plasma
frequency
the anharmonic
inthe
corrections
case of the triangular to the above dispersion.
For this purpose
it is convenient
to introduce
the Green
function
defined
(8)
CUB(Ic,t)=i(Tu,(k,r)tcp(-k,O)). where u( k, t ) is defined u(k,r)
=N-‘/*zu,(t)
Within
the harmonic
by (9)
exp(ik.R,).
approximation
G”“( k, w) is then given by
G,;“(k.w,,)=~~“:‘i’[~~+w(k)]“+~;ellIw~+wi(k)]
(10)
‘1,
where e, = i, e, = k”X e, and k^= k/k. In the presence of H,, Hz, etc., the renormalized lated as
Green
function
is calcu-
-’ - fI(k,o,,)
r~(k,w,,)-‘=~,(k,w,,)
(11)
n where II( k, a,,) is the self-energy correction. The lowest order corrections shown in figs. la and lb and are given by [9,10] II;~(k,o,)
= -12TX
~~~(~.#~)
= l&z-C
P
I’
J
as
s
d’q --+Ypy*(k,-k.q, (24”
-----+“YS(k,q, d2g (2a)
-4)
-k-q)
G,:“tq,w,,),
@“‘“(-k,
are
(12)
-q.k$-q)
XG$tq,q,) G,s”(k+q,w,+,),
(13)
where @“fl@(k, -k,q, -4) and aay8(k,, k,, k,) are the Fourier transforms of (I)@‘@ and Qavs in eq. (5) respectively. In the classical limit eqs. (12) and (13) a;; further simplified as KI;~(k,c+,)=
ll;f(k.w,)
-E
*cD”@+(k, FE 1 (24”
=FJf&
where q’ = q + k. Substituting the phonon
-k,q,
-q)
{&w,-“(q)
WY”(k,q,-k-q)@rY(-k,-q,k+q)
spectra
given in eqs. (6) and (7). we can further
r-qy( k, 0)
P4T2
“y
2( Pru:)’
WR)(k*R’)
R.R’
aR
*
a; Y
x [ 1 - eosfq~ R’)] [I - cos(q~ R’ff .
(17)
Here we have neglected the contributions from the longitudinal mode in eqs. ( 16) and (17) which are of order of magnitude smaller than of the transverse mode. Futhermore we have dropped the contribution from the positive back ground potential. Since the shear modulus is related to 1I“‘f k 1 with k I/x, we shall limit to this particular case and we find r~~~~k~~k~l~~
- ISm u:T-‘k;,
(1st
II;i(k)/kIIx=:
[ 3.5 In
(19)
where In y = 0.577... is the Euler constant. Note that eq. (19) contains a logarithmic divergence in k. It appears that all of the higher order diagrams contains the higher power of In k in the limit k tends to zero and we cannot expand the self-energy diagrams in power of k. Recalling that the shear modulus C, is given in terms of the transverse phonon velocity as C,=mv:,
we obtain C(T)=CC,(I-[3.51n~0.195Tr,‘.“~‘“)+95]
_I +o(I?-‘)I.
where we have inserted ei corresponding to that of the thermal logarithmic divergence in eq. (20) is replaced by (ln( l/rk))
_r:ln( Du,)
= ln(0.195
(20) phonon:
the
IYrJ I/?).
We should stress that the In r5 term is the most important quantum correction to the shear modulus; the other quantum corrections are given in powers of r,- i*
Finally, if we substitute in eq. (20) the rs value corresponding to the experiment by Grimes and Adams, we shall obtain a coefficient of I ~ ’ about 3 times larger than the one obtained by Morf [8] in his Monte Carlo analysis. Furthermore it appears that Morf did not notice any logaritmic dependence on I. For the moment we do not know the origin of this discrepancy, although a part of the discrepancy may be due to inaccuracy in our numerical analysis. In any case a more accurate treatment of integral (17) is certainly desirable.
References [I] [2] [3] [4] [5] [6] [7] [8] [9] [IO]
C.C. Grimes and CJ. Adams, Phys. Rev. Letters 42 (1979) 795. P.M. Platzman and H. Fukuyama. Phys. Rev. BIO (1974) 3150 J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) I 181. D.J. Thouless, .I. Phys. Cl I ( 1978) I 189. L. Bonsall and A.A. Maradudin. Phys. Rev. B15 (1977) 1959. R.W. Hackney and T.R. Brown, J. Phys. C8 (1975) 1813. R.C. Gann, S. Chakravarty and G.V. Chester. Phys. Rev. B20 ( 1979) 326. R.H. Morf, Phys. Rev. Letters 43 (1979) 931. A.A. Maradudin and A.E. Fein. Phys. Rev. I28 (1962) 2589. R.A. Cowley, Advan. Phys. I2 (1963) 421.