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Structure factors and elementary excitations of condensed matter
Physica 120B (1983) 296-299 North-Holland Publishing Company
S T R U C T U R E F A C T O R S AND E L E M E N T A R Y E X C I T A T I O N S OF CONDENSED M A T T E R A. I S I H A R A and Y. N A K A N E Statistical Physics Laboratory, Department of Physics, State University of New York at Buffalo, Buffalo, New York, 1426,0, USA
An inelastic electron scattering experiment performed at Princeton on thin films of the quasi one-dimensional organic metal tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) reveals an anomalous negative plasmon dispersion. On the other hand, a recent neutron scattering experiment on liquid 4He at Chalk River shows that the structure factor of this liquid varies with temperature and m o m e n t u m in an interesting way. In particular, it is observed that at small m o m e n t u m and above a certain temperature, the structure factor shows an anomalous dip before it starts increasing towards the first peak which is higher for higher temperatures. The structure factors in these completely different systems are related in an analogous way to their elementary excitations which determine the anomalous properties.
1. Introduction
Based on a microscopic approach, Samulski and Isihara [1] evaluated in 1977 the excitation energy and structure factor of liquid helium at finite temperature. They reported that the structure factor shows a very interesting temperature variation particularly at small m o m e n t a and also around the first peak. According to their fig. 2, which gives the structure factor for T/T~ = 0.3, 0.2, and 0.001, the first peak is higher for higher temperatures below the A-point. The figure shows also that at small m o m e n t a the structure factor flattens and approaches a small but finite value which is larger for higher temperature. More recently, Svensson, Sears, Woods, and Martel [2] studied in detail the temperature variation of the structure factor by neutron diffraction. Their data give temperature variations of the type predicted by Samulski and Isihara: The structure factor shows a higher first peak at higher temperatures below the A-point and above 2.5 K a dip before reaching the first peak. Isihara [3] has given a theoretical account of such an anomalous behavior and has reported a formula which relates the point where the dip begins with a constant in the energy dispersion relation. This constant should be positive in order to observe such a dip. Therefore, Isihara
has given in effect new support to the anomalous character of the energy dispersion and has shown how this constant could be determined by neutron diffraction. Although the theoretical results of isihara agree very well with the experimental data, it is necessary to examine the following two points. First, he has used an approximate expression for the eigenvalues Aj(q) of the particle propagator. This expression, consisting of the contributions from the excited as well as condensed particles, represents the correct limit for T = 0 for nonzero momentum. However, the contribution from the higher order terms must be investigated because they are temperature dependent. Second, the zeroth eigenvalue Ac~(q) must be treated in a special way because in the limit of vanishing momentum, the approximate eigenvalue expression becomes incorrect. Hence, in order to study the effect of Bose condensation, the role played by this particular eigenvalue has to be studied. The first motivation of the present article is in investigating these two points and to report a new improved formula for the structure factor. Our second motivation for the present article comes from an entirely different experiment. According to the recent electron scattering experiment, of Ritsko et al. [4], on crystalline
0378-4363/83/0000-0000/$03.00 © 1983 North-Holland and Yamada Science Foundation
297
A . Isihara and Y. Nakane / Structure factors and elementary excitations
films of organic charge-transfer salt T F F - T C N Q (tetrathiafulvalence-tetracyanoquinodimethane) at 300 K, the plasmon energy in the direction of the conducting axis is constant for zero m o m e n tum and decreases when m o m e n t u m increases. Such a plasmon dispersion is anomalous in comparison with the ordinary positive dispersion in bulk plasmas. W e shall show in the present article that the structure factor of such a onedimensional electron system is related to the plasmon energy, in a form analogous to that of liquid helium. We shall then discuss s o m e possible mechanisms for the negative dispersion. For simplicity, we shall adopt the natural units in which h = l, and 2m = 1 where m is the particle mass.
represents our second i m p r o v e m e n t over the previous theory [3]. Here, the first and second terms add the correct contribution to and subtract the approximate one from the first term in eq. (2.4). With this correction, the structure factor reaches the limiting value s(o) -
1
~o(O)
n/3 1 + A0u(0)
= nkTK,
(2.6)
where K is the isothermal compressibility. Introducing eq. (2.2) into eq. (2.4) and performing the summation, we arrive at a new low t e m p e r a t u r e formula for the structure factor:
2
S(q) = ~2[1 + 2f(e)]O(q) 2. Structure factor of liquid 4He
12B
We improve the previous theory first by using the eigenvalue expression Aj + AA/ instead of Aj which was used before [3]. For low temperatures, we find
Aj = 2nAj,
Aj
--
q2 q4+ (2vrj//3)2,
ZlAj = - 12(B/fl )A 2 + 16(B/fl )q2A 3j,
(2.1) (2.2)
where fl = 1/kT, q is m o m e n t u m and
q4 [1 + 2 q4 ] [ 3 e 2C qaJ f(E) [1 + f(E)]
8B q6 ~ (E2 _ q4)2 l 1 q- 2 f ( q 2 ) ] _
jr_
+ F(q).
(2.7)
Without G(q), the first term reduces itself to the previous formula of Isihara [3]. Here, f ( e ) is the Bose distribution of massless quasiparticles with energy e = q [ q Z + 2 n u ( q ) ] u2. The natural a p p e a r a n c e of such quasiparticle energies is very interesting in view of our microscopic approach which starts with actual 4He particles. The functions G(q) and F(q) are given by
(2.3)
B - rtl G5/2 H G3/2 "
G(q) = 1
B
]3
qZ[3e4 + 6e2q 4- q8l E2[E2 __ q412
(nl/n) is the fraction of the particles in the excited states, and Gs/z/G3/2 = 1.34/2.61. The structure factor shall be evaluated from F(q)=
t~C~ [61 -- 62q], 2 i-q2 167rn~1/26,/2]
,
(2.8) (T < T~) ( T > T,)
S ( q ) = y n ~i ( l + Aju
[l + A~u]2j + F ( q ) ,
(2.4)
(2.9) where
where u(q) is the Fourier transform of the interaction potential and 1
A0
F ( q ) - fin l + A 0 u
1
2
fl [q2 + 2nu(q)]
61/2= 1.105 ( ~ A T ~ )
(2.1(,)
(2.5) and 31 and 62 are the coefficients in the energy
298
A. lsihara and Y. Nakane
/ Structure factors and elementary excitations
dispersion which is f o u n d to be of the form:
S ( q ) = q~ [1 + 2f(E)]
E = c q [ 1 + ~lq 2 - 3 2 q S + • • • ] ,
where
(2.11)
c being the s o u n d velocity. C o r r e s p o n d i n g l y , the structure factor for small m o m e n t u m is SS(0) (q) _
(2.12)
1 + slq 2 + szq 3 + • • • .
T h e s e results lead us to the following conclusions. First, for T < Ta, s~ > 0. H e n c e , there is no dip. Second, for T > T~ there is a temp e r a t u r e T~ a b o v e which Sl < 0. This " i n v e r s i o n " t e m p e r a t u r e is given by
lc[
.
T, - k X/2--~l
1
2cV'~ll
~
•
(2.13)
T h e coefficient 6l in eq. (2.11) has been f o u n d to be 1.5A 2. T h e r e f o r e , we estimate ~1c 2 - 10.3 and B - 0.51. H e n c e , the correction term in eq. (2.13) is of o r d e r - 0 . 1 0 . T h a t is, we n o w estimate TI be a r o u n d 2.5 K, s o m e 10% reduction. Such a reduction is f a v o r e d by the n e u t r o n diffraction data [2].
3. Structure factor and plasmon excitation in TTF-TCNQ O n e can evaluate the structure factor of a o n e - d i m e n s i o n a l c o n d u c t o r such as T T F - T C N Q in a similar way. If the electrons are d e g e n e r a t e o n e can easily show that the first a p p r o x i m a t e expression for the eigenvalues is [ q2(q + 2pv)2 + (27rj]/3)2
1
Aj(q) = ~
In q2(q _ 2pv)2 + (27rj/fl)2
,
(3.1)
e = ~
{4pvu(q) + "n'(q2 + 4p2)} 1/2
1
(3.3)
is the excitation e n e r g y which represents plasmons. It is very interesting to observe that eq. (3.3) is of the f o r m of the first a p p r o x i m a t e structure factor for liquid 4He. By expressing this f o r m in the familiar F e y n m a n type expression in which the e n e r g y is given as a function of the structure factor, we can see the existence of boson-like excitations. T h e a p p e a r a n c e of such excitations m a y be u n d e r s t o o d f r o m a different viewpoint based on a variational m e t h o d . T h e electron eigenfunction m a y be a s s u m e d to be a p r o d u c t of the g r o u n d state free particle eigenfunction and an excited state function. Since the f o r m e r is antisymmetric, the latter should be symmetric with respect to electron exchanges. Such an a p p r o a c h has been taken by Krivnov and Ovchinikov [5] in a way parallel to the F e y n m a n theory of liquid helium. H o w e v e r , their a p p r o a c h is limited to absolute zero as in F e y n m a n ' s case. T h e o b s e r v e d plasmon dispersion in T T F T C N Q should be related to its electron behavior and distribution. Its strand structure suggests that the effective potential at large distances may be of the form a/q 2, w h e r e a is of o r d e r eZN, N being the n u m b e r of strands per unit area perpendicular to the chain axis. Also, Aj(q) m a y be given a small correction for additional electron f r e e d o m . W e assume a correction factor e x p ( - b ~ p v q 2) and e x p a n d this factor for small q to arrive at o) 2 = o~g + p2v(1 - ab)q 2 '
w h e r e Pv = ~ n is the Fermi m o m e n t u m . In view of the logarithmic function and o u r interest, we m a k e a small m o m e n t u m a p p r o a c h and neglect the corrections AAj and F in eq. (2.7) to arrive at
(3.2)
(3.4)
where o~2 = 2an. H e n c e , for ab > 1 there can be a dip. O n the o t h e r hand, the electrons in T I T T C N Q may be r e p r e s e n t e d by a tight-binding model. Williams and Bloch [6] and Brosens and
A. Isihara and Y. Nakane / Structure factors and elementary excitations
Devreese [7] have shown for such a model that the dispersion in the chain direction can indeed decrease.
Acknowledgment This work was supported by the O N R under contract No: N00014-79-C-0451.
References [1] T. Samulski and A. Isihara, Physica 86A (1977) 257. [2] E.C. Svensson, V.F. Sears, A.D.B. Woods and P. Martel, Phys. Rev. B21 (1980) 3638;
299
E.C. Svensson and A.F. Murray, Physica 108B (1981) 1317. [3] A. Isihara, Physica 106B (1981) 161; 108B (1981) 1385. [4] J.J. Ritsko, D.J. Sandman, A.J. Epstein, P.C. Gibbons, S.E. Schnatterly and J. Fields, Phys. Rev. Lett 34 (1975) 1330. [5] V. Ya. Krivnov and A.A. Ovchinikov, in Quasi OneDimensional Conductors II, ed. S. Varisic, A. Bjelis, J.R. Cooper and B. Leontic (Springer-Verlag, Berlin, 1979) p. 87. [6] P.F. Williams and A.N. Bloch, Phys. Rev. B10 (1974) 1097, Phys. Rev. Lett 36 (1976) 64. [7] F. Brosens and J.T. Devreese, Phys. Rev. B19 (1979) 762.
S T R U C T U R E F A C T O R S AND E L E M E N T A R Y E X C I T A T I O N S OF CONDENSED M A T T E R A. I S I H A R A and Y. N A K A N E Statistical Physics Laboratory, Department of Physics, State University of New York at Buffalo, Buffalo, New York, 1426,0, USA
An inelastic electron scattering experiment performed at Princeton on thin films of the quasi one-dimensional organic metal tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) reveals an anomalous negative plasmon dispersion. On the other hand, a recent neutron scattering experiment on liquid 4He at Chalk River shows that the structure factor of this liquid varies with temperature and m o m e n t u m in an interesting way. In particular, it is observed that at small m o m e n t u m and above a certain temperature, the structure factor shows an anomalous dip before it starts increasing towards the first peak which is higher for higher temperatures. The structure factors in these completely different systems are related in an analogous way to their elementary excitations which determine the anomalous properties.
1. Introduction
Based on a microscopic approach, Samulski and Isihara [1] evaluated in 1977 the excitation energy and structure factor of liquid helium at finite temperature. They reported that the structure factor shows a very interesting temperature variation particularly at small m o m e n t a and also around the first peak. According to their fig. 2, which gives the structure factor for T/T~ = 0.3, 0.2, and 0.001, the first peak is higher for higher temperatures below the A-point. The figure shows also that at small m o m e n t a the structure factor flattens and approaches a small but finite value which is larger for higher temperature. More recently, Svensson, Sears, Woods, and Martel [2] studied in detail the temperature variation of the structure factor by neutron diffraction. Their data give temperature variations of the type predicted by Samulski and Isihara: The structure factor shows a higher first peak at higher temperatures below the A-point and above 2.5 K a dip before reaching the first peak. Isihara [3] has given a theoretical account of such an anomalous behavior and has reported a formula which relates the point where the dip begins with a constant in the energy dispersion relation. This constant should be positive in order to observe such a dip. Therefore, Isihara
has given in effect new support to the anomalous character of the energy dispersion and has shown how this constant could be determined by neutron diffraction. Although the theoretical results of isihara agree very well with the experimental data, it is necessary to examine the following two points. First, he has used an approximate expression for the eigenvalues Aj(q) of the particle propagator. This expression, consisting of the contributions from the excited as well as condensed particles, represents the correct limit for T = 0 for nonzero momentum. However, the contribution from the higher order terms must be investigated because they are temperature dependent. Second, the zeroth eigenvalue Ac~(q) must be treated in a special way because in the limit of vanishing momentum, the approximate eigenvalue expression becomes incorrect. Hence, in order to study the effect of Bose condensation, the role played by this particular eigenvalue has to be studied. The first motivation of the present article is in investigating these two points and to report a new improved formula for the structure factor. Our second motivation for the present article comes from an entirely different experiment. According to the recent electron scattering experiment, of Ritsko et al. [4], on crystalline
0378-4363/83/0000-0000/$03.00 © 1983 North-Holland and Yamada Science Foundation
297
A . Isihara and Y. Nakane / Structure factors and elementary excitations
films of organic charge-transfer salt T F F - T C N Q (tetrathiafulvalence-tetracyanoquinodimethane) at 300 K, the plasmon energy in the direction of the conducting axis is constant for zero m o m e n tum and decreases when m o m e n t u m increases. Such a plasmon dispersion is anomalous in comparison with the ordinary positive dispersion in bulk plasmas. W e shall show in the present article that the structure factor of such a onedimensional electron system is related to the plasmon energy, in a form analogous to that of liquid helium. We shall then discuss s o m e possible mechanisms for the negative dispersion. For simplicity, we shall adopt the natural units in which h = l, and 2m = 1 where m is the particle mass.
represents our second i m p r o v e m e n t over the previous theory [3]. Here, the first and second terms add the correct contribution to and subtract the approximate one from the first term in eq. (2.4). With this correction, the structure factor reaches the limiting value s(o) -
1
~o(O)
n/3 1 + A0u(0)
= nkTK,
(2.6)
where K is the isothermal compressibility. Introducing eq. (2.2) into eq. (2.4) and performing the summation, we arrive at a new low t e m p e r a t u r e formula for the structure factor:
2
S(q) = ~2[1 + 2f(e)]O(q) 2. Structure factor of liquid 4He
12B
We improve the previous theory first by using the eigenvalue expression Aj + AA/ instead of Aj which was used before [3]. For low temperatures, we find
Aj = 2nAj,
Aj
--
q2 q4+ (2vrj//3)2,
ZlAj = - 12(B/fl )A 2 + 16(B/fl )q2A 3j,
(2.1) (2.2)
where fl = 1/kT, q is m o m e n t u m and
q4 [1 + 2 q4 ] [ 3 e 2C qaJ f(E) [1 + f(E)]
8B q6 ~ (E2 _ q4)2 l 1 q- 2 f ( q 2 ) ] _
jr_
+ F(q).
(2.7)
Without G(q), the first term reduces itself to the previous formula of Isihara [3]. Here, f ( e ) is the Bose distribution of massless quasiparticles with energy e = q [ q Z + 2 n u ( q ) ] u2. The natural a p p e a r a n c e of such quasiparticle energies is very interesting in view of our microscopic approach which starts with actual 4He particles. The functions G(q) and F(q) are given by
(2.3)
B - rtl G5/2 H G3/2 "
G(q) = 1
B
]3
qZ[3e4 + 6e2q 4- q8l E2[E2 __ q412
(nl/n) is the fraction of the particles in the excited states, and Gs/z/G3/2 = 1.34/2.61. The structure factor shall be evaluated from F(q)=
t~C~ [61 -- 62q], 2 i-q2 167rn~1/26,/2]
,
(2.8) (T < T~) ( T > T,)
S ( q ) = y n ~i ( l + Aju
[l + A~u]2j + F ( q ) ,
(2.4)
(2.9) where
where u(q) is the Fourier transform of the interaction potential and 1
A0
F ( q ) - fin l + A 0 u
1
2
fl [q2 + 2nu(q)]
61/2= 1.105 ( ~ A T ~ )
(2.1(,)
(2.5) and 31 and 62 are the coefficients in the energy
298
A. lsihara and Y. Nakane
/ Structure factors and elementary excitations
dispersion which is f o u n d to be of the form:
S ( q ) = q~ [1 + 2f(E)]
E = c q [ 1 + ~lq 2 - 3 2 q S + • • • ] ,
where
(2.11)
c being the s o u n d velocity. C o r r e s p o n d i n g l y , the structure factor for small m o m e n t u m is SS(0) (q) _
(2.12)
1 + slq 2 + szq 3 + • • • .
T h e s e results lead us to the following conclusions. First, for T < Ta, s~ > 0. H e n c e , there is no dip. Second, for T > T~ there is a temp e r a t u r e T~ a b o v e which Sl < 0. This " i n v e r s i o n " t e m p e r a t u r e is given by
lc[
.
T, - k X/2--~l
1
2cV'~ll
~
•
(2.13)
T h e coefficient 6l in eq. (2.11) has been f o u n d to be 1.5A 2. T h e r e f o r e , we estimate ~1c 2 - 10.3 and B - 0.51. H e n c e , the correction term in eq. (2.13) is of o r d e r - 0 . 1 0 . T h a t is, we n o w estimate TI be a r o u n d 2.5 K, s o m e 10% reduction. Such a reduction is f a v o r e d by the n e u t r o n diffraction data [2].
3. Structure factor and plasmon excitation in TTF-TCNQ O n e can evaluate the structure factor of a o n e - d i m e n s i o n a l c o n d u c t o r such as T T F - T C N Q in a similar way. If the electrons are d e g e n e r a t e o n e can easily show that the first a p p r o x i m a t e expression for the eigenvalues is [ q2(q + 2pv)2 + (27rj]/3)2
1
Aj(q) = ~
In q2(q _ 2pv)2 + (27rj/fl)2
,
(3.1)
e = ~
{4pvu(q) + "n'(q2 + 4p2)} 1/2
1
(3.3)
is the excitation e n e r g y which represents plasmons. It is very interesting to observe that eq. (3.3) is of the f o r m of the first a p p r o x i m a t e structure factor for liquid 4He. By expressing this f o r m in the familiar F e y n m a n type expression in which the e n e r g y is given as a function of the structure factor, we can see the existence of boson-like excitations. T h e a p p e a r a n c e of such excitations m a y be u n d e r s t o o d f r o m a different viewpoint based on a variational m e t h o d . T h e electron eigenfunction m a y be a s s u m e d to be a p r o d u c t of the g r o u n d state free particle eigenfunction and an excited state function. Since the f o r m e r is antisymmetric, the latter should be symmetric with respect to electron exchanges. Such an a p p r o a c h has been taken by Krivnov and Ovchinikov [5] in a way parallel to the F e y n m a n theory of liquid helium. H o w e v e r , their a p p r o a c h is limited to absolute zero as in F e y n m a n ' s case. T h e o b s e r v e d plasmon dispersion in T T F T C N Q should be related to its electron behavior and distribution. Its strand structure suggests that the effective potential at large distances may be of the form a/q 2, w h e r e a is of o r d e r eZN, N being the n u m b e r of strands per unit area perpendicular to the chain axis. Also, Aj(q) m a y be given a small correction for additional electron f r e e d o m . W e assume a correction factor e x p ( - b ~ p v q 2) and e x p a n d this factor for small q to arrive at o) 2 = o~g + p2v(1 - ab)q 2 '
w h e r e Pv = ~ n is the Fermi m o m e n t u m . In view of the logarithmic function and o u r interest, we m a k e a small m o m e n t u m a p p r o a c h and neglect the corrections AAj and F in eq. (2.7) to arrive at
(3.2)
(3.4)
where o~2 = 2an. H e n c e , for ab > 1 there can be a dip. O n the o t h e r hand, the electrons in T I T T C N Q may be r e p r e s e n t e d by a tight-binding model. Williams and Bloch [6] and Brosens and
A. Isihara and Y. Nakane / Structure factors and elementary excitations
Devreese [7] have shown for such a model that the dispersion in the chain direction can indeed decrease.
Acknowledgment This work was supported by the O N R under contract No: N00014-79-C-0451.
References [1] T. Samulski and A. Isihara, Physica 86A (1977) 257. [2] E.C. Svensson, V.F. Sears, A.D.B. Woods and P. Martel, Phys. Rev. B21 (1980) 3638;
299
E.C. Svensson and A.F. Murray, Physica 108B (1981) 1317. [3] A. Isihara, Physica 106B (1981) 161; 108B (1981) 1385. [4] J.J. Ritsko, D.J. Sandman, A.J. Epstein, P.C. Gibbons, S.E. Schnatterly and J. Fields, Phys. Rev. Lett 34 (1975) 1330. [5] V. Ya. Krivnov and A.A. Ovchinikov, in Quasi OneDimensional Conductors II, ed. S. Varisic, A. Bjelis, J.R. Cooper and B. Leontic (Springer-Verlag, Berlin, 1979) p. 87. [6] P.F. Williams and A.N. Bloch, Phys. Rev. B10 (1974) 1097, Phys. Rev. Lett 36 (1976) 64. [7] F. Brosens and J.T. Devreese, Phys. Rev. B19 (1979) 762.