- Email: [email protected]
Theory of AC conductivity of 2D Wigner solids
Physica B 249—251 (1998) 788—791
Theory of AC conductivity of 2D Wigner solids Motohiko Saitoh* Department of Physics, Graduate School of Science, Osaka University, 1-16 Machikaneyama, Toyonaka 560, Japan
Abstract First-principle calculation is made of an AC conductivity of electrons in quasi-two-dimensional Wigner solids at low temperatures, when electrons are scattered by ionized impurities. Dispersion relation of the Wigner phonons are solved within the self-consistent harmonic approximation which leads to an energy gap in the small wave vector limit. This gap is directly related to the threshold frequency of the AC conductivity, which is inversely proportional to the temperature and implies the collective localization at very low temperatures. The absorption bandwidth is primarily determined by the transverse mode of the Wigner phonons and the longitudinal mode contributes to a relatively small background. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Wigner solid; Semiconductor heterostructure
1. Introduction Transport properties of the quasi-two-dimensional Wigner solid has attracted much attention recently [1,2]. However, the theoretical understanding is rather qualitative than quantitative, based on, e.g., the CDW theory which is valid only for the degenerate electron system, while the picture of the Wigner solids applies to the non-degenerate electrons where the distance between electrons is very far as compared to the effective Bohr radius. No first-principle calculation of the conductivity was tried so far because of the difficulty of treating the strong localization. We present here the first-principle calculation of the AC conductivity at very low temperature by * Fax: #81-6-850-5764; e-mail: [email protected].
explicitly considering the localization properties on the basis of the self-consistent harmonic approximation to the Wigner phonon dispersion. If the electron scattering is dominated by the impurities, then the phonon dispersion has an energy gap in the small wave vector limit. This energy gap stabilizes the lattice in the sense that the density—density correlation has long-range order, and also leads to the vanishing of the DC conductivity.
2. Dispersion relation of the Wigner phonons We assume that quasi-two-dimensional electrons are confined to the semiconductor side of the interface, z"0, and are scattered by ionized impurities located at (r , z ) in the insulator side, where r is the j j j two-dimensional position vector and z (0. The j
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 3 1 5 - 9
M. Saitoh / Physica B 249—251 (1998) 788—791
interaction potential is given by u(r, z)" + u (r!r , z!z ), *.1 j j r j,zj
(1)
where u (r, z) is the impurity potential. Effects of *.1 the scattering by impurities will be characterized by the cummulant average: º(r)"Su(r, 0)u(0, 0)T !Su(r, 0)T Su(0, 0)T , s s s
(2)
where the angular brackets indicate the random average over the impurity positions. If the random average is characterized by Sz T "!d and j s S(z #d)2T "*2/2, then the Fourier transform of j s º(r) is calculated to be º "(pe2/i)2(n /q2)erfc(d/*), q i
(3)
where n is the surface density of impurities, i the i average of the dielectric constants of the semiconductor and the insulator, and erfc(x) is the complementary error function with the normalization erfc(0)"1. For the delta doped case, d/* may be larger than unity, and º can be very small. q The dispersion relation of the jth mode of the Wigner phonons in the presence of impurities can be calculated from the minimization principle of the free energy. Namely, we first rewrite the electron coordinates in the Hamiltonian formally in terms of the normal coordinates of the still unfixed Wigner phonons. Then, by using the finite-temperature path-integral method and the Feynman—Jensen inequality, we express the approximate free energy in terms of the Wigner phonon frequencies. The minimization of the free energy with respect to the phonon frequencies leads to [3] 1 (q ) e (p))2 j X2(p)"! + + » e*q > R j q SR m E0 q
1@T 0
dq(1!cos p ) R cos lq)/ (R, q), q
/q(R,q)"Se*q > uR(q)e~*q > u0(0)T
C A A BBD
(q ) e (p))2 + j + "exp ! 2mNp X (p) sinh(+X (p)/2¹) j ,j j ] cosh
+X (p) j !cos pR cosh +X (p) j 2¹
1 ] q! 2¹
,
(5)
where uR(q) is the positional deviation from the lattice site R at the imaginary time q, N the number of the lattice sites and summation over p is limited to the first Brillouin zone. The physical meaning of this equation is clear. Namely, the phonon frequency is determined by the product of the second derivative of the sum of the Coulombic and the impurity potentials, and the density correlation function. Note that the first derivative of the potentials determines the equilibrium positions of the lattice sites. Since the Debye—Waller factor of the density correlation function is a function of the Wigner phonon spectra, the coupled Eqs. (4) and (5) must be solved so that X (p)’s become selfj consistent. This scheme is sometimes called the self-consistent harmonic approximation. Because of impurity scattering, the energy gap v will be induced in the phonon spectra: (6)
where u (p) is the phonon frequency of the pure j Wigner crystal [3]:
1 (q ) e (p))2 j # + + º e*q > R SR q q m
P
where » "2pe2/i q is the Fourier transform of the q s Coulomb potential with i the dielectric constant s of the semiconductor, R the lattice site vector of the crystal, e (p) the polarization vector of the jth mode j phonon, m the effective electron mass, S the system area, ¹ the temperature in energy units, l"2pn¹ the Matsubara frequency, and /q(R, q) is the density-correlation function defined by
X2(p)"u2(p)#v2, j j
](1!cos p ) R)/q(R, 0)
]
789
u "u Jp/p , l 0 0 (4)
u (p)"(J2/4)u p/p . t 0 0
(7)
Here u is the longitudinal frequency at the Bril0 louin zone edge: u "(2pe2np /i m)1@2 with n the 0 0 s
790
M. Saitoh / Physica B 249—251 (1998) 788—791
surface density of electrons and p "J4pn. The 0 gap frequency v is evaluated by setting p"0 in Eq. (4) as q2 1 v2K + + º e*q > R e~l2q2@2, 2m¹ SR q q
(8)
with (9)
where the time-dependent part of /q(R, q) has been neglected and v is assumed to be much larger than u . The result for v is 0
A B
/q(R, q)Ke~l2q2@2
C
D
+q2 J (pR) cosh(+X (p)(q!1/2¹)) j + 0 , ] 1# 4mNp X (p) sinh(+X (p)/2¹) j j ,j (13) where J (x) is the Bessel function of the first kind. 0 Putting this into (12), we obtain
1 + + + K , l2" 2mNp X (p) mv ,j j
v"
important, and to the lowest order in q2 we have
AB
pe2 2 n d i erfc . i * 4p+¹
(10)
This result is valid for (pe2/i)2(n /4p+u ) erfc(d/*)< i 0 ¹. At very low temperatures, the Wigner crystal is stabilized, since /q(R, 0) is non-zero for large R.
p¹v3 v2 D(u), M(u)"! i# +u u
(14)
where D(u)"N~1+ d(u2!X (p)2) j p ,j 8 " h(Ju2/8#v2!u) 0 u2 0 (u2!v2) # h(Ju2#v2!u) h(u!v) 0 4u2 0
C
D
(15) 3. AC conductivity The AC conductivity p (u) is expressed in terms xx of the memory function M(u) as p (u)"(ne2/m)(iu#d#M(u))~1, xx
(11)
where d is a positive infinitesimal, and in the selfconsistent harmonic approximation the memory function reads [4] iq2 º e*q > R muS q
M(u)"+ + R
q
CP
]
1@2T dq(cosh +uq!1)/ (R,q) q 0
P
+u = !i sinh dt e~*+(u~*d)t/ q 2¹ 0
A
1 ] R, it# 2¹
BD
.
(12)
In order to obtain a finite Re M(u), the time dependence of the density-correlation function is
and h(x) is the step function. We arrive at the final result: ch(u!v) 1 m Re p (u)K , xx 2 (u!v)2#c2h(u!v) ne2
(16)
where c"4p¹v2/+u2. 0 4. Discussion From the result (16), we see that the threshold frequency for the AC conductivity is v as expected. The width c becomes very large as ¹ goes to zero, since vJ¹~1 (viz., collective localization). One may be tempted to correlate this threshold frequency with the pinning frequency in the CDW theory and the activation energy observed in the DC transport. But care must be taken, since the activation energy observed in the non-Ohmic DC transport may not necessarily be related to the present threshold frequency of the Ohmic AC conductivity, when the electrons are localized, since the picture of Wigner solids is valid for the system
M. Saitoh / Physica B 249—251 (1998) 788—791
where electrons are located very far and isolated, i.e., classical, while the CDW picture is applicable to the high electron-density case where electrons are quantal. The validity of the present theory will be checked only by careful AC transport measurements at very low temperatures.
791
References [1] See, for example, Surf. Sci. (1992) 263. [2] V.M. Pudalov, M. D’Iorio, S.V. Kravchenko, J.V. Campbell, Phys. Rev. Lett. 70 (1993) 1886. [3] M. Saitoh, J. Phys. Soc. Japan 55 (1986) 1311. [4] M. Saitoh, J. Phys. C 15 (1983) 6981.
Theory of AC conductivity of 2D Wigner solids Motohiko Saitoh* Department of Physics, Graduate School of Science, Osaka University, 1-16 Machikaneyama, Toyonaka 560, Japan
Abstract First-principle calculation is made of an AC conductivity of electrons in quasi-two-dimensional Wigner solids at low temperatures, when electrons are scattered by ionized impurities. Dispersion relation of the Wigner phonons are solved within the self-consistent harmonic approximation which leads to an energy gap in the small wave vector limit. This gap is directly related to the threshold frequency of the AC conductivity, which is inversely proportional to the temperature and implies the collective localization at very low temperatures. The absorption bandwidth is primarily determined by the transverse mode of the Wigner phonons and the longitudinal mode contributes to a relatively small background. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Wigner solid; Semiconductor heterostructure
1. Introduction Transport properties of the quasi-two-dimensional Wigner solid has attracted much attention recently [1,2]. However, the theoretical understanding is rather qualitative than quantitative, based on, e.g., the CDW theory which is valid only for the degenerate electron system, while the picture of the Wigner solids applies to the non-degenerate electrons where the distance between electrons is very far as compared to the effective Bohr radius. No first-principle calculation of the conductivity was tried so far because of the difficulty of treating the strong localization. We present here the first-principle calculation of the AC conductivity at very low temperature by * Fax: #81-6-850-5764; e-mail: [email protected].
explicitly considering the localization properties on the basis of the self-consistent harmonic approximation to the Wigner phonon dispersion. If the electron scattering is dominated by the impurities, then the phonon dispersion has an energy gap in the small wave vector limit. This energy gap stabilizes the lattice in the sense that the density—density correlation has long-range order, and also leads to the vanishing of the DC conductivity.
2. Dispersion relation of the Wigner phonons We assume that quasi-two-dimensional electrons are confined to the semiconductor side of the interface, z"0, and are scattered by ionized impurities located at (r , z ) in the insulator side, where r is the j j j two-dimensional position vector and z (0. The j
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 3 1 5 - 9
M. Saitoh / Physica B 249—251 (1998) 788—791
interaction potential is given by u(r, z)" + u (r!r , z!z ), *.1 j j r j,zj
(1)
where u (r, z) is the impurity potential. Effects of *.1 the scattering by impurities will be characterized by the cummulant average: º(r)"Su(r, 0)u(0, 0)T !Su(r, 0)T Su(0, 0)T , s s s
(2)
where the angular brackets indicate the random average over the impurity positions. If the random average is characterized by Sz T "!d and j s S(z #d)2T "*2/2, then the Fourier transform of j s º(r) is calculated to be º "(pe2/i)2(n /q2)erfc(d/*), q i
(3)
where n is the surface density of impurities, i the i average of the dielectric constants of the semiconductor and the insulator, and erfc(x) is the complementary error function with the normalization erfc(0)"1. For the delta doped case, d/* may be larger than unity, and º can be very small. q The dispersion relation of the jth mode of the Wigner phonons in the presence of impurities can be calculated from the minimization principle of the free energy. Namely, we first rewrite the electron coordinates in the Hamiltonian formally in terms of the normal coordinates of the still unfixed Wigner phonons. Then, by using the finite-temperature path-integral method and the Feynman—Jensen inequality, we express the approximate free energy in terms of the Wigner phonon frequencies. The minimization of the free energy with respect to the phonon frequencies leads to [3] 1 (q ) e (p))2 j X2(p)"! + + » e*q > R j q SR m E0 q
1@T 0
dq(1!cos p ) R cos lq)/ (R, q), q
/q(R,q)"Se*q > uR(q)e~*q > u0(0)T
C A A BBD
(q ) e (p))2 + j + "exp ! 2mNp X (p) sinh(+X (p)/2¹) j ,j j ] cosh
+X (p) j !cos pR cosh +X (p) j 2¹
1 ] q! 2¹
,
(5)
where uR(q) is the positional deviation from the lattice site R at the imaginary time q, N the number of the lattice sites and summation over p is limited to the first Brillouin zone. The physical meaning of this equation is clear. Namely, the phonon frequency is determined by the product of the second derivative of the sum of the Coulombic and the impurity potentials, and the density correlation function. Note that the first derivative of the potentials determines the equilibrium positions of the lattice sites. Since the Debye—Waller factor of the density correlation function is a function of the Wigner phonon spectra, the coupled Eqs. (4) and (5) must be solved so that X (p)’s become selfj consistent. This scheme is sometimes called the self-consistent harmonic approximation. Because of impurity scattering, the energy gap v will be induced in the phonon spectra: (6)
where u (p) is the phonon frequency of the pure j Wigner crystal [3]:
1 (q ) e (p))2 j # + + º e*q > R SR q q m
P
where » "2pe2/i q is the Fourier transform of the q s Coulomb potential with i the dielectric constant s of the semiconductor, R the lattice site vector of the crystal, e (p) the polarization vector of the jth mode j phonon, m the effective electron mass, S the system area, ¹ the temperature in energy units, l"2pn¹ the Matsubara frequency, and /q(R, q) is the density-correlation function defined by
X2(p)"u2(p)#v2, j j
](1!cos p ) R)/q(R, 0)
]
789
u "u Jp/p , l 0 0 (4)
u (p)"(J2/4)u p/p . t 0 0
(7)
Here u is the longitudinal frequency at the Bril0 louin zone edge: u "(2pe2np /i m)1@2 with n the 0 0 s
790
M. Saitoh / Physica B 249—251 (1998) 788—791
surface density of electrons and p "J4pn. The 0 gap frequency v is evaluated by setting p"0 in Eq. (4) as q2 1 v2K + + º e*q > R e~l2q2@2, 2m¹ SR q q
(8)
with (9)
where the time-dependent part of /q(R, q) has been neglected and v is assumed to be much larger than u . The result for v is 0
A B
/q(R, q)Ke~l2q2@2
C
D
+q2 J (pR) cosh(+X (p)(q!1/2¹)) j + 0 , ] 1# 4mNp X (p) sinh(+X (p)/2¹) j j ,j (13) where J (x) is the Bessel function of the first kind. 0 Putting this into (12), we obtain
1 + + + K , l2" 2mNp X (p) mv ,j j
v"
important, and to the lowest order in q2 we have
AB
pe2 2 n d i erfc . i * 4p+¹
(10)
This result is valid for (pe2/i)2(n /4p+u ) erfc(d/*)< i 0 ¹. At very low temperatures, the Wigner crystal is stabilized, since /q(R, 0) is non-zero for large R.
p¹v3 v2 D(u), M(u)"! i# +u u
(14)
where D(u)"N~1+ d(u2!X (p)2) j p ,j 8 " h(Ju2/8#v2!u) 0 u2 0 (u2!v2) # h(Ju2#v2!u) h(u!v) 0 4u2 0
C
D
(15) 3. AC conductivity The AC conductivity p (u) is expressed in terms xx of the memory function M(u) as p (u)"(ne2/m)(iu#d#M(u))~1, xx
(11)
where d is a positive infinitesimal, and in the selfconsistent harmonic approximation the memory function reads [4] iq2 º e*q > R muS q
M(u)"+ + R
q
CP
]
1@2T dq(cosh +uq!1)/ (R,q) q 0
P
+u = !i sinh dt e~*+(u~*d)t/ q 2¹ 0
A
1 ] R, it# 2¹
BD
.
(12)
In order to obtain a finite Re M(u), the time dependence of the density-correlation function is
and h(x) is the step function. We arrive at the final result: ch(u!v) 1 m Re p (u)K , xx 2 (u!v)2#c2h(u!v) ne2
(16)
where c"4p¹v2/+u2. 0 4. Discussion From the result (16), we see that the threshold frequency for the AC conductivity is v as expected. The width c becomes very large as ¹ goes to zero, since vJ¹~1 (viz., collective localization). One may be tempted to correlate this threshold frequency with the pinning frequency in the CDW theory and the activation energy observed in the DC transport. But care must be taken, since the activation energy observed in the non-Ohmic DC transport may not necessarily be related to the present threshold frequency of the Ohmic AC conductivity, when the electrons are localized, since the picture of Wigner solids is valid for the system
M. Saitoh / Physica B 249—251 (1998) 788—791
where electrons are located very far and isolated, i.e., classical, while the CDW picture is applicable to the high electron-density case where electrons are quantal. The validity of the present theory will be checked only by careful AC transport measurements at very low temperatures.
791
References [1] See, for example, Surf. Sci. (1992) 263. [2] V.M. Pudalov, M. D’Iorio, S.V. Kravchenko, J.V. Campbell, Phys. Rev. Lett. 70 (1993) 1886. [3] M. Saitoh, J. Phys. Soc. Japan 55 (1986) 1311. [4] M. Saitoh, J. Phys. C 15 (1983) 6981.