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Transverse plasmon in two-dimensional electrons
Volume 100A, number 2
PHYSICS LETTERS
9 January 1984
TRANSVERSE PLASMON IN TWO-DIMENSIONAL ELECTRONS Vidar GUDMUNDSSON, Tadashi TOYODA and Yasushi TAKAHASHI
Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada Received 7 October 1983
We calculated the retarded transverse current-current response function of an ideal two-dimensional electron gas and obtained a dispersion relation for the transverse plasmon in the system.
Correlations in a two-dimensional electron gas have been studied by several authors [ 1 - 3 ] . However, those studies are mainly concerned with density-density correlations and the correlations in the transverse currents, which are of special importance for understanding the electromagnetic properties of the system, have not been studied in detail. In this letter we report the exact form of the retarded response function for the transverse current in a two-dimensional ideal electron gas at zero temperature and a dispersion relation for the transverse plasmon in the system. We consider an ideal electron gas confined in the x 1 - x 2 plane by a potential V(x3), i.e., the electrons are in the lowest energy level with regard to V(x3). Throughout this paper we use the following definitions: x - (x 1 , x2), by =--a/~x~ (with v = 1,2), m = electron mass, - e = electron charge, n - electron number density, and Einstein convention for the space coordinate components v, 3,.... and for the spin variables s. The transverse current operator for the second quantized free two-dimensional electron field ~bs(X, t) is given by
~Or(x, t) = (ieti/2m)
= - i O ( t - t')(CI, F I [jOt(x, t),Iv"Or( x ,' t ,)] IqbF) --=(2zr) -3 f d 2x f d w e x p ( i k ' x - i w t ) A t u ( k , w ) ,
(2) where I~F) is the Fermi ground state, with the Fermi momentum fikF, and
O(r) =1 =0
(r > O), (r<0).
(3)
We calculated Atu~(k, co) exactly and obtained Atv(k, co) = (e2~k 2 [Trm) •(k, w)(6 u. - kukv/k 2 ) = A(k, w)(Suv - kukv/k2),
(4)
with Re 7((k. w)= v2 lu 2 - ~1 + I U2 +g_(u,v) --g+(u.v),
(5) and Im 7k(k, ¢o) = f_(u, v) - f+(u, v),
(6)
where we defined as
X (Sun - ava~./a2 ) (~o+(x, t)(bh -~x)~Os(X, t ) ) , (v = 1, 2),
A~.(x, t;x', t')
g~ (u, v) = (1/3u) sgn(o+) 0 (o 2 - 1) [(o 2 - 1)3 ] '1/2, (7) (1)
where 6vX is Kronecker's delta. Then we can define the retarded transverse current-current response function [4]
f+_(u,v)=(1/3u)O(1 - o 2) [(1 - o2)311/2 ,
(8)
u = k/kF,
(9)
v = m~/ttk2v,
v+_ = v/u + ~u
and 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
91
Volume 100A, number 2 sgn(v) = 1
(v > 0),
=-1
(v<0).
PHYSICS LETTERS
(10)
As can be expected from the well-known three-dimensional case, the result (6) shows that Im A (k, co) vanishes for 1 - (v/u + ~ u) 2 < 0. This suggests a possibility of undamped transverse electromagnetic wave propagation mode, i.e., the transverse plasmon, for small k within a "RPA type approximation". For small k and finite co, we can expand Re A (k, co) as
e2h3k4k2 [1 ReA(k'w)=
~
co
1(_~)2 +~
k2+o(k4)] w2 "
(11) This result can be used to obtain a dispersion relation for the transverse plasmon in the system. If the interaction between the current and the transverse electromagnetic field is taken into account, which is expressed by a c-number transverse vector potential Av(x, x 3, t), then the current operator becomes
Jr(X, x3, t) = ]X(X3)12 ~j° (x, t) (e2/mc)Av(x,x3,t)~o+(x,t)~s(X,t)),
+
Jr(k, k3, 6o) = P(k3)Jv(k , co),
(14)
where P(k3) is the Fourier transform of Ix(x3) f2. Since we are interested in the A v field propagating along the x 1 - x 2 plane, we consider the k 3 = 0 case. Then the self-consistent linear response equation is
jtu(k , co) =
(e2n/2nmc) F (k, w ) / t ( k , w)
- (1/2~hc)F(k, co)At~u(k,w)/~(k, co).
(13)
We assume A 3 = 0 a n d J 3 = 0. Treating A u as an external field, we can calculate the expectation value for the induced current (Jr) in terms of the retarded current-current response function and A v within the framework of the linear response theory. On the other hand, the transverse part of the induced current also contributes to A v in addition to the external current source. Since the transverse plasmon is the transverse electromagnetic wave propagating in the system in the limit of zero-external current source, in order to obtain the dispersion relation of the transverse current we can assume the external current source is zero. Then, the wave equations for A u have the current terms which depend only on the transverse part of the induced current (Jr). Thus, we have two sets of equations for (Jr) and Av, i.e., the linear response equations for (J~) and the wave equations forAy. Both sets of equations contain (Jr) and A v. By eliminating either (J~)or A , , we can get a set of equations which contain only A v or (Jr). The transverse plasmon is 92
nothing else but the non-trivial solution of such equations. We confirmed that this method, which may be called a "self-consistent linear response approximation" (SLRA), gives the correct dispersion relation of the transverse plasmon for the three-dimensional case calculated by Bohm and Pines [5] and also by Matsumoto et al. [6]. It is obvious that our SLRA is considerably simpler than those previous 3-D calculations. To calculate a possible form of the dispersion relation of the transverse plasmon we assume the Fourier transform of (J~(x, x 3 , t)) can be written approximately as
(15)
(12)
where X(X3) is the single electron ground state wavefunction of V(x3) and j°v(x, t) is defined by
/O(x, t) = (iel~/2m) (¢s (Or - ~v)¢s) •
9 January 1984
In (15),it(k, ca) is the two-dimensional transverse part of/u(k, co) and we have defined
F(k, co)=4nc f dk 3
fi(-k3)fi(k3)
c2(k 2 + k 2 ) - (co + ie) 2
(e-~O+)
(16) If there is a transverse plasmon mode, eq. (l 5) must have a nontrivial solution/"t 4: 0. Therefore, the dispersion relation of the transverse plasmon can be determined by the equation
1/F(k,~)+e2n/2rrmc+(1/2zrPtc)At(k, co)=O.
(17)
In order to solve this equation the form of p(k 3) must be given. We assume a gaussian form, i.e. o(k 3) = exp(-k32/2a2),
(18)
which gives Ix(x3)l 2 = (a/x/27r) e x p ( - ½ a2x2).
(19)
The parameter 1/a can be interpreted as the thickness of the system. Substituting (1 8) into (I 6), for co > ck we get
l~(k, ~o) = - 4rcx/~ac/(oo 2 - c2k 2) + O(a3).
(20)
Taking account of the small k behaviour of Im A t,
Volume 100A, number 2
PHYSICS LETTERS
which vanishes for 1 - (v/u +-~ u) 2 < 0, we can use the small k expansion of Re A t given by (11) for A t in (17). Then introducing (20) into (17) we obtain the dispersion relation
6:2 = 2X/'~e2na/m +(1 +rrt12n/2m2c 2) c2k 2 + O(k4), (21) where we have used the relation k2F = 2nn. It should be noted that the imaginary part of F appears in higher order powers of a, and (20) is real in the limit of small a. Furthermore, if k becomes large, Im A t also contributes and causes Landau damping. The result (21) is, therefore, valid for sufficiently large thickness and small k. The interesting features of (21) are the appearance of the gap and the k 2 dependence, which resemble the three-dimensional case in contrast to the two-dimensional longitudinal plasmon [7]. Since (21) is derived for small a, it is also interesting to see the opposite case, i.e., a --> oo, which means I×(x3)l 2 ~ ~(x3). We studied such a case and
9 January 1984
found that there cannot be a solution of (17) for co at k = 0. This may be interpreted that the gap becomes oo f o r a ~ 0. Details of the calculations as well as more extensive studies of the subject will be published elsewhere. This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4]
A. Isihara and T. Toyoda, Z. Phys. B23 (1976) 389. F. Stern, Phys. Rev. Lett. 18 (1967) 546. A.K. Rajagopal, Phys. Rev. B15 (1977) 4264. A.L. Fetter and J.D. Waleeka, Quantum theory of manyparticle systems (McGraw-Hill, New York, 1971). [5] D. Bohm and D. Pines, Phys. Rev. 82 (1951) 625. [6] H. Matsumoto et al., Fortschr. Phys. 28 (1980) 67. [7] T. Ando et aL, Rev. Mod. Phys. 54 (1982) 437.
93
PHYSICS LETTERS
9 January 1984
TRANSVERSE PLASMON IN TWO-DIMENSIONAL ELECTRONS Vidar GUDMUNDSSON, Tadashi TOYODA and Yasushi TAKAHASHI
Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada Received 7 October 1983
We calculated the retarded transverse current-current response function of an ideal two-dimensional electron gas and obtained a dispersion relation for the transverse plasmon in the system.
Correlations in a two-dimensional electron gas have been studied by several authors [ 1 - 3 ] . However, those studies are mainly concerned with density-density correlations and the correlations in the transverse currents, which are of special importance for understanding the electromagnetic properties of the system, have not been studied in detail. In this letter we report the exact form of the retarded response function for the transverse current in a two-dimensional ideal electron gas at zero temperature and a dispersion relation for the transverse plasmon in the system. We consider an ideal electron gas confined in the x 1 - x 2 plane by a potential V(x3), i.e., the electrons are in the lowest energy level with regard to V(x3). Throughout this paper we use the following definitions: x - (x 1 , x2), by =--a/~x~ (with v = 1,2), m = electron mass, - e = electron charge, n - electron number density, and Einstein convention for the space coordinate components v, 3,.... and for the spin variables s. The transverse current operator for the second quantized free two-dimensional electron field ~bs(X, t) is given by
~Or(x, t) = (ieti/2m)
= - i O ( t - t')(CI, F I [jOt(x, t),Iv"Or( x ,' t ,)] IqbF) --=(2zr) -3 f d 2x f d w e x p ( i k ' x - i w t ) A t u ( k , w ) ,
(2) where I~F) is the Fermi ground state, with the Fermi momentum fikF, and
O(r) =1 =0
(r > O), (r<0).
(3)
We calculated Atu~(k, co) exactly and obtained Atv(k, co) = (e2~k 2 [Trm) •(k, w)(6 u. - kukv/k 2 ) = A(k, w)(Suv - kukv/k2),
(4)
with Re 7((k. w)= v2 lu 2 - ~1 + I U2 +g_(u,v) --g+(u.v),
(5) and Im 7k(k, ¢o) = f_(u, v) - f+(u, v),
(6)
where we defined as
X (Sun - ava~./a2 ) (~o+(x, t)(bh -~x)~Os(X, t ) ) , (v = 1, 2),
A~.(x, t;x', t')
g~ (u, v) = (1/3u) sgn(o+) 0 (o 2 - 1) [(o 2 - 1)3 ] '1/2, (7) (1)
where 6vX is Kronecker's delta. Then we can define the retarded transverse current-current response function [4]
f+_(u,v)=(1/3u)O(1 - o 2) [(1 - o2)311/2 ,
(8)
u = k/kF,
(9)
v = m~/ttk2v,
v+_ = v/u + ~u
and 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
91
Volume 100A, number 2 sgn(v) = 1
(v > 0),
=-1
(v<0).
PHYSICS LETTERS
(10)
As can be expected from the well-known three-dimensional case, the result (6) shows that Im A (k, co) vanishes for 1 - (v/u + ~ u) 2 < 0. This suggests a possibility of undamped transverse electromagnetic wave propagation mode, i.e., the transverse plasmon, for small k within a "RPA type approximation". For small k and finite co, we can expand Re A (k, co) as
e2h3k4k2 [1 ReA(k'w)=
~
co
1(_~)2 +~
k2+o(k4)] w2 "
(11) This result can be used to obtain a dispersion relation for the transverse plasmon in the system. If the interaction between the current and the transverse electromagnetic field is taken into account, which is expressed by a c-number transverse vector potential Av(x, x 3, t), then the current operator becomes
Jr(X, x3, t) = ]X(X3)12 ~j° (x, t) (e2/mc)Av(x,x3,t)~o+(x,t)~s(X,t)),
+
Jr(k, k3, 6o) = P(k3)Jv(k , co),
(14)
where P(k3) is the Fourier transform of Ix(x3) f2. Since we are interested in the A v field propagating along the x 1 - x 2 plane, we consider the k 3 = 0 case. Then the self-consistent linear response equation is
jtu(k , co) =
(e2n/2nmc) F (k, w ) / t ( k , w)
- (1/2~hc)F(k, co)At~u(k,w)/~(k, co).
(13)
We assume A 3 = 0 a n d J 3 = 0. Treating A u as an external field, we can calculate the expectation value for the induced current (Jr) in terms of the retarded current-current response function and A v within the framework of the linear response theory. On the other hand, the transverse part of the induced current also contributes to A v in addition to the external current source. Since the transverse plasmon is the transverse electromagnetic wave propagating in the system in the limit of zero-external current source, in order to obtain the dispersion relation of the transverse current we can assume the external current source is zero. Then, the wave equations for A u have the current terms which depend only on the transverse part of the induced current (Jr). Thus, we have two sets of equations for (Jr) and Av, i.e., the linear response equations for (J~) and the wave equations forAy. Both sets of equations contain (Jr) and A v. By eliminating either (J~)or A , , we can get a set of equations which contain only A v or (Jr). The transverse plasmon is 92
nothing else but the non-trivial solution of such equations. We confirmed that this method, which may be called a "self-consistent linear response approximation" (SLRA), gives the correct dispersion relation of the transverse plasmon for the three-dimensional case calculated by Bohm and Pines [5] and also by Matsumoto et al. [6]. It is obvious that our SLRA is considerably simpler than those previous 3-D calculations. To calculate a possible form of the dispersion relation of the transverse plasmon we assume the Fourier transform of (J~(x, x 3 , t)) can be written approximately as
(15)
(12)
where X(X3) is the single electron ground state wavefunction of V(x3) and j°v(x, t) is defined by
/O(x, t) = (iel~/2m) (¢s (Or - ~v)¢s) •
9 January 1984
In (15),it(k, ca) is the two-dimensional transverse part of/u(k, co) and we have defined
F(k, co)=4nc f dk 3
fi(-k3)fi(k3)
c2(k 2 + k 2 ) - (co + ie) 2
(e-~O+)
(16) If there is a transverse plasmon mode, eq. (l 5) must have a nontrivial solution/"t 4: 0. Therefore, the dispersion relation of the transverse plasmon can be determined by the equation
1/F(k,~)+e2n/2rrmc+(1/2zrPtc)At(k, co)=O.
(17)
In order to solve this equation the form of p(k 3) must be given. We assume a gaussian form, i.e. o(k 3) = exp(-k32/2a2),
(18)
which gives Ix(x3)l 2 = (a/x/27r) e x p ( - ½ a2x2).
(19)
The parameter 1/a can be interpreted as the thickness of the system. Substituting (1 8) into (I 6), for co > ck we get
l~(k, ~o) = - 4rcx/~ac/(oo 2 - c2k 2) + O(a3).
(20)
Taking account of the small k behaviour of Im A t,
Volume 100A, number 2
PHYSICS LETTERS
which vanishes for 1 - (v/u +-~ u) 2 < 0, we can use the small k expansion of Re A t given by (11) for A t in (17). Then introducing (20) into (17) we obtain the dispersion relation
6:2 = 2X/'~e2na/m +(1 +rrt12n/2m2c 2) c2k 2 + O(k4), (21) where we have used the relation k2F = 2nn. It should be noted that the imaginary part of F appears in higher order powers of a, and (20) is real in the limit of small a. Furthermore, if k becomes large, Im A t also contributes and causes Landau damping. The result (21) is, therefore, valid for sufficiently large thickness and small k. The interesting features of (21) are the appearance of the gap and the k 2 dependence, which resemble the three-dimensional case in contrast to the two-dimensional longitudinal plasmon [7]. Since (21) is derived for small a, it is also interesting to see the opposite case, i.e., a --> oo, which means I×(x3)l 2 ~ ~(x3). We studied such a case and
9 January 1984
found that there cannot be a solution of (17) for co at k = 0. This may be interpreted that the gap becomes oo f o r a ~ 0. Details of the calculations as well as more extensive studies of the subject will be published elsewhere. This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4]
A. Isihara and T. Toyoda, Z. Phys. B23 (1976) 389. F. Stern, Phys. Rev. Lett. 18 (1967) 546. A.K. Rajagopal, Phys. Rev. B15 (1977) 4264. A.L. Fetter and J.D. Waleeka, Quantum theory of manyparticle systems (McGraw-Hill, New York, 1971). [5] D. Bohm and D. Pines, Phys. Rev. 82 (1951) 625. [6] H. Matsumoto et al., Fortschr. Phys. 28 (1980) 67. [7] T. Ando et aL, Rev. Mod. Phys. 54 (1982) 437.
93