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Exact solution of a simple exciton model in one-dimension
PHYSICA
Physica B 194-196 (1994) 1297-1298 North-Holland
E x a c t s o l u t i o n of a s i m p l e e x c i t o n m o d e l in o n e - d i m e n s i o n P. Schlottmann a D e p a r t m e n t of Physics, Florida State University, Tallahassee, Florida 32306, USA The model consists of two parabolic bands of opposite mass separated by a gap. The spinless holes in the valence band and the spinless electrons in the conduction band are locally attracted. The exact solution of the model is sketched and some low temperature properties are discussed.
We consider a simple one-dimensional model of a semiconductor consisting of two bands of spinless fermions, one valence band of effective mass m = - 1 / 2 and one conduction band of effective mass m = 1/2, separated by a gap 2A. The holes in the valence band and the electrons in the conduction band interact locally via an attractive 6function potential V,
bI ~ c T ,
H = . / d ~ at(~) [ - O ~ / 0 ~ + A - ~] a(~)
+f
- A _ .1
,
(1)
where the a-operators refer to the conduction band and the b-operators to electrons in the valence band, and tt is the chemical potential. The valence band is not bound from below, so that a m o m e n t u m cutoff pc should be introduced. In the absence of interaction (V = 0) and at T = 0 we have that in the conduction band states with p~ < # - A are occupied, and in the valence band states with p2 < - t t - A are empty. Hence, if # = 0 there are as m a n y electrons in the conduction band as holes in the valence band, which is zero if A > 0 (semiconductor) and nonzero if A < 0 (metal). A = 0 corresponds to the semimetallic situation, tt > 0 refers to more electrons (in the a-band) than holes (in the b-band) and tt < 0 to more holes than electrons. An attractive interaction (V > 0) introduces boundstates known as excitons with a binding energy of the order of V z. Assume tt = 0, than if A exceeds the binding energy the conduction band is e m p t y and the valence band is full (no holes). 0921-4526/94/$07.00
© 1994 -
SSDI 0921-4526(93)E1229-F
If A is decreased there is a threshold value A c below which the exciton band is populated. If # ~ 0, on the other hand, not all the electrons or holes are paired in boundstates and occupy a second band (of unpaired particles). The model (1) can be m a p p e d onto the twocomponent Fermi gas with attractive 6-function interaction by the following transformation
b~c~
; a~--.~c~, a ~ c t
,
(2)
which interchanges electrons and holes in the valence band. The two bands are now labeled by the spin index. Except for an irrelevant additive constant, Hamiltonian (1) is equivalent to the fermlon m a n y b o d y problem solved by Gaudin [1] and Yang [2] with V = - c by means of two nested Bethe ansatze. Below we restrict ourselves to the situation where the number of holes in the valence band is larger or equal to the number of electrons in the conduction band (# < 0). The tt > 0 case can be obtained by reversing the direction of the "spins" in (2). Note that A plays the role of chemical potential in the m a n y b o d y problem and # represents the magnetic field. The diagonalization of the gas with 6-function interaction with N - M up-spin and M down-spin electrons gives rise to a set N charge rapidities k and a set of M spin-rapidities A. All rapidities within one set have to be different in order to ensure t h a t all solutions are linearly independent. The rapidities have to satisfy the discrete Bethe ansatz equations; for an attractive interaction the solutions can be classified according to [3,4] (i) N - 2 M t real k rapidities corresponding to unpaired propagating holes, (ii) M ' ( M ' < M ) pairs of complex conjugated k values representing exciton boundstates (electron-hole pairs); each pair is associated with a real )~ rapidity, which we denote
Elsevier Science B.V. All rights reserved
1298
with ~. (iii) The remaining M - M ' rapidities of the A set form strings of length n - 1 (excited boundstates of n electrons and holes involving no charge). We denote with An the rapidity associated with the motion of their center of mass. If M,~ is the number of string-boundstates of n particles we have M = M ' + ~,,~°=t nMn. Only the classes of solutions (i) and (ii) can be represented in the groundstate. In the thermodynamic limit the rapidities become closely spaced and may be regarded as continuous variables. Since all rapidities within one class have to be different, they obey Fermi statistics. We introduce for each class of states a density of states and an energy potential (entering the Fermi distribution), which are determined by minimization of the free energy, subject to the constraints imposed by the Bethe ansatz equations. We denote the potentials by e(k) for class (i), ¢(~) for class (ii), and p,,(A) for the string states. Since all model energy parameters are large compared to the temperature, we here limit ourselves to discuss the groundstate integral equations satisfied by the potentials
/°
~(k) = (k ~ + A + . / -
d~ a~(k -
Q
¢((1
= 2(0
+ A _ v2/4)
_
/?
o
~) ¢(¢),
d~' a ~ ( i -
i')
B B
p,~(A)
~ 2~
[
dk a n ( k - A) e(k) ,
(3)
J- B
where an(x) = (nV/2~r)/(zr 2 + (nV/2)2). The integration limits B and Q correspond to the Fermi surface of the respective bands; they are related to the model parameters A and # via e(zkB) = 0 and ¢(=}=Q) = 0. In view of the Fermi statistics, a negative energy potential corresponds to occupied states in the groundstate, while a positive energy to an empty state. The energy potentials p,~(A) are always positive (unoccupied states) and hence do not have a Fermi surface. The densities of states for the various classes of rapidities can be obtained from the potentials
by differentiating with respect to A, i.e. p(k) = (Oe(k)/OA)/(2~r), o"(~ 1 = (0!l,(~l/OA)/(2r¢), and #n(A / = - - ( 0 P n ( A ) / 0 A ) / ( 2 r ) . The groundstate energy is (L is the length of the box)
ElL =
I_
B
( d k / 2 r ) e(k) +
I;
Q
(dC/r) ¢(C1 • (4)
The occupation of the exciton band is Ne®c = L f-QO d~ ~'(I1 and the one of unpaired particles
L.fB_ndk p(k); if # < 0 the number of electrons in the a-band is Ne~e and the number of holes in the b-band is N~¢ + Nu,~p. The exciton band is empty if A is larger than a critical Ac given by A c = V2/4 + (1/2 /
dk at(k) e(k) ;
(5)
B
/z and Ac decrease monotonically with the number of holes in the valence band. On the other hand, if the exciton band is partially filled but the band of unpaired holes is empty, it requires a chemical potential lower than a critical #e
.~ = - ~ +
d~ al(~) ¢(¢)
(6)
q to place the first particle into that band. Note that the elemental excitations of the system are given by energy potentials; the rapidities parametrize their momentum. If Q + B ¢ 0 the low temperature corrections to (3) and (4) are of the order o f T 2, hence giving rise to a specific heat proportional to T, except when the Fermi level is at the van Hove singularity of an empty band where a T 1/~ contribution dominates. The exciton boundstates exist at all T; there is no phase transition, but also no long range order. A more complete discussion of the results will be presented elsewhere. The support of the Department of Energy under grant DE-FG05-91ER45443 is acknowledged. REFERENCES
1. M. Gaudin, Phys. Lett. 24A (19671 55. 2. C . N . Yang, Phys. Rev. Left. 19 (19671 1312. 3. M. Takahashi, Prog. Theor. Phys. 42 (1969 / 1098. 4. C . K . Lai, Phys. Rev. Lett. 26 (19711 1472.
Physica B 194-196 (1994) 1297-1298 North-Holland
E x a c t s o l u t i o n of a s i m p l e e x c i t o n m o d e l in o n e - d i m e n s i o n P. Schlottmann a D e p a r t m e n t of Physics, Florida State University, Tallahassee, Florida 32306, USA The model consists of two parabolic bands of opposite mass separated by a gap. The spinless holes in the valence band and the spinless electrons in the conduction band are locally attracted. The exact solution of the model is sketched and some low temperature properties are discussed.
We consider a simple one-dimensional model of a semiconductor consisting of two bands of spinless fermions, one valence band of effective mass m = - 1 / 2 and one conduction band of effective mass m = 1/2, separated by a gap 2A. The holes in the valence band and the electrons in the conduction band interact locally via an attractive 6function potential V,
bI ~ c T ,
H = . / d ~ at(~) [ - O ~ / 0 ~ + A - ~] a(~)
+f
- A _ .1
,
(1)
where the a-operators refer to the conduction band and the b-operators to electrons in the valence band, and tt is the chemical potential. The valence band is not bound from below, so that a m o m e n t u m cutoff pc should be introduced. In the absence of interaction (V = 0) and at T = 0 we have that in the conduction band states with p~ < # - A are occupied, and in the valence band states with p2 < - t t - A are empty. Hence, if # = 0 there are as m a n y electrons in the conduction band as holes in the valence band, which is zero if A > 0 (semiconductor) and nonzero if A < 0 (metal). A = 0 corresponds to the semimetallic situation, tt > 0 refers to more electrons (in the a-band) than holes (in the b-band) and tt < 0 to more holes than electrons. An attractive interaction (V > 0) introduces boundstates known as excitons with a binding energy of the order of V z. Assume tt = 0, than if A exceeds the binding energy the conduction band is e m p t y and the valence band is full (no holes). 0921-4526/94/$07.00
© 1994 -
SSDI 0921-4526(93)E1229-F
If A is decreased there is a threshold value A c below which the exciton band is populated. If # ~ 0, on the other hand, not all the electrons or holes are paired in boundstates and occupy a second band (of unpaired particles). The model (1) can be m a p p e d onto the twocomponent Fermi gas with attractive 6-function interaction by the following transformation
b~c~
; a~--.~c~, a ~ c t
,
(2)
which interchanges electrons and holes in the valence band. The two bands are now labeled by the spin index. Except for an irrelevant additive constant, Hamiltonian (1) is equivalent to the fermlon m a n y b o d y problem solved by Gaudin [1] and Yang [2] with V = - c by means of two nested Bethe ansatze. Below we restrict ourselves to the situation where the number of holes in the valence band is larger or equal to the number of electrons in the conduction band (# < 0). The tt > 0 case can be obtained by reversing the direction of the "spins" in (2). Note that A plays the role of chemical potential in the m a n y b o d y problem and # represents the magnetic field. The diagonalization of the gas with 6-function interaction with N - M up-spin and M down-spin electrons gives rise to a set N charge rapidities k and a set of M spin-rapidities A. All rapidities within one set have to be different in order to ensure t h a t all solutions are linearly independent. The rapidities have to satisfy the discrete Bethe ansatz equations; for an attractive interaction the solutions can be classified according to [3,4] (i) N - 2 M t real k rapidities corresponding to unpaired propagating holes, (ii) M ' ( M ' < M ) pairs of complex conjugated k values representing exciton boundstates (electron-hole pairs); each pair is associated with a real )~ rapidity, which we denote
Elsevier Science B.V. All rights reserved
1298
with ~. (iii) The remaining M - M ' rapidities of the A set form strings of length n - 1 (excited boundstates of n electrons and holes involving no charge). We denote with An the rapidity associated with the motion of their center of mass. If M,~ is the number of string-boundstates of n particles we have M = M ' + ~,,~°=t nMn. Only the classes of solutions (i) and (ii) can be represented in the groundstate. In the thermodynamic limit the rapidities become closely spaced and may be regarded as continuous variables. Since all rapidities within one class have to be different, they obey Fermi statistics. We introduce for each class of states a density of states and an energy potential (entering the Fermi distribution), which are determined by minimization of the free energy, subject to the constraints imposed by the Bethe ansatz equations. We denote the potentials by e(k) for class (i), ¢(~) for class (ii), and p,,(A) for the string states. Since all model energy parameters are large compared to the temperature, we here limit ourselves to discuss the groundstate integral equations satisfied by the potentials
/°
~(k) = (k ~ + A + . / -
d~ a~(k -
Q
¢((1
= 2(0
+ A _ v2/4)
_
/?
o
~) ¢(¢),
d~' a ~ ( i -
i')
B B
p,~(A)
~ 2~
[
dk a n ( k - A) e(k) ,
(3)
J- B
where an(x) = (nV/2~r)/(zr 2 + (nV/2)2). The integration limits B and Q correspond to the Fermi surface of the respective bands; they are related to the model parameters A and # via e(zkB) = 0 and ¢(=}=Q) = 0. In view of the Fermi statistics, a negative energy potential corresponds to occupied states in the groundstate, while a positive energy to an empty state. The energy potentials p,~(A) are always positive (unoccupied states) and hence do not have a Fermi surface. The densities of states for the various classes of rapidities can be obtained from the potentials
by differentiating with respect to A, i.e. p(k) = (Oe(k)/OA)/(2~r), o"(~ 1 = (0!l,(~l/OA)/(2r¢), and #n(A / = - - ( 0 P n ( A ) / 0 A ) / ( 2 r ) . The groundstate energy is (L is the length of the box)
ElL =
I_
B
( d k / 2 r ) e(k) +
I;
Q
(dC/r) ¢(C1 • (4)
The occupation of the exciton band is Ne®c = L f-QO d~ ~'(I1 and the one of unpaired particles
L.fB_ndk p(k); if # < 0 the number of electrons in the a-band is Ne~e and the number of holes in the b-band is N~¢ + Nu,~p. The exciton band is empty if A is larger than a critical Ac given by A c = V2/4 + (1/2 /
dk at(k) e(k) ;
(5)
B
/z and Ac decrease monotonically with the number of holes in the valence band. On the other hand, if the exciton band is partially filled but the band of unpaired holes is empty, it requires a chemical potential lower than a critical #e
.~ = - ~ +
d~ al(~) ¢(¢)
(6)
q to place the first particle into that band. Note that the elemental excitations of the system are given by energy potentials; the rapidities parametrize their momentum. If Q + B ¢ 0 the low temperature corrections to (3) and (4) are of the order o f T 2, hence giving rise to a specific heat proportional to T, except when the Fermi level is at the van Hove singularity of an empty band where a T 1/~ contribution dominates. The exciton boundstates exist at all T; there is no phase transition, but also no long range order. A more complete discussion of the results will be presented elsewhere. The support of the Department of Energy under grant DE-FG05-91ER45443 is acknowledged. REFERENCES
1. M. Gaudin, Phys. Lett. 24A (19671 55. 2. C . N . Yang, Phys. Rev. Left. 19 (19671 1312. 3. M. Takahashi, Prog. Theor. Phys. 42 (1969 / 1098. 4. C . K . Lai, Phys. Rev. Lett. 26 (19711 1472.