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Electron tunneling through a Hubbard gap
PHYSICA ELSEVIER
Physica B 240 (1997) 128-132
Electron tunneling through a Hubbard gap Ryuichi Ugajin Sony Corporation Research Center, 174, Fu/itsuka-cho, Hodogaya-ku, Yokohama 240, Japan Received 11 October 1996
Abstract
Electronic states in a Hubbard chain of 13 sites under an overall confining potential are investigated by the Lanczos method. The Hubbard gap induced by the overall confining potential and electron tunneling through the Hubbard gap are sensitive to the number of electrons in the lower Hubbard band. Semiconductor quantum dot chains can be mapped to the Hubbard chain with confinement.
PACS: 73.20.Dx; 71.30.+h; 73.40.Gk Keywords." Hubbard gap; Tunneling; Quantum dot chain
Recent progress in fabrication technology makes it possible to fabricate a wide variety of semiconductor microstructures, e.g., quantum wires [1,2] and quantum dots [3,4]. In addition to three-dimensional fabrications made by combining epitaxial growth and lithography [5,6], new techniques, for example, one which produces whiskers, are emerging [7-9]. By setting appropriate growth conditions, very narrow crystals ("whiskers") of a compound semiconductor, whose width is of the order of 10 nm, have been made on a substrate. Whiskers can also be embedded in another kind of compound semiconductor to produce an array of quantum wires. Sequences of heterostructures, for example GaAs/AIGaAs, can be made along a whisker and applied to a one-dimensional array of quantum dots. These quantum dots made by heterostructures have different optical properties [10,11] than quantum dots made using the depletion regions 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S 0 9 2 I - 4 5 2 6 ( 9 7 ) 0 0 4 2 0 - 1
of a semiconductor [12,13] and are widely expected to be applied to new generation of optical and electronic devices [14-16]. In these quantum dot arrays, each quantum dot has a strong electron-electron interaction and the transfer energy between adjacent quantum dots can be changed by adjusting the distance between adjacent quantum dots [17,18]. When we consider electronic structures and transport properties through the quantum dot arrays, lattice models are useful because low-lying states are well simulated by appropriate lattice models [19,20]. When each quantum dot is sufficiently small, the excited state of each quantum dot may be ignored. Then the system can be mapped to a single-band Hubbard model [21 ]. Quantum dot arrays resemble arrays of atoms, and can provide new kinds of synthesized materials. For example, the density of electrons can be tuned and overall confinement potential can be applied to a chain
R. U g a j i n / P h y s i c a
of quantum dots [22]. When this overall confinement potential is achieved using the depletion regions of a semiconductor, the potential is parabolic. Let us consider a chain of quantum dots under a parabolic potential from the viewpoint of the mean-field approximation. If the density of electrons is lower than that with half-filled electrons, the system acts as a metal of correlated electrons. As the strength of the overall confining potential is increased, the density of electrons in the center of the chain becomes high, resulting in a Hubbard gap in the center [23]. Both sides of the Hubbard gap have a lower electron density, so the whole system is semi-metal in real space, providing a metal/Mott insulator/metal junction. We can expect electron tunneling through the Hubbard gap of the center. However, this scenario is not appropriate for chains with odd numbers of electrons because Pauli's principle applies in these cases. These results were obtained in our previous investigations of a Hubbard chain of nine sites with five or six electrons [22]. In this paper, electron tunneling through a Hubbard gap is discussed, based on Lanczos calculations on a Hubbard chain of 13 sites under a parabolic potential where eight, nine and ten electrons are confined. We consider a finite chain of quantum dots, which is described by a Hubbard-type model: L-I
L
j-L
a
j=L
cr
L
+U Z j
nj*njl + H . C . ,
(1)
L
where @ = @ @ . g]o creates an electron at jth site with spin a. t is the transfer between adjacent sites and U is the on-site Coulomb interaction. An overall confining potential is introduced when f2 > 0. We take L = 6 and t = 1 throughout this paper. When ten electrons are considered in the 13 sites, the system with total spin Sz = 0 has 1,656,369 states. We use the Lanczos algorithm for calculating the wave function of the ground state l TJ0) [24]. The behavior of tunneling through a Hubbard gap is analyzed using the equal-time Green's function of a single electron in the ground state aia, j =
(%IeZ~GI% ) .
(2)
B 240 (1997)
129
128 132
f~=l.2
-6
0
6
-
J
i Fig. 1. Equal-time correlation function Gi.j of a single electron in the ground state of the ten-electron chain with U = 16.
G~,j is the density of electrons with spin a at jth site. When the number of electrons is even, we will show that Gi,j = G i,j T -- - G i,j" 1 When the number of electrons is odd, we take the ground state with total spin & = - ½ out of two degenerate ground states and will show that GI,)) : Gtjand G},2)= G~j. Whether electrons are localized or delocalized can be determined by the behavior of Gi,/ (i C j). Let us start discussion from a ten-electron chain under an overall confining potential, whose behavior is similar to that of the chain of nine sites with six electrons previously reported in Ref. [22]. In Fig. 1, the equal-time Green's function Gi, j in the ground state of the ten-electron chain is shown as a function of i and j. The strength of on-site Coulomb interaction U is taken to be 16. When f 2 = 0 , Gi,j has large values where i ~ j and decays smoothly as i moves away from the value o f j . This is because metallic electrons are extended over the whole chain. When f2 > 0, Gi, j decays rapidly as i moves away from j, indicating the localization of electrons due to electron correlations. Thus, a Hubbard gap is made in the center of the chain. On the other hand, G5,-5 has a noticeable value along with a wide area where Gi,j ~'~ 0 extending between the slightly elevated point of (i,j) = (5, - 5 ) and the more elevated strip of i ~ j. This means that the electron
R. Ugq/in/ Physica B 240 (1997) 128 132
130
electron states with & = c~ for a Mott insulator of the center region, which are analogous to the states of the spin-density wave [25], are defined as
K~=1.2
j=-L N 1
× I-I
(3)
m--0
where 9 is the variational parameter of the Gutzwiller projector [26] and ~,,, (n = 0, 1.... ) is a normalization constant. When electrons are strictly localized at each site, these states may be written as
J -6
0
6
-
i Fig. 2. Equal-time correlation function Gi,/ of a single electron in the ground state of the eight-electron chain with U 16.
,~,0 / r~ C N, oC_N+I _ o .
If we take f j as an envelope function, ] I//~2N+2] ) =
Z .1¢)C~_ GI(~S,D2>
(4)
jG
at i = - 5 has tunneled to i = 5 through the Hubbard gap in the center of the chain. When ~2 is larger than 0.8, electron tunneling through the Hubbard gap is not encouraged. Hubbard gap tunneling is discouraged by the double occupancy o f the central site, which is recognized by the increase of G0,0 in Fig. 1 when ~2 is larger than 0.8. Note that three sites of the center are doubly occupied when (2 = 1.2. In Fig. 2, the equal-time Green's function Gi,j in the ground state of the eight-electron chain is shown. U is taken to be 16, again. The density of electrons is much lower than that of half-filled electrons, so electrons are more freely extended than those in the tenelectron chain with ~2 = 0, As sQ increases, electron tunneling through the Hubbard gap is seen, as indicated by G4, 4 <{0. Note that the value of G4, 4 is negative. This is related to the fact that the ground state of our chain has even parity along the chain, as is explained below. Let us look into many-electron states in a Hubbard chain with confinement. The wave function of the ruth single-electron eigenstate in the chain is denoted by @ml(m= 0, 1.... ), which is similar to the eigenfunction of a harmonic oscillator. We define an operator as q)m,o_= ~ ) _ C (pj(m)c~j,c,, SO a single-electron state ~,,o10) has parity ( - 1 ) m. Two kinds o f ( 2 N + 1)-
may be a variational wave function for a (2N + 2)electron chain with electron tunneling through the Hubbard gap. This hypothesis is suggested by the observation made in Ref. [22]. A ten-electron state having even parity
i%[,0,)
HI Z
(Pl--c~ (/)SDW\ , 4,0 /
(5)
o
may be suitable for the ground state including Hubbard gap tunneling because the state msDw\ --X,~ / has parity (-- 1 )X. On the other hand, an eight-electron state having even parity
I%%~
~ (/)SDW\ (X2 2..~ ~3 --0 3, a / (7
(6)
has G4, 4 < 0 , resulting in electron tunneling through the Hubbard gap. When a strong confining potential causes the central site to be doubly occupied, the tenelectron ground state may be \ g,-ol ~)SDW 4,0
,,
(7)
o
which has no tunneling electrons. Because the Hubbard gap becomes smaller as U decreases, electron tunneling through a Hubbard gap is
R. U,qajin/Physica B 240 (1997) 128 132
131
~=1.2
f~=1.2
Gij
-6 G
6
6
J -6
i
6
j
-
0
6 i
(a)
Fig. 3. Equal-time correlation function G,,/ of a single electron in the ground state of the eight-electron chain with U 8.
f2=1.2
G(2) 1,J
encouraged when U is small. Fig. 3 shows the equaltime Green's function in the ground state o f the eightelectron chain when U is taken to be 8, smaller than in the previous cases. Double occupancy o f the central site is induced by f2 ~>0.6. This critical strength o f confinement is weaker than that for a chain with a larger U. This is because double occupancy takes place when the energy o f the upper Hubbard band at the central site is lower than the energy o f electrons out o f the Mott insulator o f the center. We note that the double occupancy o f the three sites in the center is seen as in Fig. 1 when f2 = 1.2. However, these electrons in the upper Hubbard band extend more than those in Fig. 1. Let us turn to the case with an odd-number o f electrons in the chain, specifically, nine electrons in the chain when U = 16. The equal-time Green's functions G}I,) o f an up-spin electron in the ground state o f the
Fig. 4. Equal-time correlation functions (a) G}.!.) of an up-spin electron and (b) G!2. t,] ) of a down-spin electron in the ground state of the nine-electron chain with U = 16.
nine-electron chain and G!l,J2) o f a down-spin electron are shown in Fig. 4. While f2 is finite but small, alternating values o f --,Glli)and ~G}2i) indicate the antiferromagnetic order o f the Mott insulator. A nine-electron state ]~s~w ), just as the Mott insulator without tunneling electrons, may be appropriate for the ground state when/2 is small. However, when f2 -- 1 allows the central site to be doubly occupied, Hubbard gap tunnel-
(1) ing appears, as indicated by G4,-4 ~ 0 and G4(2)_4>~0. Note that ,~(1) .~(2) < 0 . The state o~4Co,.L](J~3,.[ At SDW) with tJ4, 4Lr4_4 S: = - 1 can be thought o f as a Mott insulating state o f eight electrons with double occupancy o f the cen~t ^'~ SDW tral site, so :~s~03,i.c0,;]4~3, ~ ) is a candidate for the ground state o f our nine-electron chain. This state
6 -6
0
i
6
-
J (b)
132
R Ugaiin/Physica B 240 (1997) 128-132
provides G~I) 4 < 0 . On the other hand, a (2N + 2)electron state
x
N ]--[ %,T ^t V,,,,; At p0), m=0
(8)
with S~ = 0 is possible for a Mott insulating state of eight electrons with double occupancy of the central site when N - 3 . If we add a tunneling electron to this eight-electron state, the electron in order to produce a should have spin ~ = - 5 nine-electron state with 5': ~ - 5' ' Thus, ~7q31,~lr3) is another candidate for the ground state of our nine-electron chain when Q - 1 . 0 . This state provides G4,-4 (2t > 0. The above discussion suggests that
~'~ ,3~5(03Tc0; (203,.L) +~7qb1,;IT3 )
(9)
is appropriate for the ground state of our nine-electron chain when f2 = 1.0. The double occupancy of more sites in the center is encouraged when /2 = 1.2 and a wavy behavior of electrons in the upper Hubbard band can be recognized as well as in Fig. 3. The system considered here is very simple, but presents a new kind of junction, i.e., a metal/Mott insulator/metal junction similar to a metal/insulator/ metal junction, e.g., A1/AI203/A1 or a semiconductor/ insulator/semiconductor junction, e.g., Si,/SiO2/Si. However, these systems have more features produced by individual electrons interacting with each other as governed by Pauli's principle. It is notable that the location of the junction can be chosen by an appropriate applied electric field - the junction is movable along the quantum dot chain. The region where the electron density is high enough to provide Mott insulating states can be transferred by a change brought about by an external electromagnetic field, as the depletion region of a semiconductor is transferred by a change brought about by the voltage difference in electrodes used in a charge-coupled device [27]. In conclusion, we have calculated equal-time Green's functions of a 13-site Hubbard chain with confinement. The overall confining potential affects
the electronic states of the chain, resulting in a Hubbard gap in the center of the chain. The nature of the Mott insulating states in the center depends on the number of electrons, particularly whether even or odd. When the strength of confinement is strong, the double occupancy of the central site reduces the number of electrons in the lower Hubbard band. Thus, a strong confinement enables chains with an odd number of electrons to exhibit electron tunneling through the Hubbard gap in the center of the chain.
References [1] T. Kono, S. Tsukamoto, Y. Nagamune, F. Sogawa, M. Nishioka, Y. Arakawa, Appl. Phys. Lett. 64 (1994) 1564. [2] Y. Nakamura, M. Tsuchiya, S. Koshiba, H. Noge, H. Sakaki, Appl. Phys. Lett. 64 (1994) 2552. [3] J.A. Lebens, C.S. Ysai, K.J. Vahala, Appl. Phys. Lett. 56 (1990) 2642. [4] J. Oshinowo, M. Nishioka, S. Ishida, Y. Arakawa, Appl. Phys. Lett. 65 (1994) 1421. [5] P.M. Petrofl, A.C. Gossard, W. Wiegmann,Appl. Phys. Lett. 45 (1984) 620. [6] Y. Fukui, S. Ando, Y. Tokura, T. Toriyama,Appl. Phys. Lett. 58 (1991) 2018. [7] M. Yazawa, K. Haraguchi, M. Koguchi, T. Katsuyama, K. Hiruma, Appl. Phys. Lett. 65 (1994) 1157. [8] T. Sato, K. Hiruma, M. Shirai, K. Tominaga, K. Haraguchi, T. Katsuyama,T. Shimada, Appl. Phys. Lett. 66 (1995) 159. [9] K. Hiruma, M. Yazawa, T. Katsuyama, K. Ogawa, K. Haraguchi, M. Koguchi, H. Kakibayashi, J. Appl. Phys. 77 (1995) 447. [10] R. Ugajin, Phys. Rev. B 51 (1994) 10714. [11] R. Ugajin, J. Appl. Phys. 77 (1995) 5509. [12] P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [13] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [14] Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939. [15] H. Sakaki, Jpn. J. Appl. Phys. 28 (1989) L314. [16] R. Ugajin, J. Appl. Phys. 76 (1994) 2833. [17] R. Ugajin,T. Suzuki, K. Nomoto, I. Hase, J. AppI. Phys. 76 (1994) 1041. [18] R. Ugajin, Phys. Rev. B 51 (1995) 11 136. [19] C.A. Stafford, S. Das Sarma, Phys. Rev. Lett. 72 (1994) 3590. [20] R. Ugajin, Phys. Rev. B 53 (1996) 10141. [21] C.A. Stafford, A.J. Millis, Phys. Rev. B 48 (1993) 1409. [22] R. Ugajin, Physica B 229 (1997) 146. [23] N.F. Mott, Philos. Mag. 6 (1961) 287; Metal insulator Transitions (Taylor & Francis Ltd, London, 1974 ). [24] H.Q. Lin, J.E. Gubemaits,Comput. Phys. 7 (1993) 400. [25] J.E. Hirsch, Phys. Rev. Lett. 53 (I984)2327. [26] W.F. Brinkman,T.M. Rice, Phys. Rev. B 2 (1970) 4302. [27] M.F. Tompsett, G.F. Amelio, G.E. Smith, Appl. Phys. Lett. 17 (1990) 111.
Physica B 240 (1997) 128-132
Electron tunneling through a Hubbard gap Ryuichi Ugajin Sony Corporation Research Center, 174, Fu/itsuka-cho, Hodogaya-ku, Yokohama 240, Japan Received 11 October 1996
Abstract
Electronic states in a Hubbard chain of 13 sites under an overall confining potential are investigated by the Lanczos method. The Hubbard gap induced by the overall confining potential and electron tunneling through the Hubbard gap are sensitive to the number of electrons in the lower Hubbard band. Semiconductor quantum dot chains can be mapped to the Hubbard chain with confinement.
PACS: 73.20.Dx; 71.30.+h; 73.40.Gk Keywords." Hubbard gap; Tunneling; Quantum dot chain
Recent progress in fabrication technology makes it possible to fabricate a wide variety of semiconductor microstructures, e.g., quantum wires [1,2] and quantum dots [3,4]. In addition to three-dimensional fabrications made by combining epitaxial growth and lithography [5,6], new techniques, for example, one which produces whiskers, are emerging [7-9]. By setting appropriate growth conditions, very narrow crystals ("whiskers") of a compound semiconductor, whose width is of the order of 10 nm, have been made on a substrate. Whiskers can also be embedded in another kind of compound semiconductor to produce an array of quantum wires. Sequences of heterostructures, for example GaAs/AIGaAs, can be made along a whisker and applied to a one-dimensional array of quantum dots. These quantum dots made by heterostructures have different optical properties [10,11] than quantum dots made using the depletion regions 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S 0 9 2 I - 4 5 2 6 ( 9 7 ) 0 0 4 2 0 - 1
of a semiconductor [12,13] and are widely expected to be applied to new generation of optical and electronic devices [14-16]. In these quantum dot arrays, each quantum dot has a strong electron-electron interaction and the transfer energy between adjacent quantum dots can be changed by adjusting the distance between adjacent quantum dots [17,18]. When we consider electronic structures and transport properties through the quantum dot arrays, lattice models are useful because low-lying states are well simulated by appropriate lattice models [19,20]. When each quantum dot is sufficiently small, the excited state of each quantum dot may be ignored. Then the system can be mapped to a single-band Hubbard model [21 ]. Quantum dot arrays resemble arrays of atoms, and can provide new kinds of synthesized materials. For example, the density of electrons can be tuned and overall confinement potential can be applied to a chain
R. U g a j i n / P h y s i c a
of quantum dots [22]. When this overall confinement potential is achieved using the depletion regions of a semiconductor, the potential is parabolic. Let us consider a chain of quantum dots under a parabolic potential from the viewpoint of the mean-field approximation. If the density of electrons is lower than that with half-filled electrons, the system acts as a metal of correlated electrons. As the strength of the overall confining potential is increased, the density of electrons in the center of the chain becomes high, resulting in a Hubbard gap in the center [23]. Both sides of the Hubbard gap have a lower electron density, so the whole system is semi-metal in real space, providing a metal/Mott insulator/metal junction. We can expect electron tunneling through the Hubbard gap of the center. However, this scenario is not appropriate for chains with odd numbers of electrons because Pauli's principle applies in these cases. These results were obtained in our previous investigations of a Hubbard chain of nine sites with five or six electrons [22]. In this paper, electron tunneling through a Hubbard gap is discussed, based on Lanczos calculations on a Hubbard chain of 13 sites under a parabolic potential where eight, nine and ten electrons are confined. We consider a finite chain of quantum dots, which is described by a Hubbard-type model: L-I
L
j-L
a
j=L
cr
L
+U Z j
nj*njl + H . C . ,
(1)
L
where @ = @ @ . g]o creates an electron at jth site with spin a. t is the transfer between adjacent sites and U is the on-site Coulomb interaction. An overall confining potential is introduced when f2 > 0. We take L = 6 and t = 1 throughout this paper. When ten electrons are considered in the 13 sites, the system with total spin Sz = 0 has 1,656,369 states. We use the Lanczos algorithm for calculating the wave function of the ground state l TJ0) [24]. The behavior of tunneling through a Hubbard gap is analyzed using the equal-time Green's function of a single electron in the ground state aia, j =
(%IeZ~GI% ) .
(2)
B 240 (1997)
129
128 132
f~=l.2
-6
0
6
-
J
i Fig. 1. Equal-time correlation function Gi.j of a single electron in the ground state of the ten-electron chain with U = 16.
G~,j is the density of electrons with spin a at jth site. When the number of electrons is even, we will show that Gi,j = G i,j T -- - G i,j" 1 When the number of electrons is odd, we take the ground state with total spin & = - ½ out of two degenerate ground states and will show that GI,)) : Gtjand G},2)= G~j. Whether electrons are localized or delocalized can be determined by the behavior of Gi,/ (i C j). Let us start discussion from a ten-electron chain under an overall confining potential, whose behavior is similar to that of the chain of nine sites with six electrons previously reported in Ref. [22]. In Fig. 1, the equal-time Green's function Gi, j in the ground state of the ten-electron chain is shown as a function of i and j. The strength of on-site Coulomb interaction U is taken to be 16. When f 2 = 0 , Gi,j has large values where i ~ j and decays smoothly as i moves away from the value o f j . This is because metallic electrons are extended over the whole chain. When f2 > 0, Gi, j decays rapidly as i moves away from j, indicating the localization of electrons due to electron correlations. Thus, a Hubbard gap is made in the center of the chain. On the other hand, G5,-5 has a noticeable value along with a wide area where Gi,j ~'~ 0 extending between the slightly elevated point of (i,j) = (5, - 5 ) and the more elevated strip of i ~ j. This means that the electron
R. Ugq/in/ Physica B 240 (1997) 128 132
130
electron states with & = c~ for a Mott insulator of the center region, which are analogous to the states of the spin-density wave [25], are defined as
K~=1.2
j=-L N 1
× I-I
(3)
m--0
where 9 is the variational parameter of the Gutzwiller projector [26] and ~,,, (n = 0, 1.... ) is a normalization constant. When electrons are strictly localized at each site, these states may be written as
J -6
0
6
-
i Fig. 2. Equal-time correlation function Gi,/ of a single electron in the ground state of the eight-electron chain with U 16.
,~,0 / r~ C N, oC_N+I _ o .
If we take f j as an envelope function, ] I//~2N+2] ) =
Z .1¢)C~_ GI(~S,D2>
(4)
jG
at i = - 5 has tunneled to i = 5 through the Hubbard gap in the center of the chain. When ~2 is larger than 0.8, electron tunneling through the Hubbard gap is not encouraged. Hubbard gap tunneling is discouraged by the double occupancy o f the central site, which is recognized by the increase of G0,0 in Fig. 1 when ~2 is larger than 0.8. Note that three sites of the center are doubly occupied when (2 = 1.2. In Fig. 2, the equal-time Green's function Gi,j in the ground state of the eight-electron chain is shown. U is taken to be 16, again. The density of electrons is much lower than that of half-filled electrons, so electrons are more freely extended than those in the tenelectron chain with ~2 = 0, As sQ increases, electron tunneling through the Hubbard gap is seen, as indicated by G4, 4 <{0. Note that the value of G4, 4 is negative. This is related to the fact that the ground state of our chain has even parity along the chain, as is explained below. Let us look into many-electron states in a Hubbard chain with confinement. The wave function of the ruth single-electron eigenstate in the chain is denoted by @ml(m= 0, 1.... ), which is similar to the eigenfunction of a harmonic oscillator. We define an operator as q)m,o_= ~ ) _ C (pj(m)c~j,c,, SO a single-electron state ~,,o10) has parity ( - 1 ) m. Two kinds o f ( 2 N + 1)-
may be a variational wave function for a (2N + 2)electron chain with electron tunneling through the Hubbard gap. This hypothesis is suggested by the observation made in Ref. [22]. A ten-electron state having even parity
i%[,0,)
HI Z
(Pl--c~ (/)SDW\ , 4,0 /
(5)
o
may be suitable for the ground state including Hubbard gap tunneling because the state msDw\ --X,~ / has parity (-- 1 )X. On the other hand, an eight-electron state having even parity
I%%~
~ (/)SDW\ (X2 2..~ ~3 --0 3, a / (7
(6)
has G4, 4 < 0 , resulting in electron tunneling through the Hubbard gap. When a strong confining potential causes the central site to be doubly occupied, the tenelectron ground state may be \ g,-ol ~)SDW 4,0
,,
(7)
o
which has no tunneling electrons. Because the Hubbard gap becomes smaller as U decreases, electron tunneling through a Hubbard gap is
R. U,qajin/Physica B 240 (1997) 128 132
131
~=1.2
f~=1.2
Gij
-6 G
6
6
J -6
i
6
j
-
0
6 i
(a)
Fig. 3. Equal-time correlation function G,,/ of a single electron in the ground state of the eight-electron chain with U 8.
f2=1.2
G(2) 1,J
encouraged when U is small. Fig. 3 shows the equaltime Green's function in the ground state o f the eightelectron chain when U is taken to be 8, smaller than in the previous cases. Double occupancy o f the central site is induced by f2 ~>0.6. This critical strength o f confinement is weaker than that for a chain with a larger U. This is because double occupancy takes place when the energy o f the upper Hubbard band at the central site is lower than the energy o f electrons out o f the Mott insulator o f the center. We note that the double occupancy o f the three sites in the center is seen as in Fig. 1 when f2 = 1.2. However, these electrons in the upper Hubbard band extend more than those in Fig. 1. Let us turn to the case with an odd-number o f electrons in the chain, specifically, nine electrons in the chain when U = 16. The equal-time Green's functions G}I,) o f an up-spin electron in the ground state o f the
Fig. 4. Equal-time correlation functions (a) G}.!.) of an up-spin electron and (b) G!2. t,] ) of a down-spin electron in the ground state of the nine-electron chain with U = 16.
nine-electron chain and G!l,J2) o f a down-spin electron are shown in Fig. 4. While f2 is finite but small, alternating values o f --,Glli)and ~G}2i) indicate the antiferromagnetic order o f the Mott insulator. A nine-electron state ]~s~w ), just as the Mott insulator without tunneling electrons, may be appropriate for the ground state when/2 is small. However, when f2 -- 1 allows the central site to be doubly occupied, Hubbard gap tunnel-
(1) ing appears, as indicated by G4,-4 ~ 0 and G4(2)_4>~0. Note that ,~(1) .~(2) < 0 . The state o~4Co,.L](J~3,.[ At SDW) with tJ4, 4Lr4_4 S: = - 1 can be thought o f as a Mott insulating state o f eight electrons with double occupancy o f the cen~t ^'~ SDW tral site, so :~s~03,i.c0,;]4~3, ~ ) is a candidate for the ground state o f our nine-electron chain. This state
6 -6
0
i
6
-
J (b)
132
R Ugaiin/Physica B 240 (1997) 128-132
provides G~I) 4 < 0 . On the other hand, a (2N + 2)electron state
x
N ]--[ %,T ^t V,,,,; At p0), m=0
(8)
with S~ = 0 is possible for a Mott insulating state of eight electrons with double occupancy of the central site when N - 3 . If we add a tunneling electron to this eight-electron state, the electron in order to produce a should have spin ~ = - 5 nine-electron state with 5': ~ - 5' ' Thus, ~7q31,~lr3) is another candidate for the ground state of our nine-electron chain when Q - 1 . 0 . This state provides G4,-4 (2t > 0. The above discussion suggests that
~'~ ,3~5(03Tc0; (203,.L) +~7qb1,;IT3 )
(9)
is appropriate for the ground state of our nine-electron chain when f2 = 1.0. The double occupancy of more sites in the center is encouraged when /2 = 1.2 and a wavy behavior of electrons in the upper Hubbard band can be recognized as well as in Fig. 3. The system considered here is very simple, but presents a new kind of junction, i.e., a metal/Mott insulator/metal junction similar to a metal/insulator/ metal junction, e.g., A1/AI203/A1 or a semiconductor/ insulator/semiconductor junction, e.g., Si,/SiO2/Si. However, these systems have more features produced by individual electrons interacting with each other as governed by Pauli's principle. It is notable that the location of the junction can be chosen by an appropriate applied electric field - the junction is movable along the quantum dot chain. The region where the electron density is high enough to provide Mott insulating states can be transferred by a change brought about by an external electromagnetic field, as the depletion region of a semiconductor is transferred by a change brought about by the voltage difference in electrodes used in a charge-coupled device [27]. In conclusion, we have calculated equal-time Green's functions of a 13-site Hubbard chain with confinement. The overall confining potential affects
the electronic states of the chain, resulting in a Hubbard gap in the center of the chain. The nature of the Mott insulating states in the center depends on the number of electrons, particularly whether even or odd. When the strength of confinement is strong, the double occupancy of the central site reduces the number of electrons in the lower Hubbard band. Thus, a strong confinement enables chains with an odd number of electrons to exhibit electron tunneling through the Hubbard gap in the center of the chain.
References [1] T. Kono, S. Tsukamoto, Y. Nagamune, F. Sogawa, M. Nishioka, Y. Arakawa, Appl. Phys. Lett. 64 (1994) 1564. [2] Y. Nakamura, M. Tsuchiya, S. Koshiba, H. Noge, H. Sakaki, Appl. Phys. Lett. 64 (1994) 2552. [3] J.A. Lebens, C.S. Ysai, K.J. Vahala, Appl. Phys. Lett. 56 (1990) 2642. [4] J. Oshinowo, M. Nishioka, S. Ishida, Y. Arakawa, Appl. Phys. Lett. 65 (1994) 1421. [5] P.M. Petrofl, A.C. Gossard, W. Wiegmann,Appl. Phys. Lett. 45 (1984) 620. [6] Y. Fukui, S. Ando, Y. Tokura, T. Toriyama,Appl. Phys. Lett. 58 (1991) 2018. [7] M. Yazawa, K. Haraguchi, M. Koguchi, T. Katsuyama, K. Hiruma, Appl. Phys. Lett. 65 (1994) 1157. [8] T. Sato, K. Hiruma, M. Shirai, K. Tominaga, K. Haraguchi, T. Katsuyama,T. Shimada, Appl. Phys. Lett. 66 (1995) 159. [9] K. Hiruma, M. Yazawa, T. Katsuyama, K. Ogawa, K. Haraguchi, M. Koguchi, H. Kakibayashi, J. Appl. Phys. 77 (1995) 447. [10] R. Ugajin, Phys. Rev. B 51 (1994) 10714. [11] R. Ugajin, J. Appl. Phys. 77 (1995) 5509. [12] P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [13] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [14] Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939. [15] H. Sakaki, Jpn. J. Appl. Phys. 28 (1989) L314. [16] R. Ugajin, J. Appl. Phys. 76 (1994) 2833. [17] R. Ugajin,T. Suzuki, K. Nomoto, I. Hase, J. AppI. Phys. 76 (1994) 1041. [18] R. Ugajin, Phys. Rev. B 51 (1995) 11 136. [19] C.A. Stafford, S. Das Sarma, Phys. Rev. Lett. 72 (1994) 3590. [20] R. Ugajin, Phys. Rev. B 53 (1996) 10141. [21] C.A. Stafford, A.J. Millis, Phys. Rev. B 48 (1993) 1409. [22] R. Ugajin, Physica B 229 (1997) 146. [23] N.F. Mott, Philos. Mag. 6 (1961) 287; Metal insulator Transitions (Taylor & Francis Ltd, London, 1974 ). [24] H.Q. Lin, J.E. Gubemaits,Comput. Phys. 7 (1993) 400. [25] J.E. Hirsch, Phys. Rev. Lett. 53 (I984)2327. [26] W.F. Brinkman,T.M. Rice, Phys. Rev. B 2 (1970) 4302. [27] M.F. Tompsett, G.F. Amelio, G.E. Smith, Appl. Phys. Lett. 17 (1990) 111.