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Hubbard chain with confinement: A model analysis of a finite chain of quantum dots
ELSEVIER
Physica B 229 (1997) 146-152
Hubbard chain with confinement: A model analysis of a finite chain of quantum dots Ryuichi Ugajin Sony Corporation Research Center, 174, Fujitsuka-cho, Hodogaya-ku, Yokohama 240, Japan
Received 2 May 1996
Abstract Correlated electron systems in a finite chain of quantum dots under an overall confining potential are discussed based on an analysis of a finite Hubbard chain of nine sites, in which five and six electrons are considered. Under this overall confining potential, a Mott Hubbard gap is induced by an increase in the density of electrons near the center of the chain, as indicated by the heat capacity. Tunneling through the Mott-Hubbard gap is found when six electrons are confined by the overall confining potential, as indicated by the equal-time Green's function of a single particle. This Hubbard gap tunneling is interpreted as electron tunneling through a barrier of electrons. Keywords: Quantum dot chain; Hubbard gap; Tunneling
1. Introduction An array of semiconductor quantum dots is of interest because m a n y - b o d y effects of electrons can be controlled by controlling the structure of the array [1, 2]. A quantum dot, i.e. a component of the array, can be realized by confining electrons in a zero-dimensional region. Confinement of electrons can be achieved in two ways: one way is to confine the electrons in the depletion region of a semiconductor [3] and the other way is to confine them by c o m p o u n d semiconductor heterostructures [4]. When we confine the electrons in the depletion region, which can be modeled by a parabolic potential [5], the typical size of the confining structure is 100 nm. M a n y - b o d y effects on heat capacity and magnetization have been reported based on numerical analyses of a small number of electrons confined by the parabolic potential [6, 7]. Optical measurements of a parabolically confined
quantum dot have been accomplished [8] and transport properties through this quantum dot have been studied to investigate charging effects [-9, 10]. A finite chain of these quantum dots has been considered to be a one-dimensional crystal [11]. This kind of confinement can be achieved by an external electric field. On the other hand, the size of the confining structure can be smaller than 15 nm if compound semiconductor heterostructures, e.g. GaAs/ A1GaAs, are used [12]. This confining structure can be modeled by a well-like potential [13]. In such a very small quantum dot, electrons cannot avoid each other, so a single-particle description is a good approximation for a few electrons in the quantum dot. If the size of the quantum dot is larger than 50nm, electron correlation becomes strong and two-electron states in a square-well quantum dot become states of the Wigner lattice [14]. If an array of coupled quantum dots in which electrons can
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R. Ugajin / Physica B 229 (1997) 146 152
tunnel through the barriers of a heterostructure between adjacent quantum dots we can expect electrons to have various phases when the array is half filled [15], e.g. a Mott insulating phase, a heavy electron phase, and a metallic phase [16]. These systems can be mapped to Hubbard-type models for low-energy excitation [17]. Although these two ways of confinement have been considered separately in previous studies, if these two ways are combined as follows, a wide variety of physical systems can be constructed. If an array of very small quantum dots of GaAs is embedded in bulk AlAs, a field effect caused by a depletion region or an external electric field causes a parabolic confinement over the array. This system can be realized using whiskers of compound semiconductors [18-21]. In this paper we consider a finite quantum dot chain where each quantum dot is assumed to have a single quantum state. These quantum dots are so small that the chain can be mapped to a finite Hubbard chain. This quantum dot chain is under a confining potential over the whole chain. The strength of this confining potential can be modulated. This paper analyzes the effect of the overall confining potential on electronic structures.
2. Hubbard chain with confinement
We consider a finite chain of quantum dots, which is described by a Hubbard-type model [22, 23]: L-1
=-
L
Ecj'~c; +'.+f2 j--L
a
r~jTr~j:+ h.c.,
has 18 564 states. Because these states can be classified by spin and parity, Hamiltonian matrices are block diagonal. We calculate the heat capacity as a function of temperature 1/fl: C(1//~) =/32((/4 2 > - (/~>2),
(2)
where ( 0 > = 2"(~b"10l~b">e-m Z,e_ m ,
(3)
and calculate the equal-time Green's function of a single particle
where Iq~> is a many-particle eigenstate. Gj~,j is the density of electrons with spin a at jth site. Whether electrons are localized or delocalized can be determined by the behavior of G~.j (i # j ) . In order to understand the properties of the ground state in detail, charge correlation functions and spin correlation functions are calculated in Appendix A using the Lanczos method. We take the conditions t = l and U = 1 6 throughout this paper, resulting in a strong correlation of electrons. These conditions can be realized in an array of semiconductor quantum dots. The strength of the on-side Coulomb interaction can be adjusted by changing the size of the quantum dot. Transfer between adjacent quantum dots can be made independent of the Coulomb interaction by setting the appropriate distance between adjacent quantum dots. The very small quantum dot under consideration makes it possible to ignore singleparticle excited states in each quantum dot.
Z j2Znj. j=-L
a
L
+ U ~
147
(1)
j=--L
where ~j, = dJ~?j,. ( ~ is the creation operator for an electron at j t h site with spin a. t is the transfer between adjacent sites and U is the on-site Coulomb interaction. An overall potential can be introduced by f2 > 0. The nth eigenenergy E, and its eigenstate [4 , > are obtained by numerical diagonalization. When L is taken to be 4, a five-electron system has 8568 states and a six-electron system
3. Effect of overall confinement on a chain of quantum dots
3.1. A chain of five electrons Let us start our discussion from a well-known case of five electrons with f2 = 0. The density of electrons is far from that of the half-filled case, so the system acts as a metal with no antiferromagnetic order, just as in an infinite chain [25]. As an overall confining potential is imposed, electrons are confined near the center of the chain, so the density
R. Ugajin/ Physica B 229 (1997) 146-152
148
of electrons in the center becomes high. When the density of electrons in this region approaches that of the half-filled case, a Mott insulating state is produced, resulting in a quasi-Hubbard gap [26]. Fig. 1 shows the heat capacity of five electrons in a chain of nine sites as a function of temperature, where Q is varied from 0 to 2.0. The reduction of the heat capacity in Fig. 1 when the temperature is about 1 indicates the appearance of the quasi-Hubbard gap induced by the overall confining potential. In an ideal one-dimensional system, excitations from the ground state can be classified by spin and charge [27]. When the temperature is very low, the heat capacity of the five-electron system has a maximum, as seen in Fig. 1. This is because a large degree of spin excitations with low energy is available in the system. A quasi-Hubbard gap is interpreted as a gap of charge excitations. The degenerate ground states of the five-electron system have spin _½, denoted as Itb+~/2). Fig. 2 shows the equal-time Green's function G~.s of the state 1~-~/2) when t 2 = 0 . G~_,,j decreases smoothly as j moves away from - 4 , and becomes negative when j is between 0 and 2. Thus there is a wavy behavior of the equal-time Green's function. On the other hand, when t2 = 1.0, antiferromagnetic behavior is seen in the equal-time Green's function, as shown in Fig. 3. An electron with down spin exists at the center of the chain, and there are
g
G.C]j a%
1 -~
-3
-2
-1
0 J
1
2
3
4
Fig. 2. Equal-time Green's function of the state Iq~_ 1/2) where five electrons are contained in a H u b b a r d chain of nine sites when t2 = 0. Open squares represent G]o, and solid diamonds
G+.
electrons with up spin at IJ I = 1. At IJ ] = 2, electrons with down spin exist. This indicates that antiferromagnetic order appears in the center of the chain, resulting in a quasi-Hubbard gap in the center.
3.2. A chain o f six electrons 3.5 I
.
.
.
.
.
I
at ,~ '~
~
2.5
a=2.o
r
I
.....f.~.k.i: ..'.." ,~"-.,.. '.,... "--.... ' : : ' 7"/.... "-., "-.,. -.,. " . , .... : ~,/.....
~;-i-.,'. / ; -,." .:
2
v ,' :":-.,,
............
.,¥, : : ,, ....
c= 1.5
,
;'x..'
':'/.
/
....
•............
..........
- .....
". . . . . . . . . . . .
-...... "................
1 0.5 0
0
1
2
3 4 Temperature
5
6
Fig. 1. Heat capacity of five electrons in a H u b b a r d claain of nine sites with the confining potential, f~ is taken to be from 0 to 2.0.
Let us now turn to a six-electron system in the Hubbard chain. Fig. 4 shows the heat capacity of six electrons in a chain of nine quantum dots as a function of temperature, where Q is varied from 0 to 3.0. While O is small, the behavior of the heat capacity is similar to that of the five-electron system. As Q becomes larger, the difference between the five- and six-electron systems becomes clearer. When a large Q enables sites in the center region to be doubly occupied by electrons, a collapse of the electron solid contained in the center region occurs. This is because a Mott insulating state becomes unstable when the density of electrons is changed from that of half-filled electrons. The second peak seen in Fig. 4 when t2 > 2 is caused by the double occupancy of the central site. When six electrons
R. Ugafin / Physica B 229 (1997) 146 152
149 I
g
l I
i
i
:
I
i
i
c~ =
.,.
.,. = }i
o
.=
G°-ld !
e-
-- - L I
r,.3
.=,
"7
G~jIT -4
i
i
i
-3
-2
-1
0
i
i
I
l
1
2
3
4
|
|
i
i
!
!
I
!
- 4 - 3 - 2 - 1 0 1 2 3 4
J
J Fig. 3. Equal-time Green's function of the state I ~ - 1/2) where five electrons are contained in a Hubbard chain of nine sites when f2 = 1.0. Open squares represent G Lt j ' and solid diamonds Gi~.i.
Fig. 5. Equal-time Green's function of six electrons in a Hubbard chain of nine sites when f2 = 0. Open squares represent G74 of the ground state I~ .... ) and solid diamonds GT.j of the state
14~o~d).
%./ r'~._
[v 0
j
f
~...______
.{2 = 0.0
I
I
I
2
3
1
4
Temperature Fig. 4. Heat capacity of six electrons in a Hubbard chain of nine sites with the confining potential, f2 is taken to be from 0 to 3.0 and successive traces are displaced upwards by 2.0.
are confined in an overall confining potential, electrons outside the Mott insulating center can tunnel through the quasi-upper Hubbard band in the center region, as discussed below.
When six electrons are considered, the ground state [4~.... ) has even parity and spin 0. The equaltime Green's function of the ground state [4 .... ) has the property of G],j = G~,I. We consider now another equal-time Green's function of the state [4~oaa), which is the least-energy state among states having odd parity and spin 0. Fig. 5 shows these two kinds of equal-time Green's functions of six electrons when f2 = 0. A distinct difference from the five-electron system is not apparent. If an overall confining potential is imposed, however, a remarkable difference is found, as shown in Fig. 6, where f2 = 1.0. In the region near the center of the chain, the behavior of the equal-time Green's function indicates a Mott insulating state similar to that in the five-electron system. Electrons are localized by the strong electron-electron correlation, and a quasi-Hubbard gap appears in the center. However, GL 3, j has a very different property. First, let us concentrate on the equal-time Green's function of the ground state [• . . . . ) , shown as open squares in Fig. 6. The amplitude of an electron from j = - 3 , i.e. GL3.j, decays as the electron approaches the center, indicated by GL3_2 and
R. Ugajin / Physica B 229 (1997) 146-152
150
|
|
|
|
!
m
E
m
l
l
r~
T T - ' - ' T, T T T , -4
-3
-2
-1
0
1
2
3
4
J Fig. 6. Equal-time Green's function of six electrons in a Hubbard chain of nine sites when ~2 = 1.0. Open squares represent GT.i of the ground state I~ .... ) and solid diamonds G74 of the state I ¢~odd).
GL 3, - 1. This means the electron is localized in the left region. However, G ~_3, 3 is very large, indicating that the localized electron in the left region tunnels through the quasi-Hubbard gap in the center region to the right region. This interpretation is supported by observation of the other equal-time Green's function of I~oaa), shown as solid diamonds in Fig. 6. When j < 2, GL3, j of [qboda) behaves similarly to that of I~ .... ), but GL 3, 3 of the I~oda) is negative, contrary to the G~_3.3 of the I~ .... ). Therefore, I~ .... ) can be interpreted as the bonding state of the two kinds of many-particle states, one of which is localized in the left-hand side of the center region and the other localized in the right-hand side of the center region. IOodd) is interpreted as the antibonding state of the two kinds of many-particle states. Therefore, the energy difference between [~e,en) and I(boda) is double the transfer energy governing the tunneling through the quasi-Hubbard gap of the center region. 4. Discussion
The transport properties through an array of quantum dots, where many electrons are confined
in each quantum dot and transfers between adjacent quantum dots are weak, have been discussed in the context of a Coulomb blockade [28]. In such a system, a finite difference in voltage between adjacent quantum dots can be induced by an applied electric field on a junction between adjacent quantum dots because of the weak transfer. These are many single-particle states in a quantum dot and the energy difference in these single-particle states is very small, so electrons in the quantum dot can react to the electromagnetic environment. The extra charge in the quantum dot induced by the movement of an electron in the adjacent quantum dot may prohibit tunneling of the electron to the quantum dot [29]. Once an electron tunnels through the junction, the electron will lose energy interacting with a mass of electrons in these quantum dots [30]. Thermodynamic variables for each quantum dot are useful to describe the system. As the quantum dots become smaller and the number of electrons in the quantum dots is reduced, the charge excitations of each component become stiff and the movement of electrons is considerably restricted. When the distance between adjacent quantum dots becomes small, coherent tunneling between adjacent quantum dots causes the formation of interdot spin-spin correlations. This system can no longer be described by an array of components which have thermodynamic variables, and should be described as a single quantum system. These systems can be described by multistate Hubbard models as used by Stafford et al. [17, 31] and interplay between coherent tunneling between adjacent quantum dots and charging effects affects transport properties of the array. The quantum dot array treated in the present paper also belongs to this kind of system, where ballistic transport theories apply if electron-electron interaction is not considered. In arrays of quantum dots, where electronelectron interaction plays an important role in transport properties, a simple description of singleelectron tunneling through a Hubbard gap, i.e. Hubbard gap tunneling, is possible when the number of electrons is even. This tunneling can be compared to the tunneling used by the Esaki diode [32], because in the Hubbard gap tunneling, an electron in the quasi-lower Hubbard band of the left-hand
R. Ugajin / Physica B 229 (1997) 146-152
side tunnels to the quasi-lower Hubbard band of the right-hand side. However, this tunneling does not happen in a five-electron system because of Pauli's principle. Electrons in quasi-lower Hubbard bands of left and right of the chain have the same spin at the center, so tunneling through the center region is not possible. In quantum mechanics, the energy of the particle is not defined locally, but globally in the system. Therefore, an electron of the energy eigenstate can tunnel through a local barrier. However, whether tunneling can take place is prescribed by the symmetries of the Hamiltonian describing the system and the symmetries imposed on quantum states. As the system becomes larger, the statistics of quantam may become less important, so tunneling through a bulk system of electrons can be described by the models previously presented [33-35].
5. Conclusion We have discussed the overall confinement effect on a finite quantum dot chain mapped to a finite Hubbard chain with confinement. We have calculated heat capacity and equal-time Green's functions, the result of which shows the existence of Mott insulating states in the center region where the density of electrons is enhanced by the overall confinement. The nature of Mott insulating states depends on the number of electrons, particularly whether the number is even or odd. This is due to the spin effect, which prohibits tunneling through the Hubbard gap. This kind of tunneling is found in a strongly correlated electron system. The tunneling, i.e. Hubbard gap tunneling, occurs when an electron tunnels through a barrier of electrons. Therefore, Pauli's principle plays a major role in determining the tunneling rate. This system can be prepared in an array of semiconductor quantum dots and reveals some features of correlated electrons.
Appendix In this appendix, we calculate charge correlation functions (~Uo[fig~fij~[7%) and spin correlation
151 |
i
i
!
i
i
X0d ¢. 0
i
X_I,j
0
6
~
v
v
v
2
3
L)
-4
-3
-2
-1
0
1
4
J Fig. 7, Charge and spin correlation functions of the ground state of six electrons when f2 = O. Open squares represent the charge correlation functions X~.j = (Tolfii.r~j~[~o) and solid diamonds the spin correlation functions Xi,j =
functions (~ol~=~ffl ~o > in the ground state of six electrons [kUo), where $~ = ½(flit -fii+)- The wave function of the ground state is obtained by the Lanczos method [-36]. These correlation functions are helpful to understand the properties of the ground state. In Fig. 7, we show the charge correlation function Xi,~= (kUol~i~r~j~[~o) indicated by open squares and double the spin correlation function Xi, i=2(~ol~qT$~lVo ) indicated by filled diamonds when f2 = 0. The charge correlation function of the nearest adjacent sites (~go In~,ni+ 1~[~o) is very small because of the strong correlation between up and down electrons. If l i - j [ > 1, the charge correlation function has a considerable value, indicating metallic behavior of electrons. The spin correlation function of the nearest adjacent sites has a minus sign, indicating antiferromagnetic correlations between nearest neighbor sites. As [i - j [ increases, the spin correlation function rapidly decreases because there is no antiferromagnetic order. Let us turn to these correlation functions when f2 = 1, as shown in Fig. 8. The behavior
152
R. Ugajin /Physica B 229 (1997) 146-152
0
,..., 0
T"-¢
V Vi
T
" T
7-:JT
m ....... -4
I
I
I
I
I
I
m
-3
-2
-1
1
2
3
4
J
Fig. 8. Charge and spin correlation functions of the ground state of six electrons when ~2 = 1.0. Open squares represent the charge correlation functions and solid diamonds the spin correlation functions.
of the spin correlation function, for example, (~o[~L2SZ2l~o)(~olSL2~Ll+2jl~o) < 0 (j = 0, 1.... ), indicates the antiferromagnetic order over the chain. It is notable that the charge correlation function (~o[~_3di3,l~Vo) is almost zero, compared to the Green's function of a single particle G*_3, 3 > 0, shown in Fig. 6. This supports our interpretation of an electron tunneling through a Hubbard gap, because a single-particle extended state has this property. For example, the state of an extended single electron I¢1) =
^'1" i0> Lcj,
(5)
J
has (¢11~;,~q,I ~t ) = fpfq and ( ¢ 1 Ir~w~a~l~1 ) ; fpz 6v.q. Variational wave functions of the manyelectron ground state having an electron tunneling through a Hubbard gap is discussed elsewhere.
References [1] R. Ugajin, T. Suzuki, K. Nomoto and I. Hase, J. Appl. Phys. 76 (1994) 1041. [2] R. Ugajin, Phys. Rev. B 51 (1995) 11 136.
[3] M.A. Reed, J.N. Randall, R. J. Aggarwal, R.J. Matyi, T.M. Moore and A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [4] H. Sakaki, Jpn. J. Appl. Phys. 28 (1989) L314. [5] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [6] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [7] P.A. Maksym and T. Chakraborty, Phys. Rev. B 45 (1992) 1947. [8] B. Meurer, D. Heitmann and K. Ploog, Phys. Rev. Lett. 68 (1992) 1371. [9] M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. [10] Y. Meir, N.S. Wingreen and P.A. Lee, Phys. Rev. Lett. 66 (1991) 3048. [11] L.P. Kouwenhoven, F.W.J. Hekking, B.J. van Wees, C.J.P.M. Harmans, C.E. Timmering and C.T. Foxon, Phys. Rev. Lett. 65 (1990) 361. [12] J. Oshinowo, M. Nishioka, S. Ishida and Y. Arakawa, Appl. Phys. Lett. 65 (1994) 1421. [13] R. Ugajin, Phys. Rev. B 51 (1995) 10714. [14] G.W. Bryant, Phys. Rev. Lett. 59 (1987) 1140. [15] R. Ugajin, J. Appl. Phys. 76 (1994) 2833. [16] R. Ugajin, Phys. Rev. B 53 (1996) 10141. [17] C.A. Stafford and S. Das Sarma, Phys. Rev. Lett. 72 (1994) 3590. [18] M. Yazawa, K. Haraguchi, M. Koguchi, T. Katsuyama and K. Hiruma, Appl. Phys. Lett. 65 (1994) 1157. [19] K. Haraguchi, T. Katsuyama, and K. Hiruma, J. Appl. Phys. 75 (1994) 4220. [20] T. Sato, K. Hiruma, M. Shirai, K. Tominaga, K. Haraguchi, T. Katsuyama and T. Shimada, Appl. Phys. Lett. 66 (1995) 159. [21] K. Hiruma, M. Yazawa, T. Katsuyama, K. Ogawa, K. Haraguchi, M. Koguchi and H. Kakibayashi, J. Appl. Phys. 77 (1995) 447. [22] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. [23] C.A. Stafford and A.J. Millis, Phys. Rev. B 48 (1993) 1409. [24] L.E. Reichl, The Transition to Chaos In Conservative Classical Systems: Quantum Manifestations (Springer, New York, 1992). [25] H. Shiba and P.A. Pincus, Phys. Rev. B 5 (1972) 1966. [26] N.F. Mott, Phil. Mag. 6 (1961) 287; Metal-Insulator Transitions (Taylor & Francis, London, 1974). [27] HJ. Schulz, Phys. Rev. Lett. 71 (1993) 1864. [28] A.A. Middleton and N.S. Wingreen, Phys. Rev. Lett. 71 (1993) 3198. [29] H. Grabert and M. H. Devoret (eds.), Single Charge Tunneling (Plenum Press, New York, 1992). [30] M. Ueda and T. Ando, Phys. Rev. Lett. 72 (1994) 1726. [31] C.A. Stafford and S. Das Sarma, preprint. [32] L. Esaki, Phys. Rev. 109 (1958) 603. [33] D. V. Averin and Yu. V. Nazarov, Phys. Rev. B 47 (1993) 9944. [34] C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220.
[35] A.I. Larkin and P.A. Lee, Phys. Rev. B 17 (1978) 1596. [36] H.Q. Lin and J.E. Gubernaits, Comput. Phys. 7 (1993) 400.
Physica B 229 (1997) 146-152
Hubbard chain with confinement: A model analysis of a finite chain of quantum dots Ryuichi Ugajin Sony Corporation Research Center, 174, Fujitsuka-cho, Hodogaya-ku, Yokohama 240, Japan
Received 2 May 1996
Abstract Correlated electron systems in a finite chain of quantum dots under an overall confining potential are discussed based on an analysis of a finite Hubbard chain of nine sites, in which five and six electrons are considered. Under this overall confining potential, a Mott Hubbard gap is induced by an increase in the density of electrons near the center of the chain, as indicated by the heat capacity. Tunneling through the Mott-Hubbard gap is found when six electrons are confined by the overall confining potential, as indicated by the equal-time Green's function of a single particle. This Hubbard gap tunneling is interpreted as electron tunneling through a barrier of electrons. Keywords: Quantum dot chain; Hubbard gap; Tunneling
1. Introduction An array of semiconductor quantum dots is of interest because m a n y - b o d y effects of electrons can be controlled by controlling the structure of the array [1, 2]. A quantum dot, i.e. a component of the array, can be realized by confining electrons in a zero-dimensional region. Confinement of electrons can be achieved in two ways: one way is to confine the electrons in the depletion region of a semiconductor [3] and the other way is to confine them by c o m p o u n d semiconductor heterostructures [4]. When we confine the electrons in the depletion region, which can be modeled by a parabolic potential [5], the typical size of the confining structure is 100 nm. M a n y - b o d y effects on heat capacity and magnetization have been reported based on numerical analyses of a small number of electrons confined by the parabolic potential [6, 7]. Optical measurements of a parabolically confined
quantum dot have been accomplished [8] and transport properties through this quantum dot have been studied to investigate charging effects [-9, 10]. A finite chain of these quantum dots has been considered to be a one-dimensional crystal [11]. This kind of confinement can be achieved by an external electric field. On the other hand, the size of the confining structure can be smaller than 15 nm if compound semiconductor heterostructures, e.g. GaAs/ A1GaAs, are used [12]. This confining structure can be modeled by a well-like potential [13]. In such a very small quantum dot, electrons cannot avoid each other, so a single-particle description is a good approximation for a few electrons in the quantum dot. If the size of the quantum dot is larger than 50nm, electron correlation becomes strong and two-electron states in a square-well quantum dot become states of the Wigner lattice [14]. If an array of coupled quantum dots in which electrons can
0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S092 1-4526(96)005 1 4-5
R. Ugajin / Physica B 229 (1997) 146 152
tunnel through the barriers of a heterostructure between adjacent quantum dots we can expect electrons to have various phases when the array is half filled [15], e.g. a Mott insulating phase, a heavy electron phase, and a metallic phase [16]. These systems can be mapped to Hubbard-type models for low-energy excitation [17]. Although these two ways of confinement have been considered separately in previous studies, if these two ways are combined as follows, a wide variety of physical systems can be constructed. If an array of very small quantum dots of GaAs is embedded in bulk AlAs, a field effect caused by a depletion region or an external electric field causes a parabolic confinement over the array. This system can be realized using whiskers of compound semiconductors [18-21]. In this paper we consider a finite quantum dot chain where each quantum dot is assumed to have a single quantum state. These quantum dots are so small that the chain can be mapped to a finite Hubbard chain. This quantum dot chain is under a confining potential over the whole chain. The strength of this confining potential can be modulated. This paper analyzes the effect of the overall confining potential on electronic structures.
2. Hubbard chain with confinement
We consider a finite chain of quantum dots, which is described by a Hubbard-type model [22, 23]: L-1
=-
L
Ecj'~c; +'.+f2 j--L
a
r~jTr~j:+ h.c.,
has 18 564 states. Because these states can be classified by spin and parity, Hamiltonian matrices are block diagonal. We calculate the heat capacity as a function of temperature 1/fl: C(1//~) =/32((/4 2 > - (/~>2),
(2)
where ( 0 > = 2"(~b"10l~b">e-m Z,e_ m ,
(3)
and calculate the equal-time Green's function of a single particle
where Iq~> is a many-particle eigenstate. Gj~,j is the density of electrons with spin a at jth site. Whether electrons are localized or delocalized can be determined by the behavior of G~.j (i # j ) . In order to understand the properties of the ground state in detail, charge correlation functions and spin correlation functions are calculated in Appendix A using the Lanczos method. We take the conditions t = l and U = 1 6 throughout this paper, resulting in a strong correlation of electrons. These conditions can be realized in an array of semiconductor quantum dots. The strength of the on-side Coulomb interaction can be adjusted by changing the size of the quantum dot. Transfer between adjacent quantum dots can be made independent of the Coulomb interaction by setting the appropriate distance between adjacent quantum dots. The very small quantum dot under consideration makes it possible to ignore singleparticle excited states in each quantum dot.
Z j2Znj. j=-L
a
L
+ U ~
147
(1)
j=--L
where ~j, = dJ~?j,. ( ~ is the creation operator for an electron at j t h site with spin a. t is the transfer between adjacent sites and U is the on-site Coulomb interaction. An overall potential can be introduced by f2 > 0. The nth eigenenergy E, and its eigenstate [4 , > are obtained by numerical diagonalization. When L is taken to be 4, a five-electron system has 8568 states and a six-electron system
3. Effect of overall confinement on a chain of quantum dots
3.1. A chain of five electrons Let us start our discussion from a well-known case of five electrons with f2 = 0. The density of electrons is far from that of the half-filled case, so the system acts as a metal with no antiferromagnetic order, just as in an infinite chain [25]. As an overall confining potential is imposed, electrons are confined near the center of the chain, so the density
R. Ugajin/ Physica B 229 (1997) 146-152
148
of electrons in the center becomes high. When the density of electrons in this region approaches that of the half-filled case, a Mott insulating state is produced, resulting in a quasi-Hubbard gap [26]. Fig. 1 shows the heat capacity of five electrons in a chain of nine sites as a function of temperature, where Q is varied from 0 to 2.0. The reduction of the heat capacity in Fig. 1 when the temperature is about 1 indicates the appearance of the quasi-Hubbard gap induced by the overall confining potential. In an ideal one-dimensional system, excitations from the ground state can be classified by spin and charge [27]. When the temperature is very low, the heat capacity of the five-electron system has a maximum, as seen in Fig. 1. This is because a large degree of spin excitations with low energy is available in the system. A quasi-Hubbard gap is interpreted as a gap of charge excitations. The degenerate ground states of the five-electron system have spin _½, denoted as Itb+~/2). Fig. 2 shows the equal-time Green's function G~.s of the state 1~-~/2) when t 2 = 0 . G~_,,j decreases smoothly as j moves away from - 4 , and becomes negative when j is between 0 and 2. Thus there is a wavy behavior of the equal-time Green's function. On the other hand, when t2 = 1.0, antiferromagnetic behavior is seen in the equal-time Green's function, as shown in Fig. 3. An electron with down spin exists at the center of the chain, and there are
g
G.C]j a%
1 -~
-3
-2
-1
0 J
1
2
3
4
Fig. 2. Equal-time Green's function of the state Iq~_ 1/2) where five electrons are contained in a H u b b a r d chain of nine sites when t2 = 0. Open squares represent G]o, and solid diamonds
G+.
electrons with up spin at IJ I = 1. At IJ ] = 2, electrons with down spin exist. This indicates that antiferromagnetic order appears in the center of the chain, resulting in a quasi-Hubbard gap in the center.
3.2. A chain o f six electrons 3.5 I
.
.
.
.
.
I
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~
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....
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..........
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". . . . . . . . . . . .
-...... "................
1 0.5 0
0
1
2
3 4 Temperature
5
6
Fig. 1. Heat capacity of five electrons in a H u b b a r d claain of nine sites with the confining potential, f~ is taken to be from 0 to 2.0.
Let us now turn to a six-electron system in the Hubbard chain. Fig. 4 shows the heat capacity of six electrons in a chain of nine quantum dots as a function of temperature, where Q is varied from 0 to 3.0. While O is small, the behavior of the heat capacity is similar to that of the five-electron system. As Q becomes larger, the difference between the five- and six-electron systems becomes clearer. When a large Q enables sites in the center region to be doubly occupied by electrons, a collapse of the electron solid contained in the center region occurs. This is because a Mott insulating state becomes unstable when the density of electrons is changed from that of half-filled electrons. The second peak seen in Fig. 4 when t2 > 2 is caused by the double occupancy of the central site. When six electrons
R. Ugafin / Physica B 229 (1997) 146 152
149 I
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i
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i
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.,.
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.=
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.=,
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i
i
i
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0
i
i
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l
1
2
3
4
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i
i
!
!
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!
- 4 - 3 - 2 - 1 0 1 2 3 4
J
J Fig. 3. Equal-time Green's function of the state I ~ - 1/2) where five electrons are contained in a Hubbard chain of nine sites when f2 = 1.0. Open squares represent G Lt j ' and solid diamonds Gi~.i.
Fig. 5. Equal-time Green's function of six electrons in a Hubbard chain of nine sites when f2 = 0. Open squares represent G74 of the ground state I~ .... ) and solid diamonds GT.j of the state
14~o~d).
%./ r'~._
[v 0
j
f
~...______
.{2 = 0.0
I
I
I
2
3
1
4
Temperature Fig. 4. Heat capacity of six electrons in a Hubbard chain of nine sites with the confining potential, f2 is taken to be from 0 to 3.0 and successive traces are displaced upwards by 2.0.
are confined in an overall confining potential, electrons outside the Mott insulating center can tunnel through the quasi-upper Hubbard band in the center region, as discussed below.
When six electrons are considered, the ground state [4~.... ) has even parity and spin 0. The equaltime Green's function of the ground state [4 .... ) has the property of G],j = G~,I. We consider now another equal-time Green's function of the state [4~oaa), which is the least-energy state among states having odd parity and spin 0. Fig. 5 shows these two kinds of equal-time Green's functions of six electrons when f2 = 0. A distinct difference from the five-electron system is not apparent. If an overall confining potential is imposed, however, a remarkable difference is found, as shown in Fig. 6, where f2 = 1.0. In the region near the center of the chain, the behavior of the equal-time Green's function indicates a Mott insulating state similar to that in the five-electron system. Electrons are localized by the strong electron-electron correlation, and a quasi-Hubbard gap appears in the center. However, GL 3, j has a very different property. First, let us concentrate on the equal-time Green's function of the ground state [• . . . . ) , shown as open squares in Fig. 6. The amplitude of an electron from j = - 3 , i.e. GL3.j, decays as the electron approaches the center, indicated by GL3_2 and
R. Ugajin / Physica B 229 (1997) 146-152
150
|
|
|
|
!
m
E
m
l
l
r~
T T - ' - ' T, T T T , -4
-3
-2
-1
0
1
2
3
4
J Fig. 6. Equal-time Green's function of six electrons in a Hubbard chain of nine sites when ~2 = 1.0. Open squares represent GT.i of the ground state I~ .... ) and solid diamonds G74 of the state I ¢~odd).
GL 3, - 1. This means the electron is localized in the left region. However, G ~_3, 3 is very large, indicating that the localized electron in the left region tunnels through the quasi-Hubbard gap in the center region to the right region. This interpretation is supported by observation of the other equal-time Green's function of I~oaa), shown as solid diamonds in Fig. 6. When j < 2, GL3, j of [qboda) behaves similarly to that of I~ .... ), but GL 3, 3 of the I~oda) is negative, contrary to the G~_3.3 of the I~ .... ). Therefore, I~ .... ) can be interpreted as the bonding state of the two kinds of many-particle states, one of which is localized in the left-hand side of the center region and the other localized in the right-hand side of the center region. IOodd) is interpreted as the antibonding state of the two kinds of many-particle states. Therefore, the energy difference between [~e,en) and I(boda) is double the transfer energy governing the tunneling through the quasi-Hubbard gap of the center region. 4. Discussion
The transport properties through an array of quantum dots, where many electrons are confined
in each quantum dot and transfers between adjacent quantum dots are weak, have been discussed in the context of a Coulomb blockade [28]. In such a system, a finite difference in voltage between adjacent quantum dots can be induced by an applied electric field on a junction between adjacent quantum dots because of the weak transfer. These are many single-particle states in a quantum dot and the energy difference in these single-particle states is very small, so electrons in the quantum dot can react to the electromagnetic environment. The extra charge in the quantum dot induced by the movement of an electron in the adjacent quantum dot may prohibit tunneling of the electron to the quantum dot [29]. Once an electron tunnels through the junction, the electron will lose energy interacting with a mass of electrons in these quantum dots [30]. Thermodynamic variables for each quantum dot are useful to describe the system. As the quantum dots become smaller and the number of electrons in the quantum dots is reduced, the charge excitations of each component become stiff and the movement of electrons is considerably restricted. When the distance between adjacent quantum dots becomes small, coherent tunneling between adjacent quantum dots causes the formation of interdot spin-spin correlations. This system can no longer be described by an array of components which have thermodynamic variables, and should be described as a single quantum system. These systems can be described by multistate Hubbard models as used by Stafford et al. [17, 31] and interplay between coherent tunneling between adjacent quantum dots and charging effects affects transport properties of the array. The quantum dot array treated in the present paper also belongs to this kind of system, where ballistic transport theories apply if electron-electron interaction is not considered. In arrays of quantum dots, where electronelectron interaction plays an important role in transport properties, a simple description of singleelectron tunneling through a Hubbard gap, i.e. Hubbard gap tunneling, is possible when the number of electrons is even. This tunneling can be compared to the tunneling used by the Esaki diode [32], because in the Hubbard gap tunneling, an electron in the quasi-lower Hubbard band of the left-hand
R. Ugajin / Physica B 229 (1997) 146-152
side tunnels to the quasi-lower Hubbard band of the right-hand side. However, this tunneling does not happen in a five-electron system because of Pauli's principle. Electrons in quasi-lower Hubbard bands of left and right of the chain have the same spin at the center, so tunneling through the center region is not possible. In quantum mechanics, the energy of the particle is not defined locally, but globally in the system. Therefore, an electron of the energy eigenstate can tunnel through a local barrier. However, whether tunneling can take place is prescribed by the symmetries of the Hamiltonian describing the system and the symmetries imposed on quantum states. As the system becomes larger, the statistics of quantam may become less important, so tunneling through a bulk system of electrons can be described by the models previously presented [33-35].
5. Conclusion We have discussed the overall confinement effect on a finite quantum dot chain mapped to a finite Hubbard chain with confinement. We have calculated heat capacity and equal-time Green's functions, the result of which shows the existence of Mott insulating states in the center region where the density of electrons is enhanced by the overall confinement. The nature of Mott insulating states depends on the number of electrons, particularly whether the number is even or odd. This is due to the spin effect, which prohibits tunneling through the Hubbard gap. This kind of tunneling is found in a strongly correlated electron system. The tunneling, i.e. Hubbard gap tunneling, occurs when an electron tunnels through a barrier of electrons. Therefore, Pauli's principle plays a major role in determining the tunneling rate. This system can be prepared in an array of semiconductor quantum dots and reveals some features of correlated electrons.
Appendix In this appendix, we calculate charge correlation functions (~Uo[fig~fij~[7%) and spin correlation
151 |
i
i
!
i
i
X0d ¢. 0
i
X_I,j
0
6
~
v
v
v
2
3
L)
-4
-3
-2
-1
0
1
4
J Fig. 7, Charge and spin correlation functions of the ground state of six electrons when f2 = O. Open squares represent the charge correlation functions X~.j = (Tolfii.r~j~[~o) and solid diamonds the spin correlation functions Xi,j =
functions (~ol~=~ffl ~o > in the ground state of six electrons [kUo), where $~ = ½(flit -fii+)- The wave function of the ground state is obtained by the Lanczos method [-36]. These correlation functions are helpful to understand the properties of the ground state. In Fig. 7, we show the charge correlation function Xi,~= (kUol~i~r~j~[~o) indicated by open squares and double the spin correlation function Xi, i=2(~ol~qT$~lVo ) indicated by filled diamonds when f2 = 0. The charge correlation function of the nearest adjacent sites (~go In~,ni+ 1~[~o) is very small because of the strong correlation between up and down electrons. If l i - j [ > 1, the charge correlation function has a considerable value, indicating metallic behavior of electrons. The spin correlation function of the nearest adjacent sites has a minus sign, indicating antiferromagnetic correlations between nearest neighbor sites. As [i - j [ increases, the spin correlation function rapidly decreases because there is no antiferromagnetic order. Let us turn to these correlation functions when f2 = 1, as shown in Fig. 8. The behavior
152
R. Ugajin /Physica B 229 (1997) 146-152
0
,..., 0
T"-¢
V Vi
T
" T
7-:JT
m ....... -4
I
I
I
I
I
I
m
-3
-2
-1
1
2
3
4
J
Fig. 8. Charge and spin correlation functions of the ground state of six electrons when ~2 = 1.0. Open squares represent the charge correlation functions and solid diamonds the spin correlation functions.
of the spin correlation function, for example, (~o[~L2SZ2l~o)(~olSL2~Ll+2jl~o) < 0 (j = 0, 1.... ), indicates the antiferromagnetic order over the chain. It is notable that the charge correlation function (~o[~_3di3,l~Vo) is almost zero, compared to the Green's function of a single particle G*_3, 3 > 0, shown in Fig. 6. This supports our interpretation of an electron tunneling through a Hubbard gap, because a single-particle extended state has this property. For example, the state of an extended single electron I¢1) =
^'1" i0> Lcj,
(5)
J
has (¢11~;,~q,I ~t ) = fpfq and ( ¢ 1 Ir~w~a~l~1 ) ; fpz 6v.q. Variational wave functions of the manyelectron ground state having an electron tunneling through a Hubbard gap is discussed elsewhere.
References [1] R. Ugajin, T. Suzuki, K. Nomoto and I. Hase, J. Appl. Phys. 76 (1994) 1041. [2] R. Ugajin, Phys. Rev. B 51 (1995) 11 136.
[3] M.A. Reed, J.N. Randall, R. J. Aggarwal, R.J. Matyi, T.M. Moore and A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [4] H. Sakaki, Jpn. J. Appl. Phys. 28 (1989) L314. [5] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [6] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [7] P.A. Maksym and T. Chakraborty, Phys. Rev. B 45 (1992) 1947. [8] B. Meurer, D. Heitmann and K. Ploog, Phys. Rev. Lett. 68 (1992) 1371. [9] M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. [10] Y. Meir, N.S. Wingreen and P.A. Lee, Phys. Rev. Lett. 66 (1991) 3048. [11] L.P. Kouwenhoven, F.W.J. Hekking, B.J. van Wees, C.J.P.M. Harmans, C.E. Timmering and C.T. Foxon, Phys. Rev. Lett. 65 (1990) 361. [12] J. Oshinowo, M. Nishioka, S. Ishida and Y. Arakawa, Appl. Phys. Lett. 65 (1994) 1421. [13] R. Ugajin, Phys. Rev. B 51 (1995) 10714. [14] G.W. Bryant, Phys. Rev. Lett. 59 (1987) 1140. [15] R. Ugajin, J. Appl. Phys. 76 (1994) 2833. [16] R. Ugajin, Phys. Rev. B 53 (1996) 10141. [17] C.A. Stafford and S. Das Sarma, Phys. Rev. Lett. 72 (1994) 3590. [18] M. Yazawa, K. Haraguchi, M. Koguchi, T. Katsuyama and K. Hiruma, Appl. Phys. Lett. 65 (1994) 1157. [19] K. Haraguchi, T. Katsuyama, and K. Hiruma, J. Appl. Phys. 75 (1994) 4220. [20] T. Sato, K. Hiruma, M. Shirai, K. Tominaga, K. Haraguchi, T. Katsuyama and T. Shimada, Appl. Phys. Lett. 66 (1995) 159. [21] K. Hiruma, M. Yazawa, T. Katsuyama, K. Ogawa, K. Haraguchi, M. Koguchi and H. Kakibayashi, J. Appl. Phys. 77 (1995) 447. [22] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. [23] C.A. Stafford and A.J. Millis, Phys. Rev. B 48 (1993) 1409. [24] L.E. Reichl, The Transition to Chaos In Conservative Classical Systems: Quantum Manifestations (Springer, New York, 1992). [25] H. Shiba and P.A. Pincus, Phys. Rev. B 5 (1972) 1966. [26] N.F. Mott, Phil. Mag. 6 (1961) 287; Metal-Insulator Transitions (Taylor & Francis, London, 1974). [27] HJ. Schulz, Phys. Rev. Lett. 71 (1993) 1864. [28] A.A. Middleton and N.S. Wingreen, Phys. Rev. Lett. 71 (1993) 3198. [29] H. Grabert and M. H. Devoret (eds.), Single Charge Tunneling (Plenum Press, New York, 1992). [30] M. Ueda and T. Ando, Phys. Rev. Lett. 72 (1994) 1726. [31] C.A. Stafford and S. Das Sarma, preprint. [32] L. Esaki, Phys. Rev. 109 (1958) 603. [33] D. V. Averin and Yu. V. Nazarov, Phys. Rev. B 47 (1993) 9944. [34] C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220.
[35] A.I. Larkin and P.A. Lee, Phys. Rev. B 17 (1978) 1596. [36] H.Q. Lin and J.E. Gubernaits, Comput. Phys. 7 (1993) 400.