Inducement of spin-pairing and correlated semi-metallic state in Mott–Hubbard quantum dot array

ARTICLE IN PRESS Physica E 41 (2009) 379–386

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Inducement of spin-pairing and correlated semi-metallic state in Mott–Hubbard quantum dot array Partha Goswami a,, Avinashi Kapoor b a b

Deshbandhu College, University of Delhi, Kalkaji, New Delhi 110019, India Department of Electronic Science, University of Delhi South Campus, Delhi, India

a r t i c l e in fo

abstract

Article history: Received 1 May 2008 Received in revised form 3 August 2008 Accepted 18 August 2008 Available online 9 September 2008

We model a quantum dot-array (with one electron per dot) comprising of NX2 coupled dots by an extended Hubbard Hamiltonian to investigate the role played by the inter-dot tunneling amplitude td, together with intra-dot (U) and inter-dot (U1) coulomb repulsions, in the singlet/triplet bound state formation and evolution of the system from an insulator-like state to a correlated semi-metallic state in the presence of magnetic field. We introduce a short-ranged inter-dot capacitive coupling U0, assumed to be non-zero for nearest-neighbor dots only, for the bound state analysis. The study shows that for the tunable parameter d ¼ (2td/U0)o1 the possibility of singlet state formation exists. As regards triplet state formation, even an infinitely small value of U0 is sufficient; the coulomb interactions (U, U1) play marginal role. The interaction U, however, is found to be important in magnetic field-induced evolution of the double quantum dot system to a correlated semi-metallic state. & 2008 Elsevier B.V. All rights reserved.

PACS: 74.90.+n Keywords: Tunneling amplitude Capacitive coupling Correlated semi-metallic state

1. Introduction

coulomb blockade condition (one electron per dot) is valid. For

d o1, both singlet and triplet states are possible. These facts It is possible to organize quantum dots (QDs) in a complex array. The array possesses features which combine the properties of solid state and atomic physics [1,2]. We consider here a system of NX2 coupled dots with one electron per dot. We model the system by a Hamiltonian involving inter-dot tunneling parameter td, intra-dot and inter-dot coulomb repulsions (U, U1) and shortranged inter-dot attraction U0 to investigate the role played by (td, U, U0) in the singlet/triplet bound state (BS) formation. The parameter td is complex when the magnetic field is present; the interaction U0 is essentially an electrostatic bonding. The approach to the problem involves setting up the Schrodinger equation in momentum space of the spin-paired fermions using the model Hamiltonian above. The integral equation obtained is solved in a manner similar to that in the well-known Cooper pair problem [3]. It is found that for the tunable parameter d ¼ (2td/U0)o1, the possibility of singlet state formation exists. As regards triplet state formation, even an infinitely small value of U0 is sufficient. The tunability of d stems from the fact that U0 could be changed by suitably tailoring the parameters, such as dot-size, inter-dot spacing, potential scenario, etc. The coulomb interactions (U,U1) do not play significant role in this aspect as long as the

 Corresponding author. Tel.: +91129 243 9099.

E-mail address: [email protected] (P. Goswami). 1386-9477/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.08.058

indicate that the system has the potential to act as the requisite structure for the basic gate operations of quantum computing. The introduction of the spin–orbit coupling (SOC) gives rise to mixing of the singlet and the triplet states. In the absence of SOC, the state of the paired fermions is either pure singlet or pure triplet. It may be mentioned that, several years ago, Loss and DiVincenzo [4,5] had mooted the proposal of the state-swap based on their investigation of a semi-phenomenological model Hamiltonian H ¼ JST(t)S1S2, where S1 and S2 are the spin-12 operators for the two localized electrons and the effective Heisenberg exchange coupling JST(t) is a function of time t. Moreover, in a QD with two active levels for transport and an even number of electrons on the dot is known, experimentally as well as theoretically [6–13], to yield singlet–triplet crossover in the presence of magnetic field due to Kondo effect. The background discussed above provides the motivation behind the problem investigated in Section 2 of this communication where we wish to show that inter-dot tunneling and interaction in a Mott–Hubbard (MH) array could also be tuned to have possibilities of singlet–triplet and pure triplet BSs in the coulomb blockade regime (CBR). This also generates the hope of showing, in future, the possibility of state-swap in a MH system with large state-swap-time (TswapU/td and Ubtd for a MH system) introducing the time-dependence explicitly in a more refined analysis. We extend the model Hamiltonian considered above with the inclusion of the single-particle terms [14]

ARTICLE IN PRESS 380

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involving explicit dependence on magnetic field in Section 3. The inter-dot tunneling amplitude is complex due to the appearance of the Peierls phase factor [14]. The purpose of the inclusion is to obtain single-dot (or single-particle) spectrum through which we wish to examine the evolution of a dot array from the insulatorlike to the correlated semi-metal-like state relaxing slightly the MH condition Ubtd in the CBR. The effect of magnetic field is directly involved here. In this regime the QDs, representing a molecule, is very weakly coupled to the source and drain which means that the charging energies U are still greater than the energies corresponding to contact, magnetic field and temperature-induced broadening of the electron levels in the dot. We have, in fact, the modest aim to examine whether additional energy level can surface in the Mottgap region of the MH system aided by magnetic field and the charging energy U. We consider a minimal system—a double QD (DQD) and corresponding Hamiltonian including Zeeman term, etc., owing to the presence of a magnetic field, to analyze the role played by the complex tunneling parameter together with coulomb interactions in the evolution of the system from the Mott insulator-like state to a correlated semi-metallic state. We find that one possible route for the evolution corresponds to the onset of magnetic field-aided charge bond ordering (CBO). We assume the tunneling event to be spin-conserving for simplicity and, therefore, introduce the Green’s functions for each spin channel separately. The Green’s functions are needed to compute the spectral function (SF) for the system under consideration where the magnetic field-induced CBO peak manifestation would occur. The signature of this evolution can possibly be seen in many experiments, such as the one on the magnetic field dependence of the differential conductance. The paper is organized as follows: in Section 2 we investigate the interplay of capacitive and tunneling couplings in singlet and triplet BS formation in a MH QD-array (with one electron per dot) comprising of NX2 coupled dots. In Section 3 we show that the interaction U, together with complex tunneling coupling td, plays an important role in the evolution of the DQD system from the insulator-like state to that of a correlated metallic state through CBO route. The paper ends with a discussion and a note on the possible uses of the investigation carried out.

A QD array may be treated as ‘‘giant artificial molecule’’. It consists of N islands with confined fermions. The inter-dot coupling, comprising of ionic and covalent components, could be tuned by varying the width of tunnel channel between them. In fact, both capacitive and tunneling couplings exist between these islands. We extend these ideas to a D-dimensional (DX1) dotarray comprising of N coupled dots (with one fermion per dot) and visualize it as a conglomerate of tiny artificial atoms whose positions are specified by a site index. We next consider an extended Hubbard-like model for this dot-array in order to investigate the singlet/triplet BS problem X X H¼  ½t d cyias cjas þ h:c: þ U nia" nia# þ U1

X 0

hiji a

nia nja 

X

ia

U aa 0 nia nja0 . 0

(1)

hijiaa0

cyias

coupling term involving NNN dots will be added later. U aa 0 is the capacitive coupling, assumed to be non-zero for NN dots only. The attraction essentially creates an electrostatic bond. Furthermore, the coupling may be assumed to be stronger for aaa0 than that for a ¼ a0 as a result of mutual attraction between electron and hole. The Hamiltonian (1) provides us a common plank to discuss the BS formation for a ¼ a0 (i.e. between electron–electron or hole–hole only) and aaa0 (i.e. excitonic (e–h) BS). Here /ijS0 corresponds to next NN (NNN) dots. We presently examine possibility of spin-singlet BS. With dot indices denoted by {l, m, y}, the total wave function in the singlet state can be written as X 0 0 p 0 jCi ¼ Cðl; mÞð1= 2ÞfFal Fam þ Fam Fal gwaa (2) s¼0 j0i, 0

l;m;aa0

p

waa s¼0 ¼ ð1= 2Þðwa" wa0 #  wa# wa0 " Þ. 0

creates a localized spin s fermion at ith The Fermi operator P y dot. The operators nias ¼ cyias cias and nia ¼ cias cias . Here i and j s (which correspond to lattice sites in a bulk MH system) are indices locating dots in the array. The tunneling is assumed to be constrained to nearest-neighbor (NN) dots /ijS. The tunneling

(3)

Here C(l, m) is the tight-binding amplitude to describe the system; F and w, respectively, correspond to the spatial and spin parts of the wave function. In terms of the fermion creation and y destruction  operators E clas and cma0 s one can write jCi ¼ P  aa0 l;m; Cðl; mÞFs ðl; mÞ where  E p X y y  aa0 fcla" cma0 #  cyla# cyma0 " gj0i. (4) Fs ðl; mÞ ¼ ð1= 2Þ aa0

We now write the Schrodinger equation for the spin-paired fermions as follows:   E X E X 0 0   Cðl; mÞFaa Cðl; mÞHFaa (5) EjCi ¼ H s ðl; mÞ ¼ s ðl; mÞ . l;m

l;m

8 < X X X Cðl; mÞ  ½t d cyias cjas þ h:c: þ U nia" nia# EjCi ¼ : hijias ia l;m 9 = E X X aa0 0  nia nja  U 0 nia nja0 Faa þU 1 s ðl; mÞ ; 0 0 hiji a

2. Singlet and triplet states

hijias

amplitude td is complex (owing to presence of a magnetic field) and corresponds to the kinetic energy. The indices a,a0 ¼ {e, h}. We include the effect of short range interactions only: U is the intradot coulomb repulsion (this is assumed to be very large Coulomb repulsion energy which will have to be paid if a dot is occupied by two fermions) and U1 is the inter-dot repulsion. A tunneling

(6)

hijiaa

The equation for C(l m) is obtained taking scalar product of 0 both the sides of the equation above with /Fsaa (l0 , m0 )|. Since aa0 0 0 /Fs (l , m )|CS ¼ C(l, m) we have X ECðl; mÞ ¼  ft d Cðj; mÞ þ t d Cðl; jÞg þ U Cðl; mÞdl;m j



X aa0 X U 0 Cðl; mÞdl;mþz þ U 1 Cðl; mÞdl;mþz . aa0 z

(7)

z

We have assumed that the dot-array is similar to a bulk solid-state environment. We may therefore define the Fourier transform (FT) X Cðk; qÞ ¼ N 1 Cðl; mÞ expfiððk=2Þ þ qÞl þ iððk=2Þ  qÞmg, l;m

where N is the number of dots in the array (at half-filling the number L of quasi-particle like fermions will be equal to N), to go over to k-space (momentum space). Taking FT is being used here as more of a theoretical tool; it may or may not be physically meaningful. We use this for the simple reason that we find the solutions of integral equation corresponding to Eq. (7) easily tractable due to close similarity with the well-known Cooper pair problem [3]. We have included excitonic BS consideration additionally this far, for electrons as well as exciton may act as

ARTICLE IN PRESS P. Goswami, A. Kapoor / Physica E 41 (2009) 379–386

the requisite structure for the basic gate operations of quantum computing. As mentioned in Section 1, the system under consideration being a dot array with one electron per dot. Hereinafter we shall focus only on singlet and triplet electronic BSs of this system with 0 U aa replaced by U0—a purely capacitive coupling. We introduce 0 P P the sums F1(K) ¼ N1 kC(K, k), F2(K, zˇ) ¼ N1 kC(K, k) exp(i k zˇ) 1P and F3(K, ´z) ¼ N z) and set up three equations kC(K, k) exp(i k ´ to determine them. Next, we obtain a condition for the solutions of these equations to exist which we wish to cast in the form P (1/G(e)) ¼ N1 k(e2ek)1. We need to look for the negative energy solution(s) of this equation to examine the two-electron BS formation condition. To this end, we multiply Eq. (7) by {exp{i((k/2)+q)l+i((k/2)q)m} and sum over all (l, m). After some algebra, we obtain from Eq. (7) ECðK; qÞ ¼ ððK=2ÞþÞq þ  ððK=2Þ  qÞgCðK; qÞ X X þ ðU=NÞ CðK; kÞ  ðz1 U 0 =2NÞ ðL1 ðq  kÞ k

X ðL2 ðq  kÞ k

þ L2 ðk  qÞÞCðK; kÞ,

integrals I0, I1 and I2 can be rewritten in the form X ð  2k Þ1 ;  ¼ E þ W 1 þ W 2 , I0 ¼ N 1 k av 2k ¼ W 1 ð1  Lav 1 ðkÞÞ þ W 2 ð1  L2 ðkÞÞ,

I1 ¼ W 1 1 ð1  EI0  W 2 J 1 Þ, I2 ¼ ðE=W 21 Þð1  EI0 Þ þ ðW 2 =W 1 ÞððEJ 1 =W 1 Þ  J 2 Þ,

(8)

P P P ˇ), where e(k) ¼  zˇtdexp(ik, zˇ) ´zt0 d exp(ik, ´z), L1(k) ¼ z1 1 zˇexp(ik z 1P L2(k) ¼ z2 z), and z1 is the number of NN, while z2 is the ´ zexp(ik ´ number of NNN. We have introduced a NNN term with tunneling amplitude t0 d above. Eq. (8) can be rewritten in terms of the sums F1(K), F2(K) and F3(K) as

(12)

where W1 ¼ 2z1td and W2 ¼ 2z2t0 d. As regards the evaluation of other integrals J1, J2, etc., we carry it out treating L2, which corresponds to NNN, to be q-independent. We find that the integrals, viz. J1, J2, J3 and In (n ¼ 1, 2,y), are expressible in terms of I0. It is tedious but straightforward to show that, under the approximations stated above, Eq. (11) may be written in the conventional form as X N1 ð  2k Þ1 ¼ ½fðb þ cÞf 1 ðEÞ þ bcBg=fðb þ cÞf 2 ðEÞ k þ bcf 1 ðEÞ þ 2bcðE þ cÞB  bcUBg,

k

þ L1 ðk  qÞÞCðK; kÞ þ ðz2 U 1 =2NÞ

381

(13)

where f1(E) ¼ [1aB(EU)], f2(E) ¼ [U(1+a+BE)BE22aEac], a ¼ [(z1U0W2L2)/W21], b ¼ (z2U1L22), c ¼ W2L2, and B ¼ [(z1U0)/W21]. Eq. (13) has been written down retaining all possible terms corresponding to NN as well as NNN. The latter ones are found to complicate the analysis without any trade-off of capturing some important piece of physics. Therefore, in what follows we approximate Eq. (13) as X f ðÞ ¼ ð1=GðÞÞ  N1 ð  2k Þ1 , (14) k

where

CðK; qÞ ¼ ðG=DÞ,

G1 ðÞ ¼  fW 21 =ðZz1 UU 0 Þg½f1 þ ðz1 UU 0 ð1  ZÞ=2W 21 g=ð  1P Þ

where

þ fW 21 =ðZz1 UU 0 Þg½f1 þ ðz1 UU 0 ð1 þ ZÞ=2W 21 Þg=ð  2P Þ, (15)

av G ¼ ½UF 1 ðKÞ  z1 U 0 Lav 1 ðqÞF 2 ðKÞ þ z2 U 1 L2 ðqÞF 3 ðKÞ,

D ¼ [Ee((K/2)+q)e*((K/2)q)], L1av(q) ¼ (12)(L1(q)+L1(q)), and 1 Lav 2 (q) ¼ (2)(L2(q)+L2(q)). In view of these results, equations for (F1, F2, F3) can be written in the neat form

1P ¼ fð1=2ÞUð1 þ ZÞ þ W 1 g, 2P ¼ fð1=2ÞUð1  ZÞ þ W 1 g, Z ¼ f1 þ ð4W 21 =z1 UU 0 Þg1=2 .

F 1 ð1  UI0 Þ þ ðz1 U 0 I1 ÞF 2 þ ðz2 U 1 J 1 ÞF 3 ¼ 0,

The function f(e) has simple poles at (e1P, e2P) and it intersects the energy axis at e ¼ W1+U+(W21/z1U0). The function has local minimum at e1 ¼ W1+U and local maximum at e2 ¼ W1+(U/2) (1+Z2). We find f(e1) ¼ U1, f(e2) ¼ (Z2U)1, and {f(e1P70+), f(e2P70+)}-7N Moreover, we find f(0) ¼ U2[{z1UU0W1 1 1 1 (1+UW1 )}] 1 +U}/{1z1U0(W1 +U

F 1 ðUI1 Þ þ ð1 þ z1 U 0 I2 ÞF 2 þ ðz2 U 1 J 2 ÞF 3 ¼ 0, F 1 ðUJ 1 Þ þ ðz1 U 0 J 2 ÞF 2 þ ð1  z2 U 1 J 3 ÞF 3 ¼ 0,

(9)

where In ¼ N1

X av ðL1 ðqÞÞn =D;

J 1 ¼ N1

q

J 2 ¼ N 1

X av ðL1 ðqÞLav 2 ðqÞÞ=D;

X av ðL2 ðqÞÞ=D, q

J3 ¼ N 1

q

X av ðL2 ðqÞÞ2 =D.

1 1 1 Þg, f ð0Þ ¼ U 2 ½fz1 UU 0 W 1 1 ð1 þ UW 1 þ Ug=f1  z1 U 0 ðW 1 þ U

(10)

q

The system of Eq. (9) will have solutions for Fj (j ¼ 1, 2, 3) provided the determinant of the coefficients of Fj in Eq. (9) is zero. We obtain ð1  UI0 Þð1 þ z1 U 0 I2 Þð1  z2 U 1 J 3 Þ þ ð1  UI0 Þðz1 z2 U 0 U 1 J 22 Þ þ ðz1 U 0 UI21 Þð1  z2 U 1 J 3 Þ þ ð2I1 z1 z2 UU 0 U 1 J 1 J 2 Þ þ ðz2 U 1 UJ 21 Þð1 þ z1 U 0 I2 Þ ¼ 0.

(16)

(11)

This is the equation to investigate the two-dot (with one electron per dot) BS formation. In the conventional Cooper pair problem [3], the strength of the attractive interaction is maximum when K ¼ 0, i.e. electron pairs with equal and opposite wave vectors. On a similar note, we assume here K ¼ 0. It is then easy to see from Ref. [9] that, in the weak magnetic field limit, one may av 1P 0 write DE[E+2tdLav 1 (q)+2t dL2 (q)] and I0 ¼ N q(1/D). Now the

1 )o1 and it is which implies that f(0)40 provided z1U0(W1 1 +U less than zero when the opposite is true. Similarly, if 2 1 z1U0(W1 )o1, f(W1)40 otherwise it is less than zero. In 1 +2U view of these conclusions it is not difficult to see that, for e2Po0 which implies U0o2td, Eq. (14) has no solution whereas, for e2P 40 which implies U042td (i.e. quite strong capacitive coupling), a negative energy solution is possible (see Figs. 1and 2). In Figs. 1 and 2 we have plotted dimensionless energy (e/U0) along x-axis and P both f(e/U0) and (N)1 kU0(e2ek)1 along y-axis. The BS energy values are given by the abscissa of the points which are intersection P of the curve (N)1 kU0(e2ek)1 (uneven-toothed curve) with the function f(e/U0) (with simple poles at (e1P/U0, e2P/U0))on the negative side of (e/U0). Figs. 1(A) and 2 correspond to N ¼ .15 and 30, respectively. The conclusion from above is that in the dot-array system comprising dots with odd electron occupation the formation of singlet BS is possible provided the inter-dot capacitive bonding is stronger than the inter-dot tunneling. It must be mentioned that, in the limit N-N, the sums in Eqs. (10) and (14) will get replaced by N-independent integrals. Therefore the conclusion above remains

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10

20

5

15

0

10

-5

5

-10 -15

0

-20 -5

-25

-10 -15

-30 -8

-6

-4

-2

0

2

4

6

8

-35

-8

-6

-4

-2

0

2

4

6

8

Fig. 2. In this figure, we have plotted dimensionless energy (e/U0) along x-axis and P both f(e/U0) and (N)1 kU0(e–2ek)1 (N ¼ 30, U ¼ 1.00 eV, td ¼ 0.18 eV, z1 ¼ 2 and U0 ¼ 0.30 eV) along y-axis. Here e2Po0 which implies U0o2td (i.e. the capacitive coupling is quite weak in comparison with the tunneling coupling). The plot shows non-feasibility of singlet BS formation in this case.

20 15 10

The equation for C(l, m) is same as Eq. (7) except the fact that the anti-symmetry of C(l, m) demands that C(m, m) should be zero now. We thus have

5 0

ECðl; mÞ ¼ 

-5

j



-10

X

U 0 Cðl; mÞjdl;mþz þ U 1

X

Cðl; mÞdl;mþz0 .

(18)

z0

z

-15

In the same way, the counterpart of Eq. (8) may be written as

-20

-8

-6

-4

-2

0

2

4

6

8

Fig. 1. In this figure, we have plotted dimensionless energy (e/U0) along x-axis and P both f(e/U0) and (N)1 kU0(e–2ek)1 (For (A) N ¼ 15, where as for (B) N ¼ 60; the rest of the parameter values are U ¼ 1.00 eV, td ¼ 0.18 eV, z1 ¼ 2 and U0 ¼ 0.47 eV )along y-axis. The singlet BS energy values are given by the abscissa of the points P which are intersection of the curve (N)1 kU0(e–2ek)1 (uneven-toothed curve) with the function f(e/U0) (with simple poles at (e1P/U0, e2P/U0)) on the negative side of (e/U0). Here e2P40 which implies U042td (i.e. the capacitive coupling is quite strong in comparison to the tunneling coupling).

the same, essentially, for an N-infinite system. Some details, such as the total number of available negative energy solutions, will differ since the right-hand-side of Eq. (14) will not be the same as that for finite N. Fig. 1(B), where we have taken N ¼ 60, confirms this fact. The values of U and td, respectively, in these figures are 1.00 and 0.18 eV with z1 ¼ 2; the interaction U0 is 0.47 eV in Fig. 1 (U042td) and 0.30 eV in Fig. 2 (U0o2td). We find that the coulomb repulsions (U,U1) do not play significant role as long as size of a QD is compatible with the assumption that there is one electron per dot. We now summarize below in brief a similar analysis carried out for triplet BS. For the triplet BS the wave functions, in terms of the tightbinding amplitude C(l, m), have the following form: X jC1 i ¼ Cðl; mÞcyl" cym" j0i, l;m

jC0 i ¼

X ft d Cðj; mÞ þ t d Cðl; jÞg

X

Cðl; mÞfcyl" cym# þ cyl# cym" gj0i,

l;m

jC1 i ¼

X l;m

Cðl; mÞcyl# cym# j0i.

(17)

ECðK; qÞ ¼ fððK=2Þ þ qÞ þ  ððK=2Þ  qÞgCðK; qÞ X  ðz1 U 0 =2NÞ k ðL1 ðq  kÞ þ L1 ðk  qÞÞCðK; kÞ X þ ðz2 U 1 =2NÞ k ðL2 ðq  kÞ þ L2 ðk  qÞÞCðK; kÞ.

(19)

After some amount of algebra, in the weak magnetic field approximation, ignoring the NNN term as before we find for triplet BS a Cooper pair-like equation ð1=U 0 Þ  FðÞ;

FðÞ ¼ N 1

X

2sin2 qa=ð  2q Þ.

(20)

q

Here ‘a’ is the separation between NN dots. Eq. (20) corresponds to triply degenerated solution(s) which have been sought graphically in Fig. 3. In this figure we have plotted dimensionless energy (e/U0) along x-axis and F(e/U0) along y-axis ( with value of the parameter N ¼ 20 and the ratio (Capacitive bonding/band width) ¼ 0.5). The BS energy values are given by the abscissas of the points which are intersection of the horizontal line (1) with the function F(e/U0) on the negative side of (e/U0). The lowest among such values of (e/U0) corresponds to ground state of triplet Cooper-like-pair. We, in fact, encounter here with no restriction involving the capacitive coupling (U0) and the tunneling coupling (td). It is thus clear from above that for U0o2td only a triplet BS is possible, whereas for U042td triplet as well as singlet states are possible. The finding shows the possibility of obtaining Loss and DiVincenzo [4,5] type state-swap in a MH system having large swap-time with the inclusion of time-dependence explicitly in a more sophisticated theoretical formulation.

ARTICLE IN PRESS P. Goswami, A. Kapoor / Physica E 41 (2009) 379–386

15 10 5 0 -5 -10 -15

-8

-6

-4

-2

0

2

4

6

8

Fig. 3. In this figure, we have plotted dimensionless energy (e/U0) along x-axis and F(e/U0) (with N ¼ 20 and the ratio (U0/W1) ¼ 0.5 where the bandwidth W1 ¼ 2z1t)along y-axis. The BS energy values are given by the abscissas of the points which are intersection of the horizontal line (1) with the function F(e/U0) on the negative side of (e/U0). The lowest among these values of (e/U0) corresponds to ground state of triplet Cooper-like-pair. We encounter here no restriction involving the capacitive coupling (U0) and the tunneling coupling (td).

3. Double quantum dot The coherent tunneling coupling and the capacitive (electrostatic) coupling are the well-known coupling mechanisms [14–17] for all possible dot-arrays including the simplest of such systems, viz. a DQD. The electrostatic coupling constants Uij, in the limit C5Cg where C is the capacitive coupling between the ith dot and neighboring dots and Cg the capacitance of a dot with relative to external gates, are given by UijU(C/Cg)|ij| [16,17]. The tunneling coupling td, on the other hand, delocalizes the electron wavefunction spreading over the neighboring dots and splitting energy states into singlet and triplet in the absence of SOC. As it is clear from above, the electrostatic bonding energy is smaller than the covalent bonding energy only when C5Cg and U is not too large compared to td. In this limit, we investigate here the possibility of inducement of semi-metal-like state in a DQD by considering a complex tunneling coupling. We have the modest aim to examine whether additional level can surface in the Mott-gap region of DQD system aided by magnetic field and the charging energy U via CBO route. It may be relevant to mention here that somewhat similar systems, such as artificial heterostructures and superlattices based on two different MH insulators, are known [18,19] to exhibit the possibility of tuning the carrier density by the electric field effect and (a superconducting transition too) at very low temperature. The simplest of all possible dot-arrays, as stated above, is the DQD first realized experimentally in planar geometry. We consider a minimal Hamiltonian for DQD in the presence of a magnetic field assumingly weak, to analyze the role played by the tunneling amplitude td (td ¼ t exp(ij) where j is the Peierls phase factor [14]) together with the coulomb repulsion U, in the evolution of the system from the Mott insulator-like state to a correlated semi-metallic state X X y H ¼  m0 ðc1s c1s þ cy2s c2s Þ þ ½t d cy1s c2s þ t d cy2s c1s  s

þ Uðn1" n1# þ n2" þ n2# Þ;

s

m0  m  D;

D ¼ _½ðoc =2Þ2 þ o20 Þ1=2 þ ð1Þs ðg mB B=2Þ.

(21) (22)

383

The first term in Eq. (21) is the intra-dot term which includes the chemical potential m, the cyclotron frequency oc, the Zeeman term (g mBB/2) and the confinement effect (e.g. for harmonic oscillator confinement potential in the absence of a magnetic field D ¼ _o0 with o0 being essentially a harmonic oscillator frequency). In the second term, the quantity td* ¼ t exp(ij).We consider a single spin-split level per dot, and set s ¼ 1, 2 in Eq. (22) to correspond to the spin up/down lowest confined QD level. We neglect corrections arising from single-particle level crossing [14] assuming the magnetic field to be weak. It may be noted that since we have also assumed here the electrostatic bonding energy to be smaller than the covalent bonding energy (which holds true when C5Cg and U is not too large compared P to td), we need not include the term ( ss0 U0n1sn2s0 ) in Eq. (21). We are interested here in single-particle spectrum, which is given by poles of the Fourier coefficient of the temperature function gaa0 s(t) ¼ /T{cas(t)cya0 s(0)}S (where a, a0 ¼ 1, 2 and T is the time-ordering operator which arranges other operators from right to left in the ascending order of time t), to fulfill the objective stated in Section 1. In what follows we summarize the technique adopted and the important steps in the calculation. The presence of atomic- and molecular-like spectra of QD array calls for appropriate theoretical tools of investigation. Since the exact diagonalization method [20,21] has its own limitations and Hartree–Fock method [22,23] suffers from sizeable systematic errors, one may treat the energy states of the array using density functional theory [24] with the exchange interaction involvement [25,26] explicitly. However, the spin-dependent properties are not exactly the focal point of the investigation at present. Besides, the strong coulomb repulsion is to be handled properly. We, therefore, prefer the equation of motion (EOM) method which is known to be reliable in the CBR and also qualitatively correct for the Kondo regime [27,28]. Since the Hamiltonian H is not diagonal one can write down the equations for the operators {c1s(t), c1s(t)n1s(t), y,} where the time evolution of an operator O is given by OðtÞ ¼ expðHtÞO expðHtÞ

(23)

in Hubbard approximation [29] to ensure that the thermal averages, such as gaa0 s(t), etc., are determined with consistency. Upon retaining terms of the type /nasS ( ¼ /cyascasS), and /naa0 sS ( ¼ /cyasca0 sS) in EOMs we find ðq=qtÞg 11" ðtÞ ¼ m0 g 11" ðtÞ  U G11" ðtÞ  t d g 21" ðtÞ  dðtÞ,

(24)

ðq=qtÞG11" ðtÞ   ðU  m0 ÞG11" ðtÞ  t d g 21" ðtÞhn1# i  t d g 11" ðtÞðhcy1# c2# i  hcy2# c1# iÞ  hn1# idðtÞ,

ðq=qtÞg 21" ðtÞ ¼ m0 g 21" ðtÞ  U G11" ðtÞ  t d g 11" ðtÞ,

(25)

(26)

ðq=qtÞG21" ðtÞ   ðU  m0 ÞG21" ðtÞ  t d g 11" ðtÞhn2# i þ t d g 21" ðtÞðhcy1# c2# i  hcy2# c1# iÞ,

(27)

where G11m(t) ¼ /T{c1m(t)n1k(t)cy1m(0)}S, G21m(t) ¼ /T{c2m(t)n2k (t)cy1m(0)}S and g21s(t) ¼ /T{c2s(t)cy1s(0)}S. The quantity m0  m_ [(oc/2)2+o20]1/2+(g mBB/2). In writing these equations we have ignored correlation between opposite spins. The Fourier coefficients of the thermal averages above are the Matsubara propagators {g11m(z), G11m(z), g21m(z), G21m(z)}, where z ¼ [(2n+1)pi/b] with n ¼ 0, 71, 72,y, with the aid of Eqs. (24)–(27) one obtains the

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e ¼ (U(1/n2kS)m0 ) in the Mott-gap-like region between the states e ¼ m0 and e ¼ (Um0 ). The Fourier coefficients (g11m(z), g21m(z)), from Eq. (28), are given by g11m(z) ¼ (D1/D) and g21m(z) ¼ (D3/D), where D is the determinant of the coefficients of {g11m(z), G11m(z), g21m(z), G21m(z}) in Eq. (28) and

following equations of these propagators: ðz þ m0 Þg 11" ðzÞ þ ðUÞG11" ðzÞ þ ðt d Þg 21" ðzÞ ¼ 1,  t d ðhcy1# c2# i  hcy2# c1# iÞg 11" ðzÞ þ ðz  ðU  m0 ÞÞG11" ðzÞ þ ðt d Þhn1# ig 21" ðzÞ ¼ hn1# i, ðt d Þg 11" ðzÞ þ ðz þ m0 Þg 21" ðzÞ þ ðUÞG21" ðzÞ ¼ 0,

D1 ¼ ðz þ m0  Uð1  hn1# iÞÞððz þ m0 Þðz þ m0  UÞ

ðt d Þhn2# ig 11" ðzÞ þ t d ðhcy1# c2# i  hcy2# c1# iÞg 21" ðzÞ þ ðz  ðU  m0 ÞÞG21" ðzÞ ¼ 0.

(28)

It must be noted that the EOMs for the other relevant Matsubara propagators {g22m(z), G22m(z), g12m(z), G12m(z)} can be obtained easily from above provided the interchange 122 and the replacement td ¼ t exp(ij)-t exp(ij) are made. For the propagators of spin-down channel, however, m0 has to be replaced by m00 m_[(oc/2)2+o20]1/2(g mBB/2). In writing the equations above we have completely ignored the correlation between intra-dot and inter-dot opposite spins thereby keeping the possibility of spin-density-wave-like and spin-bond-order-like state formations out of the purview of the present communication. Suppose, to begin with, we completely ignore the inter-dot correlations in Eq. (28). Then it is easy to see that the Matsubara propagator g11m (z) is given by Xð0Þ ð0Þ1 ðzÞ, (29) g 1 11" ðzÞ ¼ g 11" ðzÞ  12 where 0 0 g ð0Þ 11" ðzÞ ¼ ½fð1  hn1# iÞ=ðz þ m Þg þ fhn1# i=ðz þ m  UÞg,

Xð0Þ 12

ðzÞ ¼ t 2d ½fð1  hn2# iÞ=ðz þ m0 Þg þ fhn2# i=ðz þ m0  UÞg.

(30) (31)

This grossly approximate expression of the propagator g11m(z) clearly indicates that the tunneling coupling td (or the magnetic field) has the potential to alter the single-particle spectrum in a fundamental way. Taking cue from here we have solved Eq. (28) for all the propagators involved without making any approximation. In particular, for g11m(z), we obtain in the closed form ð1Þ1 g 1 11" ðzÞ ¼ g 11" ðzÞ 

XðeffÞ

ðzÞ, (32) P P(eff) (1)1 (0) (0) where the self-energy part 12 (z) ¼ g11m (z)g11m(z) 12 (z) and the propagator 12

0 1 g ð1Þ  B1 Uðz þ m0 Þ2 11" ðzÞ ¼ ½A1 ðz þ m Þ

þ A2 ðz þ m0  UÞ1  B2 Uðz þ m0  UÞ2 þ A3 ðz þ m0  Uð1  hn2# iÞÞ1 .

(33)

þ Ut d ðhn12# i  hn21# iÞÞ,

D3 ¼ t d ðz þ m0  UÞ2 þ Ut d ðhn1# i þ hn2# iÞðz þ m0  UÞ þ U 2 t d hn1# ihn2# i.

(38)

For (g22m(z), g12m(z)), td( ¼ t exp(ij)) above will get replaced by td*( ¼ t exp(ij)) as already mentioned below (28). Thus, though electron acquires the same phase j in the tunneling events 1-2 and 2-1, the terms (/n12kS/n21kS) and (/n1kS/n2kS), P P + which correspond to n exp(z 0 ) (g21m(z)g12m(z)) and n + exp(z 0 )(g11m(z)g22m(z)), respectively, will involve (t sin j)a0. P This is an outcome of the inclusion of the CBO term ( s(/ cy1sc2sS/cy2sc1sS)) and the thermal averages G11m(t),G21m(t), etc. in the EOM (24)(27). Also, we observe that the Hilbert space of the two-site Hubbard Hamiltonian H considered in this section is six dimensional with six possible basis states [30–32]. Furthermore, the Hamiltonian is exactly diagonalizable with sz ¼ 0 sector being represented by a 4  4 matrix. The orbital and the Zeeman effect emerge only in the triplet states (mm) and (kk). Now, four eigenvalues of the Hamiltonian matrix although carry no signature of magnetic flux, the CBO term P y y s(/c1sc2sS/c2sc1sS) does, as it can be easily seen that X y ðhc1s c2s i  hcy2s c1s iÞ ¼ 2i sin jZ1 Trfðq=qtÞ expðbHÞg (39) s

for a given magnetic flux. Here Z is the grand partition function of the system. The magnetic field aided inducement of an intermediate energy state in the Mott-gap region in a DQD system via CBO route, therefore, seems to be possible. An interesting question is what would be the effect on this intermediate state when the capacitive coupling involving term is also included in the Hamiltonian. We wish to address this issue in future. Presently, our task is to examine the SF to find out whether the state is sufficiently long-lived one. The single-dot SF in the spin-up channel is given by A11m(o) ¼ (p1)Im GR11m(o), where GR(o) is a retarded Green’s function given by Z 1 GR11" ðoÞ ¼ ðdo0 =2pÞfz11" ðo0 Þ=ðo  o0 þ i0þ Þg (40) 1

The coefficients {A1, A2,y} are given by A1 ¼ ½ð1  hn1# iÞ  ð2t d =UÞð1  hn1# iÞð1  hn2# iÞ1 ðhn12# iÞ  hn21# iÞ,

(34)

A2 ¼ ½hn1# i þ ð2t d =UÞhn1# ihn2# i1 ðhn12# i  hn21# iÞ,

(35)

A3 ¼ ð2t d =UÞðhn1# i  hn2# iÞð1  hn2# iÞ1 ðhn12# i  hn21# iÞhn2# i1 ,

(36)

2 Að3Þ 11"  4ðU=ðkB TÞÞðt sin j=UÞ ½ðhn1# i

B1 ¼ ðt d =UÞð1  hn1# iÞðhn12# i  hn21# iÞ, B2 ¼ ðt d =UÞhn1# iðhn12# i  hn21# iÞ.

and z11m(o) ¼ i{g11m(z)|z ¼ o–i0+g11m(z)|z ¼ o+i0+}. Similarly, SF in the spin-down channel can be obtained replacing m0 by m00. Upon using the result (x7i0+)1 ¼ [P(x1)7(1/i)pd(x)], where P represents a Cauchy’s principal value, in view of Eqs. (33)–(39) we find that the spin-up channel SF A11m(o) is given by a bunch of delta functions (a Fermi-liquid-like feature) in the form A11m(o)E (2) (3) 0 0 0 [A(1) For 11md(o+m )+A11md(o+m U)+A11md(o+m U(1/n2kS))]. (2) t5U (say, (t/U)E0.1), A(1) and A , respectively, are (1/n1kS) 11m 11m and /n1kS to the leading order. The third coefficient is

(37)

Since we have already assumed U to be not too large compared to td, (/n1kS,/n2kS)a1. We shall now consider the terms (/n12kS/n21kS) and (/n1kS/n2kS) in Eqs. (36) and (37). The aim is to explain that these terms are non-zero only in the presence of a magnetic field. As it is clear from Eqs. (33)–(37), the terms lead to emergence of an intermediate energy state

 hn2# ið1  hn1# iÞ=hn2# i,

(41)

where kB is the Boltzmann constant and T is the dot-array temperature. We have assumed here (U/(kBT))b1 (e.g. with U ¼ 1 eV and (kBT)E0.02 eV we obtain (U/(kBT))E50). Since the (2) (3) coefficients (A(1) 11m, A11m, A11m) are non-zero, all the three states 0 0 e ¼ m , e ¼ (Um ) and e ¼ (U(1/n2kS)m0 ) are sufficiently long-lived ones.

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140

140

120

120

100

100

80

80

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60

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-0.8

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140 120 100 80 60 40 20 0 -1

0)E[A(1) 11m

/U)+0.5)+A(2) 11m

(3) Fig. 4. These figures depict plots of SF, given by A11m(o/U, j ¼ d((o d((o/U)–0.5) and A11m(o/U,j d((o/U)+0.5)+A(2) 11md((o/U)–0.5)+A11m(j) d((o/U)0.02)], as functions of (o/U) for increasing Peierls phase factor j. The delta functions are replaced by appropriate Lorentzians mentioned in the text. The doublepeaked curve corresponds to a SF obtainable from Eqs. (29) and (30). The figures represent the emergence of correlated semi-metal-like feature from insulator-like-state for a DQD due to the presence of a magnetic field. A plot of spectral function for phi equal to (pi/6) (A); (pi/3) (B) and (pi/2) (C).

We now replace the delta functions above by the well-known result d(x) ¼ LimG-0(1/p)(G/(x2+G2)) to obtain a graphical representation of A11m(o). In Fig. 4, we have depicted A11m(o/U)-vs. -(o/U) sketches for j increasing. The double-peaked curves correspond to the SF obtainable from Eqs. (29) and (30) while the curves with three peaks correspond to A11m(o) with three delta functions. The values of the parameters are t ¼ 0.18 eV, U ¼ 1 eV and (kBT) ¼ 0.02 eV. Additionally, we have /n1kS ¼ 0.50, /n2kS ¼ 0.48 and m0 ¼ (U/2). We find that as j increases due to enhancement in magnetic flux through the system, taller peak for the intermediate energy state e ¼ (U(1–/n2kS)m0 ) is obtained owing to enhanced A(3) 11m.

4. Discussion In Section 2 we considered an extended Hubbard Hamiltonian for two-fermion BS in an array of NX2 dots. The Hamiltonian

a0)E[A(1) 11m

provides us a common plank to discuss BS formation for two electrons (or two holes) and excitons. We transformed our basis from the coordinate space to the momentum space, and solved the Schrodinger equation explicitly. We showed that, depending on capacitive coupling U0 between neighboring dots, the ‘‘BS energies’’ (negative eigen-energies) appear in the singlet and triplet pair for large U0, and only in the triplet pair for small U0. We have included excitonic BS consideration additionally, for electrons as well as exciton may act as the requisite structure for the basic gate operations of quantum computing. In the Hubbard approximation framework we have shown a major gain in the form of the emergence of quasi-metallic state (QMS) from insulating state in a DQD system, due to application of magnetic field, in Section 3. The gain is an outcome of the P inclusion of the CBO terms ( s(/cyasca0 sS–/cya0 scasS)a0) and the thermal averages G11m(t), G21m(t), etc. in the EOM (24)–(27). The appearance of QMS was quite an expected one as the possibility of external field driven, similar, metal–insulator

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transition in QD super-lattices had been reported [33–35] in the past. This gain indicates the usefulness of correlated DQD as a sensor. The effect has the potential of modulating excitations locally and can be applied to new field-effect devices.

[15] [16] [17] [18] [19]

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