- Email: [email protected]
Coherent conductance in an alternating dot: exact results
ARTICLE IN PRESS
Physica E 27 (2005) 227–234 www.elsevier.com/locate/physe
Coherent conductance in an alternating dot: exact results Mohammad Mardaani, Keivan Esfarjani Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Received 1 November 2004; accepted 17 November 2004 Available online 23 January 2005
Abstract In this paper we have calculated the conductance of a periodic quantum dot attached to metallic leads, within the tight-binding (TB) model and in the ballistic regime. We have calculated the Green’s function (GF), density of states (DOS) and the coherent transmission coefficient (TC) fully analytically for an alternating quantum dot (A-QD). The quasi-gap, bound states energies, the energy and dot-size dependence of the GF and conductance for the system are also derived. Finally, we show analytically the conductance can be switched between insulating (OFF) and conducting (ON) states by applying a gate voltage. r 2004 Elsevier B.V. All rights reserved. PACS: 05.60.Gg; 81.07.Ta; 81.07.Vh Keywords: Green’s Function; Quantum transport; Alternating dots; Bound states
1. Introduction During the past few years, the study of electronic transport through quantum dots (QDs) and quantum wires have been major areas in nanoelectronics and nanostructure physics [1–9]. These systems play an important role in the development of molecular wires for nanoelectronic devices. Experimental research on quantum and molecular wires has increased over the past few Corresponding author.
E-mail address: [email protected] (M. Mardaani).
years looking into the possibility of rectification and other related phenomena [4–7]. There has also been an increasing interest in deriving analytical results on transport in molecular wires, single molecule, and nanocrystal devices [10–14]. Different methods based on the Landauer formula [3,15] which states that the conductance is proportional to the transmission coefficient, have been developed for the study of transport phenomena. One class of methods is based on transfer matrix [3,4,16], and the other makes use of the Green’s function formalism [3,14,17–20]. We rely on the GF method for studying the coherent transport properties in quantum dot systems. It relates the
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.11.012
ARTICLE IN PRESS 228
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
lead-to-lead current to the transmission probability for an electron to scatter through the quantum dot. By coherence mechanism, we mean that the dot length is smaller than the phase coherence length so that electrons do not experience any inelastic scattering during transport except in contacts. In our previous work [14], we studied coherence conductance for a uniform dot and uniform nanocrystal by the GF method. In the present work, we study the coherent transport properties of an alternating dot in the presence of metallic leads, within the TB model and in the ballistic regime. The organization of this paper is as follows. In Section 2, the tight-binding one-electron Hamiltonian for A-QD is described. In this section, the transmission coefficient and the local density of states, which generally depend on the Fermi energy, dot size, dot-lead coupling, and gate voltage, are analytically calculated. The TC calculations in symmetrical case are explained in much detail. Three characteristic energy ranges are identified for this problem. Finally, we obtain a nonlinear equation which gives the bound state energies. The paper is ended by a conclusion in Section 3. In appendix some useful relations are derived which relate the GF, TC, and DOS to the parity of the number of atoms in the chain for an A-QD.
…
(1)
-1,0;
1,2, 3
…
N; N+1 …
where H L ; H DL ; H D ; H DR ; H R refer to the leftlead, dot to left-lead, isolated dot, dot to rightlead, and right-lead Hamiltonians, respectively. Fig. 1 shows the geometry of an A-QD attached to two metallic leads. The first and the last atoms in the dot are, respectively, labeled by 1 and N. In the TB approximation, and assuming nearest-neighbor interaction, the Hamiltonian of the leads is defined as follows: X H LðRÞ ¼ tLðRÞ ðcyz czþ1 þ h:c:Þ; (2) z
where H LðRÞ and tLðRÞ describe the semi-infinite uniform left (right)-lead Hamiltonians and the hopping integral, respectively. For simplicity, the atomic on-site energies of both leads are set to zero. The dot-lead Hamiltonian is defined as H DLðRÞ ¼ tDLðRÞ ðcy0ðNÞ c1ðNþ1Þ þ h:c:Þ;
(3)
where tDLðRÞ refer to the contact hopping integrals which depend on distance between the dot and the leads [3]. The dot Hamiltonian is given by N X z¼1
In this section, we describe our model and assumptions, and derive a formula allowing one to calculate the GF and the conductance of a A-QD analytically. Consequently, in what follows, we will assume a chain composed of a central region (dot) and two general leads, and will derive the analytic expression for the GF and the TC in the resonance and tunneling regions, assuming the dot is periodic with two atoms per unit cell. In the general form, there will be various on-site energies and hopping integrals in each region, plus two hoppings from dot to the two leads. We use the following generalized Hamiltonian for the system:
Right-Lead
…
Fig. 1. Geometry of an A-QD attached to two leads.
HD ¼
2. The theoretical model
H ¼ H L þ H DL þ H D þ H DR þ H R ;
Periodic Dot Left-Lead
z cyz cz þ
N 1 X
tz;zþ1 ðcyz czþ1 þ h:c:Þ;
(4)
z¼1
where z ¼ 0 þ uð1Þzþ1 and tz;zþ1 ¼ tD refer to the on-site energy and hopping integral for nearest neighbor sites, respectively. From Eq. (4), we can obtain the retarded GF of the dot in contact with leads using the following partitioning formula. For a general system divided into two parts A and B, it is easy to show that the total (A+B) GF function projected onto the Hilbert space of the dot (called A below), can be written as [19] 1 G1 A ¼ G 0A H AB G 0B H BA ;
(5)
where G0A and G A are GFs of the A system in the absence and presence of the B system, respectively, H AB is the interaction Hamiltonian between A
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
and B. The last term in Eq. (5) is also called the self-energy due to system B. For the lead–dot–lead configuration defined above, we have ½G1 ij ¼ ð þ i0þ i SL di1 SR diN Þdij tD ðdi;jþ1 þ di;j1 Þ;
ð6Þ
where SL and SR refer to the self-energies due to the left and right leads. It is known that energy bands of the leads are given by 2tR cosðkR aÞ and 2tL cosðkL aÞ; so that self-energies can be calculated as follows [14]: t2DL exp½ikL ðÞa; tL t2 SR ðÞ ¼ DR exp½ikR ðÞa; tR
SL ðÞ ¼
ð7Þ
where a refers to the lattice constant in the leads. For the calculation of the TC, G1N and the imaginary part of SLðRÞ are needed [3] TðÞ ¼ 4 ImðSL ÞImðSR ÞjG 1N j2 :
(8)
At zero temperature and zero bias, the conductance is just 2e2 =h times the transmission coefficient.
229
where M ðN¼2nÞ ðx; a; y; bÞ 0 x a 1 B B 1 y B B :: ¼B B :: B B 0 0 @ 0 0
::
0
::
0
::
::
::
x
::
1
0
1
C C C C :: C C C 1 C A yb 0
;
2n 2n
in which x ¼ ðz 0 uÞ=tD ; y ¼ ðz 0 þ uÞ=tD ; z ¼ þ i0þ ; a ¼ SL =tD ; b ¼ SR =tD ; and n is the number of unit cells. G 1;2n has the following form: G1; 2n ¼
1 ; tD D2n ðx; a; y; bÞ
(11)
where D2n ðx; a; y; bÞ ¼ Det½M 2n ðx; a; y; bÞ: The explicit form of D is given in the appendix. Now, from Eq. (8) the transmission coefficient can be obtained as TðÞ ¼
4 t2DL t2DR j sinðkL aÞ sinðkR aÞj : t2D tL tR j D2n ðx; a; y; bÞj2
(12)
In order to calculate the local DOS, diagonal elements of the GF are needed: 3. Analytical results for TC and local DOS In this model we assume the dot is periodic with two atoms per unit cell with different on-site energies HD ¼
N X
½0 þ uð1Þzþ1 cyz cz
1 DOSj ðÞ ¼ ImG jj ðÞ; p
depending on parity of j, the form of diagonal elements of the GF is as follows (appendix): G2jþ1; 2jþ1 ¼
z¼1
þ tD
N 1 X
ðcyz czþ1 þ h:cÞ:
ð9Þ G2j; 2j ¼
z¼1
The inverse GF of the A-QD in the presence of leads, can be written in the following form: G 1 D ¼ tD M N ðx; a; y; bÞ;
TðÞ ¼
(10)
t2D j
(13)
D2j ðx; a; y; 0ÞD2n2j1 ðy; 0; y; bÞ ; tD D2n ðx; a; y; bÞ
D2j1 ðx; a; x; 0ÞD2n2j ðx; 0; y; bÞ : tD D2n ðx; a; y; bÞ
(14)
From Eqs. (8) and (11), using the results of appendix, we can obtain the transmission coefficient as follows:
4 ImðSL ÞImðSR Þ ; SL SR 1 0 u 0 þu D2n ðxÞ þ SLt2SR S0Lu S0Rþu D2n2 ðxÞj2 D
(15)
ARTICLE IN PRESS 230
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
where D2n ðxÞ D2n ðx; 0; y; 0Þ depends only on the parameters of the isolated dot, D2n ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ x þ x2 1 x þ x2 1 ¼ ; pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 x þ x2 1 x þ x2 1 in which x ¼ ½ð 0D Þ2 u2 2t2D =ð2t2D Þ: D2n ðxÞ has oscillating and exponential behaviors depending on whether the Fermi energy lies inside (jxjp1) or outside (jxj41) dot energy band. Eq. (15) is a general equation for an A-QD attached to two metallic leads within a nearest-neighbor TB model. This formula can also be applied to a periodic superlattice attached to two uniform nanocrystals [14]. In the following, for simplicity, we assume that right and left leads are similar (tL ¼ tR ) and furthermore the dot-left-lead hopping is equal to the dot-right-lead hopping term (tDL ¼ tDR ). All atomic on-site energies in the considered system are set to zero (symmetrical case). The transmission coefficient depends on the dot size, gate voltage, and specially on the Fermi energy. Its analytical form, using the recursive relations of appendix, is given by
Three energy ranges can been defined using x: In the following equation, the relation between the parameter x and the Fermi energy is shown: 8 ½0; juj; xo 1; > > > h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii > < jxjp1; juj; u2 þ 4t2D ; (16) jj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q h i > > > 2 2 > u þ 4tD ; 2jtL j ; x4 þ 1; : jxjp1 and jxj41; respectively, indicate that the Fermi energy lies inside and outside the dot energy band.
4. Dependence on dot size and energy For energy ranges outside the dot band it can easily be seen that the behavior of G 1;2n which appears in the transmission, as a function of energy is exponential. A decay length may be identified in this limit. In units of the lattice constant, it is the inverse of the coefficient of 2n 1 in log½G 1; 2n : ð2n 1Þa G1; 2n ! exp ; (17) lðÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi where jxj41 and lðÞ ¼ a= log jjxj þ x2 1j: Likewise, the TC will have a similar behavior since
8 ð2k sin W cosh jÞ2 > > > ; > > ½ðcos W kmÞ coshð2n þ 1Þj ðk2 cos W kmÞ coshð2n 1Þj2 þ sin2 W½coshð2n þ 1Þj þ k2 coshð2n 1Þj2 > > > > > > xo 1; > > > > > > ð2k sin W sin jÞ2 > < ; 2 2 2 TðÞ ¼ ½ðcos W kmÞ sinð2n þ 1Þj þ ðk2 cos W kmÞ sinð2n 1Þj þ sin W½sinð2n þ 1Þj k2 sinð2n 1Þj > > > > jxjp1; > > > > > > ð2k sin W sinh jÞ2 > > ; > > > ½ðcos W kmÞ sinhð2n þ 1Þj þ ðk2 cos W kmÞ sinhð2n 1Þj2 þ sin2 W½sinhð2n þ 1Þj k2 sinhð2n 1Þj2 > > > : x4 þ 1;
where k ¼ t2DL =ðtD tL Þ; m ¼ ðx þ yÞ=ðxyÞ; x ¼ ð uÞ=tD ; y ¼ ð þ uÞ=tD ; x ¼ ðxy 2Þ=2; W ¼ arccos½=ð2tL Þ; 8 jxjp1; < arccos x pffiffiffiffiffiffiffiffiffiffiffiffiffi 2j ¼ : log x þ x2 1 jxj41:
it is quadratic in G 1;2n : In the dot gap, it can easily be seen that the behavior of TðÞ is as follows: 2 u2 2t2D TðÞ! 2t2D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi9ðN1Þ s 2 = 2 2 2 u 2tD þ 1 : ; 2t2D
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
231
8 7 6 log[DOS(E)]
5 4 3 2 1 N = 4, 8,12
0 Dot E-Band
-1 -2 -1.5
-1
Dot E-Band
Gap: Eg = 2|u|
-0.5
0 E
0.5
1
1.5
Fig. 2. log½DOSðÞ vs. (Fermi energy) for an A-QD connected to similar uniform leads. Here, (N ¼ 4,8,12), tL ¼ tR ¼ 2tD ¼ 1; u ¼ 0:5; and dot-lead hoppings are 0.25. (Note: in the figures, has been changed to E).
Fig. 4. TðV g ; uÞ: The TC at The conductance switches conducting (ON) states by Here N ¼ 6; tD ¼ tLðRÞ ¼ 1; energies are set to zero.
zero Fermi energy for an A-QD. between insulating (OFF) and applying the gate voltage (V g ). tDL ¼ tDR ¼ 0:5; and all on-site
depending on whether the Fermi level falls in the dot band or outside it. Fig. 4 shows the TC as a function of gate voltage V g and u parameters for F ¼ 0:
0 N=4
-5 log[T(E)]
N=8
4.1. Bound states
-10 N = 12
-15 -20 -25 -1.5
Resonance
Tunneling
Resonance
Dot E-Band
Gap
Dot E-Band
-1
-0.5
0 E
0.5
1
1.5
Fig. 3. Same as Fig. 2 for log½TðÞ vs. :
Figs. 2 and 3 show log½DOSðÞ and log½TðÞ for three different dot sizes, respectively, N ¼ 4; 8; 12 (note n ¼ N=2). Inside the dot energy band DOSðÞ and TðÞ have oscillating behavior. We identify this as the resonance regime. There are N peaks in TðÞ and DOSðÞ curves, which are distributed symmetrically around the A-QD onsite energy. Outside dot energy band (specially in the gap) DOSðÞ is almost independent of dot size and TðÞ is decreasing exponentially with size [13,14]. If we apply a uniform gate voltage to the dot, on-site energies of the dot will shift and the system can become either metallic or insulating
If the dot energy band falls outside the leads energy bands, then we can have formation of bound states within the dot. The bound state energies are the poles of the GF of the dot or the roots of the following equation (note that a and b are functions of the energy): a b F ðÞ ¼ 1 D2n ðxÞ x y a b ð18Þ þ ab D2n2 ðxÞ ¼ 0: x y It can be shown that the roots of Eq. (18) satisfy the following nonlinear equations: tan gm
am bm xamm bym sin 2fm m ¼ ; 1 xamm bym þ am bm xamm bym cos 2fm m
m
ð19Þ where xm ¼ ðm uÞ=tD ; ym ¼ ðm þ uÞ=tD ; am ; bm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m ðt2DLðRÞ =tLðRÞ tD Þðj 2tLðRÞ j ð2tLðRÞ Þ2 1Þ; and
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
232
tDL= 0 (
F( E) = Det [ (E I- H) ]
3
) 0.5 (
) 1.0 (
+ 2 | tL |
− u 2 + 4tD2 − 2 | tL | −|u |
4 )
Bound States Region
R
|u | Tunneling
R
u 2 + 4tD2
Bound States Region
2 Lead Energy Band
1 Dot Energy Band
0
Quasi GAP
Dot Energy Band
Energy
-1
Fig. 6. All energy regions are identified for an A-QD attached to identical leads. Here, R refers to the resonance region.
-2 -3 1
1.2
1.4
1.6
1.8
2
Fig. 5. Bound states energies of a 20-atom A-QD attached to two similar leads. They are the roots of F ðÞ: The three curves are obtained for different dot-lead hoppings where u ¼ 0:5; tD ¼ 2tL ¼ 2tR ¼ 1; and 0L ¼ 0D ¼ 0R ¼ 0:
Finally, in Fig. 6 all energy regions are identified for the A-QD attached to the identical leads. If jj belongs to ½juj; 2jtL j and ½0; juj; the conductance has, respectively, resonance (resonance states region), and decaying behavior (tunneling region). Also, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2jtL j; u2 þ 4t2D ; corresponds to the bound states
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ u2 þ 4t2D cos2 fm ;
region. It should be noted that when u ¼ 0 in the A-QD, model is reduced to the case of a uniform dot, which was studied in Ref. [14].
E
mp þ gm : fm ¼ N þ1
In Fig. 5, the function F has been plotted as a function of the energy for a symmetrical system (tL ¼ tR ) where all on-site energies are equal, and the hopping of the dot is twice that of the leads (tD ¼ 2tL ). The three curves correspond to three different values of dot-to-lead hoppings. The bound state energies are the intersection of the curve of F ðÞ with the energy axis. F is an even function in energy and the same bound states also exist in the [2; 1] energy range. The energy eigenvalues of an isolated A-QD are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp ; (20) E m ¼ u2 þ 4t2D cos2 N þ1 where m ¼ 1; 2; . . . ; N=2 and, the energy gap is 2juj: We reproduce the results found by Sprung et al. [16]. The eigenvectors of the isolated dot are also given by the following: 8 pffiffiffiffiffiffiffiffi jmp > > > < jym j sin N þ 1 ; j ¼ 2k 1; cm; j ¼ pffiffiffiffiffiffiffiffiffi jmp > > > j sin ; j ¼ 2k: þ jx m : N þ1 (21)
5. Conclusion In this work, TC and DOS of a periodic dot attached to two uniform leads have been calculated fully analytically. The quasi gap, bound state energies, the energy and dot-size dependence of the conductance for this system have been derived. Accordingly, different energy regions have been identified. It was shown that within the dot and the wire energy ranges, the TC and DOS have an oscillating behavior. Outside the dot band, the transmission decays exponentially with dot-size, but the DOS decays independently of it. One can make a switch by applying a gate voltage to the dot. There is no flow of current at zero gate voltage, where the Fermi energy lies in the quasi gap region. But if the gate voltage is such that the Fermi energy is within the dot band, the transmission will not be exponentially small anymore, and current will flow. Most quantum switches work on this basis, namely in systems which either display a band gap or have a different energy band width than the attached leads.
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
Acknowledgements
M 2nþ1 ðx; a; x; bÞ 0
We thank H. Cheraghchi and A.A. Shokri for useful discussions and Dr. A.T. Rezakhani for a careful reading of this paper. Appendix To compute the TC of an A-QD we first need the following relation for even parity: G 1; 2n ¼
(22)
Atomic occupation numbers are obtained from the integral of diagonal elements of G. For even parity, diagonal elements of the GF are given by D2j1 ðx; a; x; 0ÞD2n2j ðx; 0; y; bÞ ; tD D2n ðx; a; y; bÞ D2j ðx; a; y; 0ÞD2n2j1 ðy; 0; y; bÞ G 2jþ1; 2jþ1 ¼ : tD D2n ðx; a; y; bÞ G 2j; 2j ¼
ð23Þ Likewise, for a dot with an odd number of atoms, we have 1 ; tD D2nþ1 ðx; a; x; bÞ
(24)
D2j1 ðx; a; x; 0ÞD2nþ12j ðx; 0; x; bÞ ; tD D2nþ1 ðx; a; x; bÞ D2j ðx; a; y; 0ÞD2n2j ðy; 0; x; bÞ G 2jþ1; 2jþ1 ¼ : tD D2nþ1 ðx; a; x; bÞ
ð25Þ
We define D Det½M: The following matrices are the inverse of the dot GF in the presence of the leads for dots with even and odd atom numbers, respectively:
0
0
B B 1 B B B 0 B B B :: B B B 0 @ 0
y
0
::
1 ::
1
x
::
::
::
::
0
0
::
0
0
::
0
0
1
C C C C 0 0 C C C :: :: C C C y 1 C A 1 x b 0
0
:
ð2nþ1Þ ð2nþ1Þ
0
::
0
1
::
0
x
::
0
::
::
::
0
::
x
0
:: 1
0
1
C C C C 0 C C C :: C C C 1 C A yb
We need to calculate the following determinants in order to calculate the TC and the local DOS. The following equations relate the determinant of the inverse of the GF of an A-QD attached to leads to those of an isolated dot, a b D2n ðx; a; y; bÞ ¼ 1 D2n ðx; yÞ x y a b þ ab D2n2 ðx; yÞ; ð28Þ x y D2nþ1 ðx; a; x; bÞ ¼
ab 2 x þ a b D2n ðx; yÞ x y ab 1 D2n2 ðx; yÞ: ð29Þ þ x y
It is easy to show that D satisfies the following recursion relation (D2n ðxÞ D2n ðx; 0; y; 0ÞÞ:
G 2j; 2j ¼
M 2n ðx; a; y; bÞ 0 x a 1 B B 1 y B B B 0 1 B B B :: :: B B B 0 0 @
x a 1
ð27Þ
1 : tD D2n ðx; a; y; bÞ
G 1; 2nþ1 ¼
233
0
;
2n 2n
ð26Þ
D2n ðxÞ ¼ 2xD2n2 ðxÞ D2n4 ðxÞ;
(30)
where D0 ðxÞ 1 and D2 ðxÞ 1: For an isolated A-QD, D2n ðxÞ has the following form: D2n ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ x þ x2 1 x þ x2 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi þ1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ; x þ x2 1 x þ x2 1 8 coshð2n þ 1Þj > > ; xo 1; ð1Þn > > cosh j > > > > < sinð2n þ 1Þj ; jxjp1; D2n ðxÞ ¼ sin j > > > > > > sinhð2n þ 1Þj > > ; x41; : sinh j
ARTICLE IN PRESS 234
where 8 < arccos x; pffiffiffiffiffiffiffiffiffiffiffiffiffi 2j ¼ 2 : log x þ x 1 ;
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
jxjp1; jxj41;
x ¼ ðxy 2Þ=2; x ¼ ð uÞ=tD ; and y ¼ ð þ uÞ=tD : One thus has the determinant of the inverse GF of an A-QD of even or odd parities. References [1] P. Harrison, Quantum Wells, Wires and Dots, John Wiley, New York, 2000. [2] D.K. Ferry, S.M. Goognick, Transport in Nanostractures, Cambridge University Press, Cambridge, 1997. [3] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1997. [4] C. Joachim, J.K. Gimzewski, A. Aviram, Nature (London) 408 (2000) 521; M. Magoga, C. Joachim, Phys. Rev. B 56 (1997) 4722; C. Joachim, J.F. Vinuesa, Europhys. Lett. 33 (1996) 635. [5] A. Aviram, M. Ratner, Molecular Electronics: Science and Technology, New York Academy of Sciences, New York, 1998. [6] M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, J.M. Tour, Science 278 (1997) 252; M.A. Reed, Proc. IEEE 97 (1999) 652;
[7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19] [20]
J. Chen, M.A. Reed, A.M. Rawlett, J.M. Tour, Science 286 (1999) 1550; Z.J. Donhauser, B.A. Mantooth, K.F. Kelly, L.A. Bumm, J.D. Monnell, J.J. Stapleton, D.W. Price, A.M. Rawlett, D.L. Allara, J.M. Tour, P.S. Weiss, Science 292 (2001) 2303. P. Ball, Nature (London) 404 (2000) 918. A.A. Farajian, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 82 (1999) 5084. K. Tagami, M. Tsukada, T. Massumoto, T. Kawai, Phys. Rev. B 67 (2003) 245324. Y. Xue, M.A. Ratner, Phys. Rev. B 68 (2003) 115406. H.X. He, X.L. Li, N.J. Tao, Phys. Rev. B. 68 (2003) 045302. A. Opniko, Y. Klymenko, L. Malysheva, Phys. Rev. B 62 (2003) 10482. G. Cuniberti, G. Fagas, K. Richter, Chem. Phys. 281 (2002) 465. M. Mardaani, K. Esfarjani, Physica E 25 (2004) 119. R. Landauer, Phys. Lett. 85 A (1986) 91; R. Landauer, IBM J. Res. Dev. 32 (1988) 306; M. Buttiker, IBM J. Res. Dev. 32 (1988) 317. D.W.L. Sprung, J.D. Sigetich, Hua Wu, J. Martorell, Am. J. Phys. 68 (2000) 715. C. Caroli, R. Combescot, P. Nozieres, D. Saint-James, J. Phys. C 4 (1971) 916. D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1981) 6851. P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. H. Pastawski, Phys. Rev. B 44 (1991) 6329.
Physica E 27 (2005) 227–234 www.elsevier.com/locate/physe
Coherent conductance in an alternating dot: exact results Mohammad Mardaani, Keivan Esfarjani Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Received 1 November 2004; accepted 17 November 2004 Available online 23 January 2005
Abstract In this paper we have calculated the conductance of a periodic quantum dot attached to metallic leads, within the tight-binding (TB) model and in the ballistic regime. We have calculated the Green’s function (GF), density of states (DOS) and the coherent transmission coefficient (TC) fully analytically for an alternating quantum dot (A-QD). The quasi-gap, bound states energies, the energy and dot-size dependence of the GF and conductance for the system are also derived. Finally, we show analytically the conductance can be switched between insulating (OFF) and conducting (ON) states by applying a gate voltage. r 2004 Elsevier B.V. All rights reserved. PACS: 05.60.Gg; 81.07.Ta; 81.07.Vh Keywords: Green’s Function; Quantum transport; Alternating dots; Bound states
1. Introduction During the past few years, the study of electronic transport through quantum dots (QDs) and quantum wires have been major areas in nanoelectronics and nanostructure physics [1–9]. These systems play an important role in the development of molecular wires for nanoelectronic devices. Experimental research on quantum and molecular wires has increased over the past few Corresponding author.
E-mail address: [email protected] (M. Mardaani).
years looking into the possibility of rectification and other related phenomena [4–7]. There has also been an increasing interest in deriving analytical results on transport in molecular wires, single molecule, and nanocrystal devices [10–14]. Different methods based on the Landauer formula [3,15] which states that the conductance is proportional to the transmission coefficient, have been developed for the study of transport phenomena. One class of methods is based on transfer matrix [3,4,16], and the other makes use of the Green’s function formalism [3,14,17–20]. We rely on the GF method for studying the coherent transport properties in quantum dot systems. It relates the
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.11.012
ARTICLE IN PRESS 228
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
lead-to-lead current to the transmission probability for an electron to scatter through the quantum dot. By coherence mechanism, we mean that the dot length is smaller than the phase coherence length so that electrons do not experience any inelastic scattering during transport except in contacts. In our previous work [14], we studied coherence conductance for a uniform dot and uniform nanocrystal by the GF method. In the present work, we study the coherent transport properties of an alternating dot in the presence of metallic leads, within the TB model and in the ballistic regime. The organization of this paper is as follows. In Section 2, the tight-binding one-electron Hamiltonian for A-QD is described. In this section, the transmission coefficient and the local density of states, which generally depend on the Fermi energy, dot size, dot-lead coupling, and gate voltage, are analytically calculated. The TC calculations in symmetrical case are explained in much detail. Three characteristic energy ranges are identified for this problem. Finally, we obtain a nonlinear equation which gives the bound state energies. The paper is ended by a conclusion in Section 3. In appendix some useful relations are derived which relate the GF, TC, and DOS to the parity of the number of atoms in the chain for an A-QD.
…
(1)
-1,0;
1,2, 3
…
N; N+1 …
where H L ; H DL ; H D ; H DR ; H R refer to the leftlead, dot to left-lead, isolated dot, dot to rightlead, and right-lead Hamiltonians, respectively. Fig. 1 shows the geometry of an A-QD attached to two metallic leads. The first and the last atoms in the dot are, respectively, labeled by 1 and N. In the TB approximation, and assuming nearest-neighbor interaction, the Hamiltonian of the leads is defined as follows: X H LðRÞ ¼ tLðRÞ ðcyz czþ1 þ h:c:Þ; (2) z
where H LðRÞ and tLðRÞ describe the semi-infinite uniform left (right)-lead Hamiltonians and the hopping integral, respectively. For simplicity, the atomic on-site energies of both leads are set to zero. The dot-lead Hamiltonian is defined as H DLðRÞ ¼ tDLðRÞ ðcy0ðNÞ c1ðNþ1Þ þ h:c:Þ;
(3)
where tDLðRÞ refer to the contact hopping integrals which depend on distance between the dot and the leads [3]. The dot Hamiltonian is given by N X z¼1
In this section, we describe our model and assumptions, and derive a formula allowing one to calculate the GF and the conductance of a A-QD analytically. Consequently, in what follows, we will assume a chain composed of a central region (dot) and two general leads, and will derive the analytic expression for the GF and the TC in the resonance and tunneling regions, assuming the dot is periodic with two atoms per unit cell. In the general form, there will be various on-site energies and hopping integrals in each region, plus two hoppings from dot to the two leads. We use the following generalized Hamiltonian for the system:
Right-Lead
…
Fig. 1. Geometry of an A-QD attached to two leads.
HD ¼
2. The theoretical model
H ¼ H L þ H DL þ H D þ H DR þ H R ;
Periodic Dot Left-Lead
z cyz cz þ
N 1 X
tz;zþ1 ðcyz czþ1 þ h:c:Þ;
(4)
z¼1
where z ¼ 0 þ uð1Þzþ1 and tz;zþ1 ¼ tD refer to the on-site energy and hopping integral for nearest neighbor sites, respectively. From Eq. (4), we can obtain the retarded GF of the dot in contact with leads using the following partitioning formula. For a general system divided into two parts A and B, it is easy to show that the total (A+B) GF function projected onto the Hilbert space of the dot (called A below), can be written as [19] 1 G1 A ¼ G 0A H AB G 0B H BA ;
(5)
where G0A and G A are GFs of the A system in the absence and presence of the B system, respectively, H AB is the interaction Hamiltonian between A
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
and B. The last term in Eq. (5) is also called the self-energy due to system B. For the lead–dot–lead configuration defined above, we have ½G1 ij ¼ ð þ i0þ i SL di1 SR diN Þdij tD ðdi;jþ1 þ di;j1 Þ;
ð6Þ
where SL and SR refer to the self-energies due to the left and right leads. It is known that energy bands of the leads are given by 2tR cosðkR aÞ and 2tL cosðkL aÞ; so that self-energies can be calculated as follows [14]: t2DL exp½ikL ðÞa; tL t2 SR ðÞ ¼ DR exp½ikR ðÞa; tR
SL ðÞ ¼
ð7Þ
where a refers to the lattice constant in the leads. For the calculation of the TC, G1N and the imaginary part of SLðRÞ are needed [3] TðÞ ¼ 4 ImðSL ÞImðSR ÞjG 1N j2 :
(8)
At zero temperature and zero bias, the conductance is just 2e2 =h times the transmission coefficient.
229
where M ðN¼2nÞ ðx; a; y; bÞ 0 x a 1 B B 1 y B B :: ¼B B :: B B 0 0 @ 0 0
::
0
::
0
::
::
::
x
::
1
0
1
C C C C :: C C C 1 C A yb 0
;
2n 2n
in which x ¼ ðz 0 uÞ=tD ; y ¼ ðz 0 þ uÞ=tD ; z ¼ þ i0þ ; a ¼ SL =tD ; b ¼ SR =tD ; and n is the number of unit cells. G 1;2n has the following form: G1; 2n ¼
1 ; tD D2n ðx; a; y; bÞ
(11)
where D2n ðx; a; y; bÞ ¼ Det½M 2n ðx; a; y; bÞ: The explicit form of D is given in the appendix. Now, from Eq. (8) the transmission coefficient can be obtained as TðÞ ¼
4 t2DL t2DR j sinðkL aÞ sinðkR aÞj : t2D tL tR j D2n ðx; a; y; bÞj2
(12)
In order to calculate the local DOS, diagonal elements of the GF are needed: 3. Analytical results for TC and local DOS In this model we assume the dot is periodic with two atoms per unit cell with different on-site energies HD ¼
N X
½0 þ uð1Þzþ1 cyz cz
1 DOSj ðÞ ¼ ImG jj ðÞ; p
depending on parity of j, the form of diagonal elements of the GF is as follows (appendix): G2jþ1; 2jþ1 ¼
z¼1
þ tD
N 1 X
ðcyz czþ1 þ h:cÞ:
ð9Þ G2j; 2j ¼
z¼1
The inverse GF of the A-QD in the presence of leads, can be written in the following form: G 1 D ¼ tD M N ðx; a; y; bÞ;
TðÞ ¼
(10)
t2D j
(13)
D2j ðx; a; y; 0ÞD2n2j1 ðy; 0; y; bÞ ; tD D2n ðx; a; y; bÞ
D2j1 ðx; a; x; 0ÞD2n2j ðx; 0; y; bÞ : tD D2n ðx; a; y; bÞ
(14)
From Eqs. (8) and (11), using the results of appendix, we can obtain the transmission coefficient as follows:
4 ImðSL ÞImðSR Þ ; SL SR 1 0 u 0 þu D2n ðxÞ þ SLt2SR S0Lu S0Rþu D2n2 ðxÞj2 D
(15)
ARTICLE IN PRESS 230
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
where D2n ðxÞ D2n ðx; 0; y; 0Þ depends only on the parameters of the isolated dot, D2n ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ x þ x2 1 x þ x2 1 ¼ ; pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 x þ x2 1 x þ x2 1 in which x ¼ ½ð 0D Þ2 u2 2t2D =ð2t2D Þ: D2n ðxÞ has oscillating and exponential behaviors depending on whether the Fermi energy lies inside (jxjp1) or outside (jxj41) dot energy band. Eq. (15) is a general equation for an A-QD attached to two metallic leads within a nearest-neighbor TB model. This formula can also be applied to a periodic superlattice attached to two uniform nanocrystals [14]. In the following, for simplicity, we assume that right and left leads are similar (tL ¼ tR ) and furthermore the dot-left-lead hopping is equal to the dot-right-lead hopping term (tDL ¼ tDR ). All atomic on-site energies in the considered system are set to zero (symmetrical case). The transmission coefficient depends on the dot size, gate voltage, and specially on the Fermi energy. Its analytical form, using the recursive relations of appendix, is given by
Three energy ranges can been defined using x: In the following equation, the relation between the parameter x and the Fermi energy is shown: 8 ½0; juj; xo 1; > > > h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii > < jxjp1; juj; u2 þ 4t2D ; (16) jj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q h i > > > 2 2 > u þ 4tD ; 2jtL j ; x4 þ 1; : jxjp1 and jxj41; respectively, indicate that the Fermi energy lies inside and outside the dot energy band.
4. Dependence on dot size and energy For energy ranges outside the dot band it can easily be seen that the behavior of G 1;2n which appears in the transmission, as a function of energy is exponential. A decay length may be identified in this limit. In units of the lattice constant, it is the inverse of the coefficient of 2n 1 in log½G 1; 2n : ð2n 1Þa G1; 2n ! exp ; (17) lðÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi where jxj41 and lðÞ ¼ a= log jjxj þ x2 1j: Likewise, the TC will have a similar behavior since
8 ð2k sin W cosh jÞ2 > > > ; > > ½ðcos W kmÞ coshð2n þ 1Þj ðk2 cos W kmÞ coshð2n 1Þj2 þ sin2 W½coshð2n þ 1Þj þ k2 coshð2n 1Þj2 > > > > > > xo 1; > > > > > > ð2k sin W sin jÞ2 > < ; 2 2 2 TðÞ ¼ ½ðcos W kmÞ sinð2n þ 1Þj þ ðk2 cos W kmÞ sinð2n 1Þj þ sin W½sinð2n þ 1Þj k2 sinð2n 1Þj > > > > jxjp1; > > > > > > ð2k sin W sinh jÞ2 > > ; > > > ½ðcos W kmÞ sinhð2n þ 1Þj þ ðk2 cos W kmÞ sinhð2n 1Þj2 þ sin2 W½sinhð2n þ 1Þj k2 sinhð2n 1Þj2 > > > : x4 þ 1;
where k ¼ t2DL =ðtD tL Þ; m ¼ ðx þ yÞ=ðxyÞ; x ¼ ð uÞ=tD ; y ¼ ð þ uÞ=tD ; x ¼ ðxy 2Þ=2; W ¼ arccos½=ð2tL Þ; 8 jxjp1; < arccos x pffiffiffiffiffiffiffiffiffiffiffiffiffi 2j ¼ : log x þ x2 1 jxj41:
it is quadratic in G 1;2n : In the dot gap, it can easily be seen that the behavior of TðÞ is as follows: 2 u2 2t2D TðÞ! 2t2D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi9ðN1Þ s 2 = 2 2 2 u 2tD þ 1 : ; 2t2D
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
231
8 7 6 log[DOS(E)]
5 4 3 2 1 N = 4, 8,12
0 Dot E-Band
-1 -2 -1.5
-1
Dot E-Band
Gap: Eg = 2|u|
-0.5
0 E
0.5
1
1.5
Fig. 2. log½DOSðÞ vs. (Fermi energy) for an A-QD connected to similar uniform leads. Here, (N ¼ 4,8,12), tL ¼ tR ¼ 2tD ¼ 1; u ¼ 0:5; and dot-lead hoppings are 0.25. (Note: in the figures, has been changed to E).
Fig. 4. TðV g ; uÞ: The TC at The conductance switches conducting (ON) states by Here N ¼ 6; tD ¼ tLðRÞ ¼ 1; energies are set to zero.
zero Fermi energy for an A-QD. between insulating (OFF) and applying the gate voltage (V g ). tDL ¼ tDR ¼ 0:5; and all on-site
depending on whether the Fermi level falls in the dot band or outside it. Fig. 4 shows the TC as a function of gate voltage V g and u parameters for F ¼ 0:
0 N=4
-5 log[T(E)]
N=8
4.1. Bound states
-10 N = 12
-15 -20 -25 -1.5
Resonance
Tunneling
Resonance
Dot E-Band
Gap
Dot E-Band
-1
-0.5
0 E
0.5
1
1.5
Fig. 3. Same as Fig. 2 for log½TðÞ vs. :
Figs. 2 and 3 show log½DOSðÞ and log½TðÞ for three different dot sizes, respectively, N ¼ 4; 8; 12 (note n ¼ N=2). Inside the dot energy band DOSðÞ and TðÞ have oscillating behavior. We identify this as the resonance regime. There are N peaks in TðÞ and DOSðÞ curves, which are distributed symmetrically around the A-QD onsite energy. Outside dot energy band (specially in the gap) DOSðÞ is almost independent of dot size and TðÞ is decreasing exponentially with size [13,14]. If we apply a uniform gate voltage to the dot, on-site energies of the dot will shift and the system can become either metallic or insulating
If the dot energy band falls outside the leads energy bands, then we can have formation of bound states within the dot. The bound state energies are the poles of the GF of the dot or the roots of the following equation (note that a and b are functions of the energy): a b F ðÞ ¼ 1 D2n ðxÞ x y a b ð18Þ þ ab D2n2 ðxÞ ¼ 0: x y It can be shown that the roots of Eq. (18) satisfy the following nonlinear equations: tan gm
am bm xamm bym sin 2fm m ¼ ; 1 xamm bym þ am bm xamm bym cos 2fm m
m
ð19Þ where xm ¼ ðm uÞ=tD ; ym ¼ ðm þ uÞ=tD ; am ; bm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m ðt2DLðRÞ =tLðRÞ tD Þðj 2tLðRÞ j ð2tLðRÞ Þ2 1Þ; and
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
232
tDL= 0 (
F( E) = Det [ (E I- H) ]
3
) 0.5 (
) 1.0 (
+ 2 | tL |
− u 2 + 4tD2 − 2 | tL | −|u |
4 )
Bound States Region
R
|u | Tunneling
R
u 2 + 4tD2
Bound States Region
2 Lead Energy Band
1 Dot Energy Band
0
Quasi GAP
Dot Energy Band
Energy
-1
Fig. 6. All energy regions are identified for an A-QD attached to identical leads. Here, R refers to the resonance region.
-2 -3 1
1.2
1.4
1.6
1.8
2
Fig. 5. Bound states energies of a 20-atom A-QD attached to two similar leads. They are the roots of F ðÞ: The three curves are obtained for different dot-lead hoppings where u ¼ 0:5; tD ¼ 2tL ¼ 2tR ¼ 1; and 0L ¼ 0D ¼ 0R ¼ 0:
Finally, in Fig. 6 all energy regions are identified for the A-QD attached to the identical leads. If jj belongs to ½juj; 2jtL j and ½0; juj; the conductance has, respectively, resonance (resonance states region), and decaying behavior (tunneling region). Also, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½2jtL j; u2 þ 4t2D ; corresponds to the bound states
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ u2 þ 4t2D cos2 fm ;
region. It should be noted that when u ¼ 0 in the A-QD, model is reduced to the case of a uniform dot, which was studied in Ref. [14].
E
mp þ gm : fm ¼ N þ1
In Fig. 5, the function F has been plotted as a function of the energy for a symmetrical system (tL ¼ tR ) where all on-site energies are equal, and the hopping of the dot is twice that of the leads (tD ¼ 2tL ). The three curves correspond to three different values of dot-to-lead hoppings. The bound state energies are the intersection of the curve of F ðÞ with the energy axis. F is an even function in energy and the same bound states also exist in the [2; 1] energy range. The energy eigenvalues of an isolated A-QD are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp ; (20) E m ¼ u2 þ 4t2D cos2 N þ1 where m ¼ 1; 2; . . . ; N=2 and, the energy gap is 2juj: We reproduce the results found by Sprung et al. [16]. The eigenvectors of the isolated dot are also given by the following: 8 pffiffiffiffiffiffiffiffi jmp > > > < jym j sin N þ 1 ; j ¼ 2k 1; cm; j ¼ pffiffiffiffiffiffiffiffiffi jmp > > > j sin ; j ¼ 2k: þ jx m : N þ1 (21)
5. Conclusion In this work, TC and DOS of a periodic dot attached to two uniform leads have been calculated fully analytically. The quasi gap, bound state energies, the energy and dot-size dependence of the conductance for this system have been derived. Accordingly, different energy regions have been identified. It was shown that within the dot and the wire energy ranges, the TC and DOS have an oscillating behavior. Outside the dot band, the transmission decays exponentially with dot-size, but the DOS decays independently of it. One can make a switch by applying a gate voltage to the dot. There is no flow of current at zero gate voltage, where the Fermi energy lies in the quasi gap region. But if the gate voltage is such that the Fermi energy is within the dot band, the transmission will not be exponentially small anymore, and current will flow. Most quantum switches work on this basis, namely in systems which either display a band gap or have a different energy band width than the attached leads.
ARTICLE IN PRESS M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
Acknowledgements
M 2nþ1 ðx; a; x; bÞ 0
We thank H. Cheraghchi and A.A. Shokri for useful discussions and Dr. A.T. Rezakhani for a careful reading of this paper. Appendix To compute the TC of an A-QD we first need the following relation for even parity: G 1; 2n ¼
(22)
Atomic occupation numbers are obtained from the integral of diagonal elements of G. For even parity, diagonal elements of the GF are given by D2j1 ðx; a; x; 0ÞD2n2j ðx; 0; y; bÞ ; tD D2n ðx; a; y; bÞ D2j ðx; a; y; 0ÞD2n2j1 ðy; 0; y; bÞ G 2jþ1; 2jþ1 ¼ : tD D2n ðx; a; y; bÞ G 2j; 2j ¼
ð23Þ Likewise, for a dot with an odd number of atoms, we have 1 ; tD D2nþ1 ðx; a; x; bÞ
(24)
D2j1 ðx; a; x; 0ÞD2nþ12j ðx; 0; x; bÞ ; tD D2nþ1 ðx; a; x; bÞ D2j ðx; a; y; 0ÞD2n2j ðy; 0; x; bÞ G 2jþ1; 2jþ1 ¼ : tD D2nþ1 ðx; a; x; bÞ
ð25Þ
We define D Det½M: The following matrices are the inverse of the dot GF in the presence of the leads for dots with even and odd atom numbers, respectively:
0
0
B B 1 B B B 0 B B B :: B B B 0 @ 0
y
0
::
1 ::
1
x
::
::
::
::
0
0
::
0
0
::
0
0
1
C C C C 0 0 C C C :: :: C C C y 1 C A 1 x b 0
0
:
ð2nþ1Þ ð2nþ1Þ
0
::
0
1
::
0
x
::
0
::
::
::
0
::
x
0
:: 1
0
1
C C C C 0 C C C :: C C C 1 C A yb
We need to calculate the following determinants in order to calculate the TC and the local DOS. The following equations relate the determinant of the inverse of the GF of an A-QD attached to leads to those of an isolated dot, a b D2n ðx; a; y; bÞ ¼ 1 D2n ðx; yÞ x y a b þ ab D2n2 ðx; yÞ; ð28Þ x y D2nþ1 ðx; a; x; bÞ ¼
ab 2 x þ a b D2n ðx; yÞ x y ab 1 D2n2 ðx; yÞ: ð29Þ þ x y
It is easy to show that D satisfies the following recursion relation (D2n ðxÞ D2n ðx; 0; y; 0ÞÞ:
G 2j; 2j ¼
M 2n ðx; a; y; bÞ 0 x a 1 B B 1 y B B B 0 1 B B B :: :: B B B 0 0 @
x a 1
ð27Þ
1 : tD D2n ðx; a; y; bÞ
G 1; 2nþ1 ¼
233
0
;
2n 2n
ð26Þ
D2n ðxÞ ¼ 2xD2n2 ðxÞ D2n4 ðxÞ;
(30)
where D0 ðxÞ 1 and D2 ðxÞ 1: For an isolated A-QD, D2n ðxÞ has the following form: D2n ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðnþ1=2Þ x þ x2 1 x þ x2 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi þ1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ; x þ x2 1 x þ x2 1 8 coshð2n þ 1Þj > > ; xo 1; ð1Þn > > cosh j > > > > < sinð2n þ 1Þj ; jxjp1; D2n ðxÞ ¼ sin j > > > > > > sinhð2n þ 1Þj > > ; x41; : sinh j
ARTICLE IN PRESS 234
where 8 < arccos x; pffiffiffiffiffiffiffiffiffiffiffiffiffi 2j ¼ 2 : log x þ x 1 ;
M. Mardaani, K. Esfarjani / Physica E 27 (2005) 227–234
jxjp1; jxj41;
x ¼ ðxy 2Þ=2; x ¼ ð uÞ=tD ; and y ¼ ð þ uÞ=tD : One thus has the determinant of the inverse GF of an A-QD of even or odd parities. References [1] P. Harrison, Quantum Wells, Wires and Dots, John Wiley, New York, 2000. [2] D.K. Ferry, S.M. Goognick, Transport in Nanostractures, Cambridge University Press, Cambridge, 1997. [3] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1997. [4] C. Joachim, J.K. Gimzewski, A. Aviram, Nature (London) 408 (2000) 521; M. Magoga, C. Joachim, Phys. Rev. B 56 (1997) 4722; C. Joachim, J.F. Vinuesa, Europhys. Lett. 33 (1996) 635. [5] A. Aviram, M. Ratner, Molecular Electronics: Science and Technology, New York Academy of Sciences, New York, 1998. [6] M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, J.M. Tour, Science 278 (1997) 252; M.A. Reed, Proc. IEEE 97 (1999) 652;
[7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19] [20]
J. Chen, M.A. Reed, A.M. Rawlett, J.M. Tour, Science 286 (1999) 1550; Z.J. Donhauser, B.A. Mantooth, K.F. Kelly, L.A. Bumm, J.D. Monnell, J.J. Stapleton, D.W. Price, A.M. Rawlett, D.L. Allara, J.M. Tour, P.S. Weiss, Science 292 (2001) 2303. P. Ball, Nature (London) 404 (2000) 918. A.A. Farajian, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 82 (1999) 5084. K. Tagami, M. Tsukada, T. Massumoto, T. Kawai, Phys. Rev. B 67 (2003) 245324. Y. Xue, M.A. Ratner, Phys. Rev. B 68 (2003) 115406. H.X. He, X.L. Li, N.J. Tao, Phys. Rev. B. 68 (2003) 045302. A. Opniko, Y. Klymenko, L. Malysheva, Phys. Rev. B 62 (2003) 10482. G. Cuniberti, G. Fagas, K. Richter, Chem. Phys. 281 (2002) 465. M. Mardaani, K. Esfarjani, Physica E 25 (2004) 119. R. Landauer, Phys. Lett. 85 A (1986) 91; R. Landauer, IBM J. Res. Dev. 32 (1988) 306; M. Buttiker, IBM J. Res. Dev. 32 (1988) 317. D.W.L. Sprung, J.D. Sigetich, Hua Wu, J. Martorell, Am. J. Phys. 68 (2000) 715. C. Caroli, R. Combescot, P. Nozieres, D. Saint-James, J. Phys. C 4 (1971) 916. D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1981) 6851. P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. H. Pastawski, Phys. Rev. B 44 (1991) 6329.