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Analytical results on coherent conductance in a general periodic quantum dot: Transfer matrix method
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Physica E 28 (2005) 150–161 www.elsevier.com/locate/physe
Analytical results on coherent conductance in a general periodic quantum dot: Transfer matrix method Mohammad Mardaania,b,c,, Ali A. Shokria,b,d, Keivan Esfarjania a Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Computational Physical Sciences Research Laboratory, Department of Nano-Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5531, Tehran, Iran c Faculty of Science, Physics Group, Shahrekord University, Shahrekord, Iran d Department of Physics, Tehran Payame-Noor University, Fallahpour St. Nejatollahi St., Tehran, Iran
b
Received 22 February 2005; accepted 9 March 2005 Available online 17 May 2005
Abstract In this work, we study the conductance of a general periodic quantum dot (QD) attached to ideal semi-infinite uniform metallic leads (nanocrystals), fully analytically. We propose a new general formula which relates conductance to transfer matrix (TM) for an isolated cell in the periodic dot. The equation describes exactly the dependence of the transmission coefficient (TC) on Fermi energy, dot-size, dot–lead coupling, and gate voltage for an arbitrary periodic dot. Then, we derive a nonlinear equation which gives the resonance, bound, and surface state energies. Finally, the TC has been calculated for gapless, single, and double gap models exactly. Moreover, we have also calculated the effects of the cross-section of the leads, which were separated by a polymer chain on the conductance. Our calculations can be generalized to any type of QD and quantum wire (QW) within the one-electron approximation, and, can be applied to, e.g., molecular, polymer, and nanocrystal junctions, where these results may be useful in designing future molecular electronic devices. r 2005 Elsevier B.V. All rights reserved. PACS: 05.60.Gg; 81.07.Ta; 81.07.Vb Keywords: Coherent conductance; Periodic quantum dot; Transfer matrix; Analytical results
Corresponding author.
E-mail address: [email protected] (M. Mardaani). 1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.03.005
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1. Introduction In the past few years, the study of electronic transport through quantum dots (QDs), quantum wires (QWs), and molecular wires have been major areas of research in mesoscopic physics [1–3]. The size and shape of QDs and QWs can be precisely controlled with today’s technology [1,2,4]. QDs can be connected to electrodes and are, therefore, excellent tools to study atomic-like properties. The transport properties of a QD can be measured by coupling it to leads and passing current through the dot. In fact, there is a vast and recent literature on this subject [1–3]. Here, at low temperatures, we ignore the inelastic scattering process, thus, conductance through a QD is characterized by quantum coherence. We rely on the Landauer’s approach [5] for studying the coherent transport properties in QD and QW systems. The Landauer’s approach states that electronic conductance is proportional to transmission coefficient (TC). Different methods based on this approach have been developed for the study of transport phenomena. One class of methods is based on the transfer matrix (TM), e.g. [1,3,6–12], and the others make use of Green’s function (GF) formalism, e.g. [3,13–20]. In our previous works [20], we have calculated the coherent conductance for uniform and alternating dot fully analytically by GF method. In the present paper, we use an analytical method based on the tight binding (TB) model and transfer matrix method to derive dependence of conductance on Fermi energy, dot-size, and gate voltage, for a general periodic dot connected to two uniform ideal leads at T ¼ 0 K. The organization of this paper is as follows. In Section 2, the TB one-electron Hamiltonian for a general QD is described. Here, a detailed description of the TM approach in the TB model, assuming nearestneighbor interaction, is given. In this section, for a general dot, the TC, which depends on the Fermi energy, QD size, dot–leads coupling, and gate voltage was derived analytically. The TC calculations in the periodic dot case are explained in much detail in Section 3. Also, in this section we obtain a nonlinear equation which gives the resonant and bound states energies and surface state energies if they exit. In Section 4, we calculate the TC for gapless, single gap, and double gap dots in more details. The paper ends with the conclusion in Section 5.
2. Description of transfer matrix method We consider a chain composed of a general QD attached to two uniform semi-infinite leads. For simplicity, the two metallic leads are assumed to be ideal. Also, the elements of TM for the general QD are calculated by coherent mechanism. It means that the dot length is smaller than the phase coherence length [3]. The generalized Hamiltonian which describes the system is given by H ¼ H L þ H DL þ H D þ H DR þ H R ;
ð1Þ
where H L;D;R describes the Hamiltonian for the left lead, dot and the right lead, and also, H DLðRÞ refers to the Hamiltonian for the coupling between the general QD and the left (right) lead. The Hamiltonian of left (right) lead is defined as X y H LðRÞ ¼ tLðRÞ ðci ciþ1 þ h:c:Þ; ð2Þ i
where tLðRÞ refers to the hopping integral between s atomic orbitals (assumed to be orthogonal), for the left and right lead, i varies between ð1; 0 and ½N þ 1; þ1Þ, respectively, and N refers to the number of total atoms in the dot. The QD is coupled to the leads via the contact Hamiltonian, H DLðRÞ as follows: H DLðRÞ ¼ tDLðRÞ ðcy0ðNÞ c1ðNþ1Þ þ h:c:Þ;
ð3Þ
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where tDLðRÞ refers to the contact hopping integral between dot and the left (right) lead. Also, the dot Hamiltonian is given by " # N N1 X X y y H D ¼ tD 0i ci ci þ bi;iþ1 ðci ciþ1 þ h:c:Þ ; ð4Þ i¼1
i¼1
where 0i and bi;iþ1 refer to the on-site energy and hopping integral in units of tD , respectively. We write the time-independent Schro¨dinger equation, HC ¼ eC, for the QD coupled to the leads as 0i ci þ bi;iþ1 ciþ1 þ bi1;i ci1 ¼ zci ;
ð5Þ
where i ¼ 1; . . . ; þ1, ci is the amplitude of the wave function at site i, and z ¼ e=tD . Eq. (5) can be written in the following form: ! ! ciþ1 ci ¼ PðiÞ ; ð6Þ ci ci1 ðiÞ ðiÞ ðiÞ where PðiÞ 11 ¼ ðz 0i Þ=bi;iþ1 , P12 ¼ bi1;i =bi;iþ1 , P21 ¼ 1, and P22 ¼ 0. Thus, after multiplication, we can obtain a recurrent equation for the whole dot including dot-lead contacts as ! ! cNþ2 c0 ðRÞ ðLÞ ¼P P P ; ð7Þ cNþ1 c1
where PðLÞ and PðRÞ refer to the TM of the left and the right contacts, respectively, and P stands for PðNÞ PðN1Þ Pð1Þ . It is clear that the discrepancy between ci , ci 1 , in the leads is a phase factor, because the leads are ideal. Also, it is known that right and left leads energy bands are described by eLðRÞ ¼ 2tLðRÞ cos kLðRÞ a ¼ 2tLðRÞ cos yLðRÞ , therefore, after this simplification, we have 0 1 0 1 tL tD 0 0 B tDL C B tDR C B C B C ; PðRÞ ¼ eiyR B . PðLÞ ¼ eiyL B C tDL A tDR C @ @ A 0 0 tD tR In order to calculate the transmission and reflection coefficients, the wave functions in the 1 and 0 sites of the left lead, and N and N þ 1 sites of the right lead are needed: c0 ¼ 1 þ r;
c1 ¼ eiyL þ reiyL
and cNþ1 ¼ teiðNþ1ÞyR ;
cNþ2 ¼ teiðNþ2ÞyR ,
where r and t are the amplitudes of the reflected and transmitted electrons, respectively. The relation between transmission coefficient and transmission amplitude is given by [3] vR tR sin yR tt ; T ¼ tt ¼ ð8Þ vL tL sin yL where t is the conjugate of t. If we solve for t and r from Eq. (7) and substitute t in Eq. (8), the following analytical form for TC in the considered system is obtained: TðzÞ ¼
t2D tL tR jP11 þ ðt2DL =tD tL ÞP12 eiyL
4t2DL t2DR j sin yL sin yR j ; ðt2DR =tD tR ÞP21 eiyR ðt2DL =tD tL Þðt2DR =tD tR ÞP22 eiðyR þyL Þ j2
ð9Þ
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where P refers to the TM for isolated dot. As it can be seen, to calculate the TC, it is only needed to evaluate the TM for an isolated dot. Therefore Eq. (9) is a general equation for a dot attached to two leads within a nearest-neighbor TB model. The conductance of a single-channel system is just 2e2 =h times the TC at zerotemperature and zero-bias in the linear regime [5].
3. Conductance in general periodic dot: exact results A periodic dot is identified with periodic on-site terms and/or periodic hopping integrals, that is in Eq. (4) we have 0i ¼ 0iþN c and/or bi;iþ1 ¼ bN c þi;N c þiþ1 . The TM for isolated periodic dot is as follows: P ¼ Mn ;
n ¼ N=N c ;
ð10Þ
where N, N c , n, and M refer to the number of total atoms, the number of atoms in unit cell, the number of cells in the QD, and the TM for a unit cell, respectively. M is calculated from the TM of the unit cell as the following: M ¼ QðN C Þ QðN C 1Þ . . . Qð1Þ ;
ð11Þ
where QðiÞ stands for the TM of the sites i and i 1 within the unit cell. After substituting M in Eq. (10), from Eq. (9), the TC which depends on Fermi energy, dot–leads coupling, and number of unit cells, takes the following analytical form: T n ðzÞ ¼
4jImðsL ÞImðsR Þj ; jf n ðzÞ½M 11 þ sL M 12 sR M 21 sL sR M 22 þ f n1 ðzÞ½sL sR 1j2
ð12Þ
in which z ¼ e=tD , sLðRÞ ¼ kLðRÞ eiyLðRÞ , kLðRÞ ¼ t2DLðRÞ =tD tLðRÞ , cos yLðRÞ ¼ ðe e0LðRÞ Þ=2tLðRÞ , and f n ðzÞ depends on the number of cells and the trace of the TM of the isolated unit cell, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ln TrðMÞ TrðMÞ 2 þ f n ðzÞ ¼ ; l¼ 1. 2 2 l l1 The terms sL and sR can be identified with the dot self-energies due to the left and right leads, respectively. The denominator of Eq. (12) can also be identified with the inverse of jG 1N ðzÞj2 as we also have TðzÞ ¼ 4 ImðsL ÞImðsR ÞjG 1N ðzÞj2 . These can also be analytically calculated within the GF formalism in our previous works [20]. If the Fermi energy lies inside the dot energy band, ðjTrðMÞjp2Þ, otherwise, ðjTrðMÞj42Þ. In the former case, the TC has oscillating behaviors in both size and energy, but in the latter, the TC decays exponentially with dot size. Therefore, the dot energy band is determined by the following equation: TrðMÞ 2 cos fðzÞ ¼ 0:
ð13Þ
Thus, with the above consideration, f n ðzÞ takes the following more explicit forms: ( sin nfðzÞ= sin fðzÞ : jTrðMÞjp2; f n ðzÞ ¼ sinh nfðzÞ= sinh fðzÞ : jTrðMÞj42:
ð14Þ
Then, from Eqs. (12) and (14), the following results can be simply obtained for the limiting cases: T n ðzÞ ¼
16jImðsL ÞImðsR Þsinh2 fðzÞjl2ðn1Þ ; jlðM11 þ sL M12 sR M21 sL sR M22 Þ þ ðsL sR 1Þj2
jTrðMÞj42;
nb1
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and T n ðzÞ ¼
4jImðsL ÞImðsR ÞjjTrðMÞj2ðn2Þ ; jTrðMÞ½M11 þ sL M12 sR M21 sL sR M22 þ ðsL sR 1Þj2
jTrðMÞjb2,
and if jTrðMÞj ! 2, then T n ðzÞ ! n2 . Here, for simplicity, we consider a symmetric case, it is assumed that the right and left leads are similar (yL ¼ yR ¼ y) and further, the dot-left-lead hopping is equal the dot-right hopping term (tDL ¼ tDR ). Also, all atomic on-site energies in the leads are set to zero. With these assumptions, the TC (Eq. (12)) can be obtained as a simple analytical form T n ðzÞ ¼
4k2 sin2 y ; X 2n ðzÞ þ Y 2n ðzÞ
ð15Þ
where X n ðzÞ ¼ f n ðzÞ½M11 þ kðM12 M21 Þ cos y k2 M22 cos 2y þ f n1 ðzÞðk2 cos 2y 1Þ and Y n ðzÞ ¼ f n ðzÞ½kðM12 M21 Þ sin y k2 M22 sin 2y þ f n1 ðzÞk2 sin 2y, in which k ¼ t2DL =tD tL and cos y ¼ ztD =2tL . As is seen, in the symmetric case, the dependence of T n ðzÞ on z is even. If Fermi energy lies inside the dot and leads energy bands, the roots of X n ðzÞ ¼ 0 give the energies of resonance peaks, and the following relation describes conductance for a periodic dot at the resonance energies: T n ðzm Þ ¼
4 sin2 fm ; fsin nfm ðMm;12 Mm;21 Þ þ 2k cos ym ½sinðn 1Þfm Mm;22 sin nfm g2
ð16Þ
where Mm;ij ¼ Mij ðzm Þ, cos ym ¼ em =ð2tL Þ, cos fm ¼ Tr Mðzm Þ, zm ¼ em =tD , and em is resonance state energy. In the half-filled case, where the Fermi energy is zero (eF ¼ 0 ¼ z), T n ð0Þ has the following simple form: T n ð0Þ ¼
½f n ð0ÞðM0;11 þ
k2 M
4k2 ; 2 2 2 2 2 0;22 Þ f n1 ð0Þðk þ 1Þ þ f n ð0Þk ðM0;12 M0;21 Þ
ð17Þ
where M0;ij Mij ðz ¼ 0Þ, and in the dot energy band f n ð0Þ is equal to n . If the energy lies inside the dot energy band, but outside the leads energy band, then bound states can be formed. The bound state energies are the poles of the GF [20] of the dot, or the poles of the TC which correspond to the roots of following equation, obtained by considering jTrðMÞjp2: ZðzÞ ¼ f n ðzÞ½M11 þ sL M12 sR M21 sL sR M22 þ f n1 ðzÞ½sL sR 1 ¼ 0, where sLðRÞ is sLðRÞ
3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e e0LðRÞ e e0LðRÞ 4 ¼ 15. tD tLðRÞ 2tLðRÞ 2tLðRÞ t2DLðRÞ
2
ð18Þ
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It can be shown that the roots satisfy the following nonlinear equation (in this case, the eigenvectors are extended): tan gm ¼
ð1 sL sR Þ sin fm , M11 þ sL M12 sR M21 sL sR M22 ð1 sL sR Þ cos fm
(19)
where fm ¼ ðmp gm Þ=n, 2 cos fm ¼ Tr Mðzm Þ. Under some specific conditions, namely (jtL ¼ tR jojtD j5jtDL ¼ tDR j), if the energy lies outside the dot energy band (jTrðMÞj42), there can be formation of surface states. In this case, some of the roots of Eq. (18) fall outside the dot energy range, and these states are localized at the boundaries of the dot [7,12]. In the next section, we study three types (gapless, single, and double gap) of periodic dot, and we derive the TC exactly inside and outside the dot energy range.
4. Periodic dot: 0, 1 and 2 gap model 4.1. Conductance of uniform (gapless) dot From Eq. (4), if on-site energies and hopping integrals in the dot are fixed and independent of site index bi;iþ1 ¼ 1 and 0i ¼ 0 ¼ 0, we have a uniform dot. The TM can be obtained for a cell of uniform dot as z 1 MðzÞ ¼ . (20) 1 0 From Eq. (13), the dot energy band for uniform QD is identified by z ¼ 2tD cos f. When the energy lies inside the dot energy band, the TM can be calculated from Eq. (10), as follows: ! sinðn þ 1Þf sin nf 1 PðzÞ ¼ f n ðzÞMðzÞ f n1 ðzÞI ¼ , ð21Þ sin f sin nf sinðn 1Þf where I is the 2 2 identity matrix. Thus, substituting the elements of P matrix in Eq. (9), the TC of a uniform dot takes the following form: TðzÞ ¼
4j sin yL sin yR jsin2 ft2DL t2DR =ðt2D tL tR Þ . j sinðn þ 1Þf þ sin nfððt2DL =tD tL ÞeiyL þ ðt2DR =tD tR ÞeiyR Þ ðt2DL =tD tL Þðt2DR =tD tR Þ sinðn 1ÞfeiðyR þyL Þ j2
1
0
0.8
-2
0.6
-4
0.4
-6
0.2
-8 -2
-1
0 E
1
2
log T(E)
T(E)
(22)
-2
-1
0 E
1
2
Fig. 1. TðEÞ and log TðEÞ vs. E for a uniform dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85, and the dot–lead contact hopping is 0.5.
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In the symmetric case, at zero energy, the TC only depends on n and k is as follows: T n ð0Þ ¼
4k2 . ðk 1Þ2 ½k þ 1 þ nð1 kÞ2 þ 4k2 ½nðk 1Þ k2
Fig. 1 shows TðzÞ and log TðzÞ for a uniform (gapless) dot connected to ideal uniform leads for the mentioned values of the parameters. When Fermi energy is outside the dot energy band, the TC can be obtained from Eq. (22), by the simple replacement of sin by sinh. Eq. (22) has also been calculated by another method (GF) in our previous work [20]. 4.2. Conductance of AB (single gap) dot The AB dot can be formed by two different atoms, A and B or one atom with two different bonds. This model for example can be applied to the carbon chain [ðCH ¼ CH Þn ]. Fig. 2 shows an illustration of an AB dot attached to the uniform metallic leads. We identify the Hamiltonian of AB dot by periodic on-site and/or hopping energies, as follows: ( ) N N 1 X X ½0 þ dð1Þiþ1 cyi ci þ ½1 þ nð1Þiþ1 ðcyi ciþ1 þ h:c:Þ . ð23Þ H D ¼ tD i¼1
i¼1
In the following, we calculate the conductance for an AB dot analytically. From Eq. (12), in order to obtain the TC, we need the TM for isolated AB unit cell. By noting that 0 ¼ 0 and n2 a1, the TM can be written as follows: ! z2 d2 ð1 þ nÞ2 ðz dÞð1 nÞ 1 MðzÞ ¼ . ð24Þ 1 n2 ðz þ dÞð1 nÞ ð1 nÞ2 If we replace MðzÞ in Eq. (12), the TC can be derived analytically, in all energy regions: T n ðzÞ ¼
4k2 sin2 yð1 n2 Þ2 , jf n ½d2 þ ð1 þ nÞ2 z2 þ 2zkeiy ð1 nÞ k2 e2iy ð1 nÞ2 þ f n1 ð1 k2 e2iy Þj2
(25)
where k ¼ t2DL =tD tL , y ¼ arc cos ztD =2tL , f n ¼ ðln ln Þ=ðl l1 Þ, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 z2 d2 2ð1 þ nÞ2 z d2 2ð1 þ nÞ2 þ l¼ 1. 2ð1 n2 Þ 2ð1 n2 Þ From Eq. (16), conductance at the resonance energy is obtained as T n ðzm Þ ¼
ð1 n2 Þ2 sin2 fm , z2m j sin nfm ð1 nÞ ðt2DL =2t2L Þfð1 n2 Þ sinðn 1Þfm þ ð1 nÞ2 sin nfm gj2
where 2 cos fm ¼ Tr Mðzm Þ and zm refers to resonance energy. Fig. 3 shows TðEÞ and log TðEÞ vs. E (Fermi energy) for an AB dot connected to ideal uniform leads in the resonance and tunneling regions (note that z ¼ =tD ). Fig. 4 shows the TC as a function of on-site (d) Left Lead
Site No., 0
AB - Dot
(1,2)... (1,2)j
…
Right Lead
(1,2)n, 2n+1, …
Fig. 2. Geometry of an AB dot attached to two metallic leads.
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0
0.8
-2
0.6
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0.4
-6
0.2
-8
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1
-2
2
157
log T(E)
T(E)
M. Mardaani et al. / Physica E 28 (2005) 150–161
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0 E
1
2
1 0.8 0.6 0.4 0.2 0 -1
0.4 0.2 0 -0.5
1 0.8 0.6 0.4 0.2 0 -1
0.4 0.2 0 -0.5
-0.2
0
δ
ν
T
T
Fig. 3. TðEÞ and log TðEÞ vs. E for a single gap dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85, n ¼ 0:25, and dot–lead contact hopping is 0.5.
0.5
-0.4
0
δ
ν
-0.2 0.5
1
-0.4 1
Fig. 4. Tðd; nÞ for the Fermi energies z ¼ 0 (left) and 0:5 (right), for an AB dot connected to ideal uniform leads. Here, tL ¼ tR ¼ tD ¼ 1:0, and dot–lead contact hopping is 0.5.
and hopping (n) parameters, for the 0 and 0.5 of Fermi energy. As is evident, Tðd; nÞ has a sharp behavior with the hopping parameter n. From Eq. (19) the following eigenenergies for the isolated AB dot are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f mp gm zm ¼ d2 þ 4n2 þ 4ð1 n2 Þcos2 m ; fm ¼ 2 n and tan gm ¼
ð1 þ nÞ sin fm . 1 n þ ð1 þ nÞ cos fm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, the energy gap of the system is obtained as E g ¼ 2tD d2 þ 4n2 . If n ¼ 0, then M22 ¼ 1 and gm ¼ fm =2, thus, zm takes the following simple form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp . zm ¼ d2 þ 4 cos2 2n þ 1
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Left Lead
Double Gap Dot
Site No., 0 (1,2,3,
4)..(1,2,3,
Right Lead
4)j.. (1,2,3, 4)n, 4n+1, …
1
0
0.8
-2
0.6
-4
0.4
-6
0.2
-8
-2
-1
0 E
1
2
log T(E)
T(E)
Fig. 5. Geometry of a periodic-dot (double gap dot) attached to two metallic leads.
-2
-1
0 E
1
2
Fig. 6. TðEÞ and log TðEÞ vs. E for a double gap dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85 , n ¼ 0:25, and dot–lead contact hopping is 0.5.
4.3. Conductance of double gap dot Finally, we consider a special periodic QD with two gaps in its density of states. Fig. 5 shows the geometry of the considered QD attached to two uniform metallic leads. Such a configuration can be the result of displacements due to an acostic wave of wave length l ¼ 2a (Section 4.2: single gap case) or l ¼ 4a (double gap case). The electron speed being large, the transversal time of the dot is small, and of the order of femto second (fs), one can assume the process of atomic vibrations as adiabatic. We thus consider the time-dependent Hamiltonian matrix elements as slowly varying functions of time, and solve the electronic problem for each value of t. In the TB approximation and by assuming nearest-neighbor interaction, the Hamiltonian of the isolated QD is defined in the following form: " # N N 1 X X y y H D ¼ tD 0 ci ci þ ð1 þ nl i Þðci ciþ1 þ h:c:Þ , ð26Þ i¼1
i¼1
where l i ¼ þ1, if mod ði; 4Þ ¼ 1 or 2, and l i ¼ 1, if mod ði; 4Þ ¼ 0 or 3. The TM of the isolated cell, noting that 0 ¼ 0 and n2 a1, has the following form: ! z4 ð3n2 þ 2n þ 3Þz2 þ ð1 n2 Þ2 ð1 nÞ½z3 2ð1 þ n2 Þz 1 . MðzÞ ¼ ð1 n2 Þ2 ð1 nÞ½z3 2ð1 þ nÞ2 z ð1 nÞ2 ½ð1 þ nÞ2 z2 If we substitute elements of MðzÞ in Eq. (12), the TC can be calculated analytically. In zero Fermi energy, from Eq. (12), the TC becomes T n ð0Þ ¼
4jImðsL ÞImðsR Þj . jsL sR 1j2
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Fig. 7. TðE; nÞ: in double gap dot, the number of cells is two, all on-site energies are set to zero, and, tL ¼ tR ¼ tD ¼ 1:0; tDLðRÞ ¼ 0:5. The oscillation in the boundaries of the figure are not physical (due to some unavoidable errors).
It is seen that T n ð0Þ is independent of the number of cells and is fixed. From Eq. (16), conductance in the resonance energy is as follows: T¼
ð1 n2 Þ4 sin2 fm =z2m , j sin nfm ð1 nÞðz2m þ 2nÞ ðt2DL =2t2L Þfð1 n2 Þ2 sinðn 1Þfm ð1 nÞ2 ½ð1 þ nÞ2 z2m sin nfm gj2
where 2 cos fm ¼ Tr Mðzm Þ and zm refers to resonance energy. Fig. 6 shows the plot of the TC and log TC vs. Fermi energy inside and outside (gap) dot energy band. As is evident from Fig. 7, where the TC is plotted vs. Fermi energy and n, the gap strongly depends on n. From Eq. (18), the energy spectrum of the isolated dot is obtained as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!ffi u u f zm ¼ t2 ð1 þ n2 Þ 4n2 þ ð1 n2 Þcos2 m , 2 where ð1 þ nÞ2 sin fm . z2m ð1 þ nÞ2 ð1 cos fm Þ pffiffiffi Thus in this system the energy gap is E g ¼ 2 2ntD . fm ¼
mp gm ; n
tan gm ¼
4.4. Effect of lead’s thickness on transport Here, we consider a simple cubic periodic dot attached to an ideal simple cubic uniform leads. For simplicity, we assume all cross-sections within the dot and the leads obey the same symmetry properties, i.e., each cross-section can be diagonalized by the same unitary transformation [20]. Therefore, the problem is reduced to N x N y strictly one-dimensional chains, where N x N y is the total number of atoms in the cross-section of this system.
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T(E)
160
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2
-1.5
-1
-0.5
0 E
0.5
1
1.5
2
Fig. 8. TðEÞ vs. E for a single gap dot (polymer chain) connected to two ideal uniform leads with square cross-section (N x ¼ N y ¼ 1: ðiÞ ðiÞ thin curve, and N x ¼ N y ¼ 2: wide curve). Here, tðiÞ x ¼ ty ¼ tx ¼ 1:0; where i refers to the left (right)-lead index. tD;i ¼ 1:0½1 þ 0:25ð1Þiþ1 , and dot–lead contact hopping is 0.5. 0.6 0.5
T(E)
0.4 0.3 0.2 0.1 0
-2
-1.5
-1
-0.5
0 E
0.5
1
1.5
2
Fig. 9. TðEÞ vs. E for a single gap dot connected to an ideal simple cubic leads with different cross-section (N x ¼ N y ¼ 5: dash curve, iþ1 ðiÞ and N x ¼ N y ¼ 10: solid curve). Here, tðiÞ , and dot–lead contact hopping is 0.5. x ¼ ty ¼ 1:0; tD;i ¼ 1:0 ½1 þ 0:25ð1Þ
With the above considerations, the Hamiltonians of (a single gap) dot (see Section 4.2) coupled to the simple cubic leads, after diagonalization of transverse degrees of freedom can be written as follows: X y X y H LðRÞ ¼ eðLÞ ðcmn;i cmn;iþ1 þ h:c:Þ, ð27Þ mn cmn;i cmn;i þ tLðRÞ mn;i
HD ¼
X mn;i
y eðDÞ mn cmn;i cmn;i þ tD
mn;i
X
½1 þ ð1Þiþ1 nðcymn;i cmn;iþ1 þ h:c:Þ,
ð28Þ
mn;i
ðiÞ ðiÞ ðiÞ where the on-site energies for ðm; nÞ mode are eðiÞ mn ¼ e0 þ 2tx cosðmp=ðN x þ 1ÞÞ þ 2ty cosðnp=ðN y þ 1ÞÞ, and i refers becomes P to the dot or left (right) lead. In this case, the total conductance of the system ðDÞ TðzÞ ¼ mn T mn ðzÞ where T mn ðzÞ can be calculated by Eq. (25). In the special case, tðDÞ ¼ t x y ¼ 0, the periodic dot is formed of the N x N y disconnected and parallel periodic polymer chains bridging the two simple cubic leads. Therefore, the conductance of a single chain is almost 1=ðN x N y Þ times of conductance of N x N y parallel polymer chains [20]. Figs. 8 and 9 show the conductance for a periodic chain such as ðCH ¼ CH Þn , which is attached to the same right and left leads (with square cross-sections) for arbitrary N x ¼ N y ¼ 1; 2; 5 and 10. Here, all on-site energies have been set to 0 in this system, and also, all hopping energies of the leads are set to 1. It is clear from Figs. 8 and 9 that by increasing of the cross-section the conductance is decreased. The reason for
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this effect in comparison to an ideal chain (Section 4.2) is related to the difference between dot and leads energy band. We have discussed the single gap dot; our results can be, however, easily generalized to other cases (as in Sections 4.1 and 4.2). This allows us to consider coherent transport in a more realistic setup. The effect of lead thickness, which modifies the dot self-energy due to leads, can also be analytically calculated in the GF formalism [20].
5. Conclusion We have studied the conductance of a general dot in the one-electron approximation. We have derived a general formula for the calculation of conductance in the tight-binding approach and assuming nearestneighbor interaction based on the transfer matrix method. Specially, the TC has been studied in much detail for a general periodic dot attached to two uniform leads. The equation describes the relation between the TC (conductance) and the TM for isolated cell in a periodic dot. It was shown that when the Fermi energy is within the periodic dot energy band, the TC has an oscillating behavior. At the band edge of the dot, the behavior is scaled as 1=n2cell . Outside the dot band, it decays exponentially with dot size. Next, we have derived a nonlinear equation which gives the bound state and surface state energies for the considered dot. Finally, as examples, we have obtained the TC for gapless, single and double gap dots in more details. Also, the effect of the cross-section of the leads has been considered in our study. The results may describe a realistic system such as a polymer chain coupled to leads with arbitrary cross-section. Generally, our results can also be used for molecular wires and periodic superlattices [20] attached to uniform leads.
Acknowledgements We thank Dr. A.T. Rezakhani for useful discussion and critical reading of this paper. References [1] P. Harrison, Quantum Wells, Wires and Dots, Wiley, New York, 2000. [2] D.K. Ferry, S.M. Goognick, Transport in Nanostructures, Cambridge University Press, Cambridge, 1997; G. Speyer, R. Akis, D.K. Ferry, Physica E 19 (2003) 145. [3] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [4] H.X. He, X.L. Li, N.J. Tao, Phys. Rev. B 68 (2003) 045302. [5] R. Landauer, Phys. Lett. 85 A (1986) 91. [6] C. Joachim, J.F. Vinuesa, Europhys. Lett. 33 (1996) 635. [7] D.W.L. Sprung, J.D. Sigetich, H. Wu, J. Martorell, Amer. J. Phys. 68 (2000) 715. [8] M.H. Tyc, W. Salejda, Physica A 303 (2002) 493. [9] C. Pacher, E. Gornik, Physica E 21 (2004) 783. [10] J. Heinrichs, J. Phys.: Condens. Mat. 12 (2000) 5565. [11] A.F. Sadreev, I. Rotter, J. Phys. A: Math. Gen. 36 (2003) 11413. [12] S. Chattopadhyah, A. Chakrabarti, J. Phys.: Condens. Mat. 16 (2004) 313. [13] C. Caroli, C.R. Combescot, P. Nozieres, D. Saint-James, J. Phys. C 4 (1971) 916. [14] D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1981) 6851. [15] P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. [16] H. Pastawski, Phys. Rev. B 44 (1991) 6329. [17] Y. Xue, M.A. Ratner, Phys. Rev. B 68 (2003) 115406 and references there in. [18] A. Opniko, Y. Klymenko, L. Malysheva, Phys. Rev. B 62 (2003) 10482. [19] G. Cuniberti, G. Fagas, K. Richter, Chem. Phys. 281 (2002) 465. [20] M. Mardaani, K. Esfarjani, Physica E 25 (2004) 119; M. Mardaani, K. Esfarjani, Physica E 27 (2005) 227.
Physica E 28 (2005) 150–161 www.elsevier.com/locate/physe
Analytical results on coherent conductance in a general periodic quantum dot: Transfer matrix method Mohammad Mardaania,b,c,, Ali A. Shokria,b,d, Keivan Esfarjania a Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran Computational Physical Sciences Research Laboratory, Department of Nano-Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5531, Tehran, Iran c Faculty of Science, Physics Group, Shahrekord University, Shahrekord, Iran d Department of Physics, Tehran Payame-Noor University, Fallahpour St. Nejatollahi St., Tehran, Iran
b
Received 22 February 2005; accepted 9 March 2005 Available online 17 May 2005
Abstract In this work, we study the conductance of a general periodic quantum dot (QD) attached to ideal semi-infinite uniform metallic leads (nanocrystals), fully analytically. We propose a new general formula which relates conductance to transfer matrix (TM) for an isolated cell in the periodic dot. The equation describes exactly the dependence of the transmission coefficient (TC) on Fermi energy, dot-size, dot–lead coupling, and gate voltage for an arbitrary periodic dot. Then, we derive a nonlinear equation which gives the resonance, bound, and surface state energies. Finally, the TC has been calculated for gapless, single, and double gap models exactly. Moreover, we have also calculated the effects of the cross-section of the leads, which were separated by a polymer chain on the conductance. Our calculations can be generalized to any type of QD and quantum wire (QW) within the one-electron approximation, and, can be applied to, e.g., molecular, polymer, and nanocrystal junctions, where these results may be useful in designing future molecular electronic devices. r 2005 Elsevier B.V. All rights reserved. PACS: 05.60.Gg; 81.07.Ta; 81.07.Vb Keywords: Coherent conductance; Periodic quantum dot; Transfer matrix; Analytical results
Corresponding author.
E-mail address: [email protected] (M. Mardaani). 1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.03.005
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1. Introduction In the past few years, the study of electronic transport through quantum dots (QDs), quantum wires (QWs), and molecular wires have been major areas of research in mesoscopic physics [1–3]. The size and shape of QDs and QWs can be precisely controlled with today’s technology [1,2,4]. QDs can be connected to electrodes and are, therefore, excellent tools to study atomic-like properties. The transport properties of a QD can be measured by coupling it to leads and passing current through the dot. In fact, there is a vast and recent literature on this subject [1–3]. Here, at low temperatures, we ignore the inelastic scattering process, thus, conductance through a QD is characterized by quantum coherence. We rely on the Landauer’s approach [5] for studying the coherent transport properties in QD and QW systems. The Landauer’s approach states that electronic conductance is proportional to transmission coefficient (TC). Different methods based on this approach have been developed for the study of transport phenomena. One class of methods is based on the transfer matrix (TM), e.g. [1,3,6–12], and the others make use of Green’s function (GF) formalism, e.g. [3,13–20]. In our previous works [20], we have calculated the coherent conductance for uniform and alternating dot fully analytically by GF method. In the present paper, we use an analytical method based on the tight binding (TB) model and transfer matrix method to derive dependence of conductance on Fermi energy, dot-size, and gate voltage, for a general periodic dot connected to two uniform ideal leads at T ¼ 0 K. The organization of this paper is as follows. In Section 2, the TB one-electron Hamiltonian for a general QD is described. Here, a detailed description of the TM approach in the TB model, assuming nearestneighbor interaction, is given. In this section, for a general dot, the TC, which depends on the Fermi energy, QD size, dot–leads coupling, and gate voltage was derived analytically. The TC calculations in the periodic dot case are explained in much detail in Section 3. Also, in this section we obtain a nonlinear equation which gives the resonant and bound states energies and surface state energies if they exit. In Section 4, we calculate the TC for gapless, single gap, and double gap dots in more details. The paper ends with the conclusion in Section 5.
2. Description of transfer matrix method We consider a chain composed of a general QD attached to two uniform semi-infinite leads. For simplicity, the two metallic leads are assumed to be ideal. Also, the elements of TM for the general QD are calculated by coherent mechanism. It means that the dot length is smaller than the phase coherence length [3]. The generalized Hamiltonian which describes the system is given by H ¼ H L þ H DL þ H D þ H DR þ H R ;
ð1Þ
where H L;D;R describes the Hamiltonian for the left lead, dot and the right lead, and also, H DLðRÞ refers to the Hamiltonian for the coupling between the general QD and the left (right) lead. The Hamiltonian of left (right) lead is defined as X y H LðRÞ ¼ tLðRÞ ðci ciþ1 þ h:c:Þ; ð2Þ i
where tLðRÞ refers to the hopping integral between s atomic orbitals (assumed to be orthogonal), for the left and right lead, i varies between ð1; 0 and ½N þ 1; þ1Þ, respectively, and N refers to the number of total atoms in the dot. The QD is coupled to the leads via the contact Hamiltonian, H DLðRÞ as follows: H DLðRÞ ¼ tDLðRÞ ðcy0ðNÞ c1ðNþ1Þ þ h:c:Þ;
ð3Þ
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where tDLðRÞ refers to the contact hopping integral between dot and the left (right) lead. Also, the dot Hamiltonian is given by " # N N1 X X y y H D ¼ tD 0i ci ci þ bi;iþ1 ðci ciþ1 þ h:c:Þ ; ð4Þ i¼1
i¼1
where 0i and bi;iþ1 refer to the on-site energy and hopping integral in units of tD , respectively. We write the time-independent Schro¨dinger equation, HC ¼ eC, for the QD coupled to the leads as 0i ci þ bi;iþ1 ciþ1 þ bi1;i ci1 ¼ zci ;
ð5Þ
where i ¼ 1; . . . ; þ1, ci is the amplitude of the wave function at site i, and z ¼ e=tD . Eq. (5) can be written in the following form: ! ! ciþ1 ci ¼ PðiÞ ; ð6Þ ci ci1 ðiÞ ðiÞ ðiÞ where PðiÞ 11 ¼ ðz 0i Þ=bi;iþ1 , P12 ¼ bi1;i =bi;iþ1 , P21 ¼ 1, and P22 ¼ 0. Thus, after multiplication, we can obtain a recurrent equation for the whole dot including dot-lead contacts as ! ! cNþ2 c0 ðRÞ ðLÞ ¼P P P ; ð7Þ cNþ1 c1
where PðLÞ and PðRÞ refer to the TM of the left and the right contacts, respectively, and P stands for PðNÞ PðN1Þ Pð1Þ . It is clear that the discrepancy between ci , ci 1 , in the leads is a phase factor, because the leads are ideal. Also, it is known that right and left leads energy bands are described by eLðRÞ ¼ 2tLðRÞ cos kLðRÞ a ¼ 2tLðRÞ cos yLðRÞ , therefore, after this simplification, we have 0 1 0 1 tL tD 0 0 B tDL C B tDR C B C B C ; PðRÞ ¼ eiyR B . PðLÞ ¼ eiyL B C tDL A tDR C @ @ A 0 0 tD tR In order to calculate the transmission and reflection coefficients, the wave functions in the 1 and 0 sites of the left lead, and N and N þ 1 sites of the right lead are needed: c0 ¼ 1 þ r;
c1 ¼ eiyL þ reiyL
and cNþ1 ¼ teiðNþ1ÞyR ;
cNþ2 ¼ teiðNþ2ÞyR ,
where r and t are the amplitudes of the reflected and transmitted electrons, respectively. The relation between transmission coefficient and transmission amplitude is given by [3] vR tR sin yR tt ; T ¼ tt ¼ ð8Þ vL tL sin yL where t is the conjugate of t. If we solve for t and r from Eq. (7) and substitute t in Eq. (8), the following analytical form for TC in the considered system is obtained: TðzÞ ¼
t2D tL tR jP11 þ ðt2DL =tD tL ÞP12 eiyL
4t2DL t2DR j sin yL sin yR j ; ðt2DR =tD tR ÞP21 eiyR ðt2DL =tD tL Þðt2DR =tD tR ÞP22 eiðyR þyL Þ j2
ð9Þ
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where P refers to the TM for isolated dot. As it can be seen, to calculate the TC, it is only needed to evaluate the TM for an isolated dot. Therefore Eq. (9) is a general equation for a dot attached to two leads within a nearest-neighbor TB model. The conductance of a single-channel system is just 2e2 =h times the TC at zerotemperature and zero-bias in the linear regime [5].
3. Conductance in general periodic dot: exact results A periodic dot is identified with periodic on-site terms and/or periodic hopping integrals, that is in Eq. (4) we have 0i ¼ 0iþN c and/or bi;iþ1 ¼ bN c þi;N c þiþ1 . The TM for isolated periodic dot is as follows: P ¼ Mn ;
n ¼ N=N c ;
ð10Þ
where N, N c , n, and M refer to the number of total atoms, the number of atoms in unit cell, the number of cells in the QD, and the TM for a unit cell, respectively. M is calculated from the TM of the unit cell as the following: M ¼ QðN C Þ QðN C 1Þ . . . Qð1Þ ;
ð11Þ
where QðiÞ stands for the TM of the sites i and i 1 within the unit cell. After substituting M in Eq. (10), from Eq. (9), the TC which depends on Fermi energy, dot–leads coupling, and number of unit cells, takes the following analytical form: T n ðzÞ ¼
4jImðsL ÞImðsR Þj ; jf n ðzÞ½M 11 þ sL M 12 sR M 21 sL sR M 22 þ f n1 ðzÞ½sL sR 1j2
ð12Þ
in which z ¼ e=tD , sLðRÞ ¼ kLðRÞ eiyLðRÞ , kLðRÞ ¼ t2DLðRÞ =tD tLðRÞ , cos yLðRÞ ¼ ðe e0LðRÞ Þ=2tLðRÞ , and f n ðzÞ depends on the number of cells and the trace of the TM of the isolated unit cell, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ln TrðMÞ TrðMÞ 2 þ f n ðzÞ ¼ ; l¼ 1. 2 2 l l1 The terms sL and sR can be identified with the dot self-energies due to the left and right leads, respectively. The denominator of Eq. (12) can also be identified with the inverse of jG 1N ðzÞj2 as we also have TðzÞ ¼ 4 ImðsL ÞImðsR ÞjG 1N ðzÞj2 . These can also be analytically calculated within the GF formalism in our previous works [20]. If the Fermi energy lies inside the dot energy band, ðjTrðMÞjp2Þ, otherwise, ðjTrðMÞj42Þ. In the former case, the TC has oscillating behaviors in both size and energy, but in the latter, the TC decays exponentially with dot size. Therefore, the dot energy band is determined by the following equation: TrðMÞ 2 cos fðzÞ ¼ 0:
ð13Þ
Thus, with the above consideration, f n ðzÞ takes the following more explicit forms: ( sin nfðzÞ= sin fðzÞ : jTrðMÞjp2; f n ðzÞ ¼ sinh nfðzÞ= sinh fðzÞ : jTrðMÞj42:
ð14Þ
Then, from Eqs. (12) and (14), the following results can be simply obtained for the limiting cases: T n ðzÞ ¼
16jImðsL ÞImðsR Þsinh2 fðzÞjl2ðn1Þ ; jlðM11 þ sL M12 sR M21 sL sR M22 Þ þ ðsL sR 1Þj2
jTrðMÞj42;
nb1
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and T n ðzÞ ¼
4jImðsL ÞImðsR ÞjjTrðMÞj2ðn2Þ ; jTrðMÞ½M11 þ sL M12 sR M21 sL sR M22 þ ðsL sR 1Þj2
jTrðMÞjb2,
and if jTrðMÞj ! 2, then T n ðzÞ ! n2 . Here, for simplicity, we consider a symmetric case, it is assumed that the right and left leads are similar (yL ¼ yR ¼ y) and further, the dot-left-lead hopping is equal the dot-right hopping term (tDL ¼ tDR ). Also, all atomic on-site energies in the leads are set to zero. With these assumptions, the TC (Eq. (12)) can be obtained as a simple analytical form T n ðzÞ ¼
4k2 sin2 y ; X 2n ðzÞ þ Y 2n ðzÞ
ð15Þ
where X n ðzÞ ¼ f n ðzÞ½M11 þ kðM12 M21 Þ cos y k2 M22 cos 2y þ f n1 ðzÞðk2 cos 2y 1Þ and Y n ðzÞ ¼ f n ðzÞ½kðM12 M21 Þ sin y k2 M22 sin 2y þ f n1 ðzÞk2 sin 2y, in which k ¼ t2DL =tD tL and cos y ¼ ztD =2tL . As is seen, in the symmetric case, the dependence of T n ðzÞ on z is even. If Fermi energy lies inside the dot and leads energy bands, the roots of X n ðzÞ ¼ 0 give the energies of resonance peaks, and the following relation describes conductance for a periodic dot at the resonance energies: T n ðzm Þ ¼
4 sin2 fm ; fsin nfm ðMm;12 Mm;21 Þ þ 2k cos ym ½sinðn 1Þfm Mm;22 sin nfm g2
ð16Þ
where Mm;ij ¼ Mij ðzm Þ, cos ym ¼ em =ð2tL Þ, cos fm ¼ Tr Mðzm Þ, zm ¼ em =tD , and em is resonance state energy. In the half-filled case, where the Fermi energy is zero (eF ¼ 0 ¼ z), T n ð0Þ has the following simple form: T n ð0Þ ¼
½f n ð0ÞðM0;11 þ
k2 M
4k2 ; 2 2 2 2 2 0;22 Þ f n1 ð0Þðk þ 1Þ þ f n ð0Þk ðM0;12 M0;21 Þ
ð17Þ
where M0;ij Mij ðz ¼ 0Þ, and in the dot energy band f n ð0Þ is equal to n . If the energy lies inside the dot energy band, but outside the leads energy band, then bound states can be formed. The bound state energies are the poles of the GF [20] of the dot, or the poles of the TC which correspond to the roots of following equation, obtained by considering jTrðMÞjp2: ZðzÞ ¼ f n ðzÞ½M11 þ sL M12 sR M21 sL sR M22 þ f n1 ðzÞ½sL sR 1 ¼ 0, where sLðRÞ is sLðRÞ
3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e e0LðRÞ e e0LðRÞ 4 ¼ 15. tD tLðRÞ 2tLðRÞ 2tLðRÞ t2DLðRÞ
2
ð18Þ
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It can be shown that the roots satisfy the following nonlinear equation (in this case, the eigenvectors are extended): tan gm ¼
ð1 sL sR Þ sin fm , M11 þ sL M12 sR M21 sL sR M22 ð1 sL sR Þ cos fm
(19)
where fm ¼ ðmp gm Þ=n, 2 cos fm ¼ Tr Mðzm Þ. Under some specific conditions, namely (jtL ¼ tR jojtD j5jtDL ¼ tDR j), if the energy lies outside the dot energy band (jTrðMÞj42), there can be formation of surface states. In this case, some of the roots of Eq. (18) fall outside the dot energy range, and these states are localized at the boundaries of the dot [7,12]. In the next section, we study three types (gapless, single, and double gap) of periodic dot, and we derive the TC exactly inside and outside the dot energy range.
4. Periodic dot: 0, 1 and 2 gap model 4.1. Conductance of uniform (gapless) dot From Eq. (4), if on-site energies and hopping integrals in the dot are fixed and independent of site index bi;iþ1 ¼ 1 and 0i ¼ 0 ¼ 0, we have a uniform dot. The TM can be obtained for a cell of uniform dot as z 1 MðzÞ ¼ . (20) 1 0 From Eq. (13), the dot energy band for uniform QD is identified by z ¼ 2tD cos f. When the energy lies inside the dot energy band, the TM can be calculated from Eq. (10), as follows: ! sinðn þ 1Þf sin nf 1 PðzÞ ¼ f n ðzÞMðzÞ f n1 ðzÞI ¼ , ð21Þ sin f sin nf sinðn 1Þf where I is the 2 2 identity matrix. Thus, substituting the elements of P matrix in Eq. (9), the TC of a uniform dot takes the following form: TðzÞ ¼
4j sin yL sin yR jsin2 ft2DL t2DR =ðt2D tL tR Þ . j sinðn þ 1Þf þ sin nfððt2DL =tD tL ÞeiyL þ ðt2DR =tD tR ÞeiyR Þ ðt2DL =tD tL Þðt2DR =tD tR Þ sinðn 1ÞfeiðyR þyL Þ j2
1
0
0.8
-2
0.6
-4
0.4
-6
0.2
-8 -2
-1
0 E
1
2
log T(E)
T(E)
(22)
-2
-1
0 E
1
2
Fig. 1. TðEÞ and log TðEÞ vs. E for a uniform dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85, and the dot–lead contact hopping is 0.5.
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In the symmetric case, at zero energy, the TC only depends on n and k is as follows: T n ð0Þ ¼
4k2 . ðk 1Þ2 ½k þ 1 þ nð1 kÞ2 þ 4k2 ½nðk 1Þ k2
Fig. 1 shows TðzÞ and log TðzÞ for a uniform (gapless) dot connected to ideal uniform leads for the mentioned values of the parameters. When Fermi energy is outside the dot energy band, the TC can be obtained from Eq. (22), by the simple replacement of sin by sinh. Eq. (22) has also been calculated by another method (GF) in our previous work [20]. 4.2. Conductance of AB (single gap) dot The AB dot can be formed by two different atoms, A and B or one atom with two different bonds. This model for example can be applied to the carbon chain [ðCH ¼ CH Þn ]. Fig. 2 shows an illustration of an AB dot attached to the uniform metallic leads. We identify the Hamiltonian of AB dot by periodic on-site and/or hopping energies, as follows: ( ) N N 1 X X ½0 þ dð1Þiþ1 cyi ci þ ½1 þ nð1Þiþ1 ðcyi ciþ1 þ h:c:Þ . ð23Þ H D ¼ tD i¼1
i¼1
In the following, we calculate the conductance for an AB dot analytically. From Eq. (12), in order to obtain the TC, we need the TM for isolated AB unit cell. By noting that 0 ¼ 0 and n2 a1, the TM can be written as follows: ! z2 d2 ð1 þ nÞ2 ðz dÞð1 nÞ 1 MðzÞ ¼ . ð24Þ 1 n2 ðz þ dÞð1 nÞ ð1 nÞ2 If we replace MðzÞ in Eq. (12), the TC can be derived analytically, in all energy regions: T n ðzÞ ¼
4k2 sin2 yð1 n2 Þ2 , jf n ½d2 þ ð1 þ nÞ2 z2 þ 2zkeiy ð1 nÞ k2 e2iy ð1 nÞ2 þ f n1 ð1 k2 e2iy Þj2
(25)
where k ¼ t2DL =tD tL , y ¼ arc cos ztD =2tL , f n ¼ ðln ln Þ=ðl l1 Þ, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 z2 d2 2ð1 þ nÞ2 z d2 2ð1 þ nÞ2 þ l¼ 1. 2ð1 n2 Þ 2ð1 n2 Þ From Eq. (16), conductance at the resonance energy is obtained as T n ðzm Þ ¼
ð1 n2 Þ2 sin2 fm , z2m j sin nfm ð1 nÞ ðt2DL =2t2L Þfð1 n2 Þ sinðn 1Þfm þ ð1 nÞ2 sin nfm gj2
where 2 cos fm ¼ Tr Mðzm Þ and zm refers to resonance energy. Fig. 3 shows TðEÞ and log TðEÞ vs. E (Fermi energy) for an AB dot connected to ideal uniform leads in the resonance and tunneling regions (note that z ¼ =tD ). Fig. 4 shows the TC as a function of on-site (d) Left Lead
Site No., 0
AB - Dot
(1,2)... (1,2)j
…
Right Lead
(1,2)n, 2n+1, …
Fig. 2. Geometry of an AB dot attached to two metallic leads.
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0
0.8
-2
0.6
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0.4
-6
0.2
-8
-2
-1
0 E
1
-2
2
157
log T(E)
T(E)
M. Mardaani et al. / Physica E 28 (2005) 150–161
-1
0 E
1
2
1 0.8 0.6 0.4 0.2 0 -1
0.4 0.2 0 -0.5
1 0.8 0.6 0.4 0.2 0 -1
0.4 0.2 0 -0.5
-0.2
0
δ
ν
T
T
Fig. 3. TðEÞ and log TðEÞ vs. E for a single gap dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85, n ¼ 0:25, and dot–lead contact hopping is 0.5.
0.5
-0.4
0
δ
ν
-0.2 0.5
1
-0.4 1
Fig. 4. Tðd; nÞ for the Fermi energies z ¼ 0 (left) and 0:5 (right), for an AB dot connected to ideal uniform leads. Here, tL ¼ tR ¼ tD ¼ 1:0, and dot–lead contact hopping is 0.5.
and hopping (n) parameters, for the 0 and 0.5 of Fermi energy. As is evident, Tðd; nÞ has a sharp behavior with the hopping parameter n. From Eq. (19) the following eigenenergies for the isolated AB dot are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f mp gm zm ¼ d2 þ 4n2 þ 4ð1 n2 Þcos2 m ; fm ¼ 2 n and tan gm ¼
ð1 þ nÞ sin fm . 1 n þ ð1 þ nÞ cos fm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, the energy gap of the system is obtained as E g ¼ 2tD d2 þ 4n2 . If n ¼ 0, then M22 ¼ 1 and gm ¼ fm =2, thus, zm takes the following simple form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp . zm ¼ d2 þ 4 cos2 2n þ 1
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Left Lead
Double Gap Dot
Site No., 0 (1,2,3,
4)..(1,2,3,
Right Lead
4)j.. (1,2,3, 4)n, 4n+1, …
1
0
0.8
-2
0.6
-4
0.4
-6
0.2
-8
-2
-1
0 E
1
2
log T(E)
T(E)
Fig. 5. Geometry of a periodic-dot (double gap dot) attached to two metallic leads.
-2
-1
0 E
1
2
Fig. 6. TðEÞ and log TðEÞ vs. E for a double gap dot connected to ideal uniform leads. Here, tL ¼ tR ¼ 1; tD ¼ 0:85 , n ¼ 0:25, and dot–lead contact hopping is 0.5.
4.3. Conductance of double gap dot Finally, we consider a special periodic QD with two gaps in its density of states. Fig. 5 shows the geometry of the considered QD attached to two uniform metallic leads. Such a configuration can be the result of displacements due to an acostic wave of wave length l ¼ 2a (Section 4.2: single gap case) or l ¼ 4a (double gap case). The electron speed being large, the transversal time of the dot is small, and of the order of femto second (fs), one can assume the process of atomic vibrations as adiabatic. We thus consider the time-dependent Hamiltonian matrix elements as slowly varying functions of time, and solve the electronic problem for each value of t. In the TB approximation and by assuming nearest-neighbor interaction, the Hamiltonian of the isolated QD is defined in the following form: " # N N 1 X X y y H D ¼ tD 0 ci ci þ ð1 þ nl i Þðci ciþ1 þ h:c:Þ , ð26Þ i¼1
i¼1
where l i ¼ þ1, if mod ði; 4Þ ¼ 1 or 2, and l i ¼ 1, if mod ði; 4Þ ¼ 0 or 3. The TM of the isolated cell, noting that 0 ¼ 0 and n2 a1, has the following form: ! z4 ð3n2 þ 2n þ 3Þz2 þ ð1 n2 Þ2 ð1 nÞ½z3 2ð1 þ n2 Þz 1 . MðzÞ ¼ ð1 n2 Þ2 ð1 nÞ½z3 2ð1 þ nÞ2 z ð1 nÞ2 ½ð1 þ nÞ2 z2 If we substitute elements of MðzÞ in Eq. (12), the TC can be calculated analytically. In zero Fermi energy, from Eq. (12), the TC becomes T n ð0Þ ¼
4jImðsL ÞImðsR Þj . jsL sR 1j2
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Fig. 7. TðE; nÞ: in double gap dot, the number of cells is two, all on-site energies are set to zero, and, tL ¼ tR ¼ tD ¼ 1:0; tDLðRÞ ¼ 0:5. The oscillation in the boundaries of the figure are not physical (due to some unavoidable errors).
It is seen that T n ð0Þ is independent of the number of cells and is fixed. From Eq. (16), conductance in the resonance energy is as follows: T¼
ð1 n2 Þ4 sin2 fm =z2m , j sin nfm ð1 nÞðz2m þ 2nÞ ðt2DL =2t2L Þfð1 n2 Þ2 sinðn 1Þfm ð1 nÞ2 ½ð1 þ nÞ2 z2m sin nfm gj2
where 2 cos fm ¼ Tr Mðzm Þ and zm refers to resonance energy. Fig. 6 shows the plot of the TC and log TC vs. Fermi energy inside and outside (gap) dot energy band. As is evident from Fig. 7, where the TC is plotted vs. Fermi energy and n, the gap strongly depends on n. From Eq. (18), the energy spectrum of the isolated dot is obtained as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!ffi u u f zm ¼ t2 ð1 þ n2 Þ 4n2 þ ð1 n2 Þcos2 m , 2 where ð1 þ nÞ2 sin fm . z2m ð1 þ nÞ2 ð1 cos fm Þ pffiffiffi Thus in this system the energy gap is E g ¼ 2 2ntD . fm ¼
mp gm ; n
tan gm ¼
4.4. Effect of lead’s thickness on transport Here, we consider a simple cubic periodic dot attached to an ideal simple cubic uniform leads. For simplicity, we assume all cross-sections within the dot and the leads obey the same symmetry properties, i.e., each cross-section can be diagonalized by the same unitary transformation [20]. Therefore, the problem is reduced to N x N y strictly one-dimensional chains, where N x N y is the total number of atoms in the cross-section of this system.
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2
-1.5
-1
-0.5
0 E
0.5
1
1.5
2
Fig. 8. TðEÞ vs. E for a single gap dot (polymer chain) connected to two ideal uniform leads with square cross-section (N x ¼ N y ¼ 1: ðiÞ ðiÞ thin curve, and N x ¼ N y ¼ 2: wide curve). Here, tðiÞ x ¼ ty ¼ tx ¼ 1:0; where i refers to the left (right)-lead index. tD;i ¼ 1:0½1 þ 0:25ð1Þiþ1 , and dot–lead contact hopping is 0.5. 0.6 0.5
T(E)
0.4 0.3 0.2 0.1 0
-2
-1.5
-1
-0.5
0 E
0.5
1
1.5
2
Fig. 9. TðEÞ vs. E for a single gap dot connected to an ideal simple cubic leads with different cross-section (N x ¼ N y ¼ 5: dash curve, iþ1 ðiÞ and N x ¼ N y ¼ 10: solid curve). Here, tðiÞ , and dot–lead contact hopping is 0.5. x ¼ ty ¼ 1:0; tD;i ¼ 1:0 ½1 þ 0:25ð1Þ
With the above considerations, the Hamiltonians of (a single gap) dot (see Section 4.2) coupled to the simple cubic leads, after diagonalization of transverse degrees of freedom can be written as follows: X y X y H LðRÞ ¼ eðLÞ ðcmn;i cmn;iþ1 þ h:c:Þ, ð27Þ mn cmn;i cmn;i þ tLðRÞ mn;i
HD ¼
X mn;i
y eðDÞ mn cmn;i cmn;i þ tD
mn;i
X
½1 þ ð1Þiþ1 nðcymn;i cmn;iþ1 þ h:c:Þ,
ð28Þ
mn;i
ðiÞ ðiÞ ðiÞ where the on-site energies for ðm; nÞ mode are eðiÞ mn ¼ e0 þ 2tx cosðmp=ðN x þ 1ÞÞ þ 2ty cosðnp=ðN y þ 1ÞÞ, and i refers becomes P to the dot or left (right) lead. In this case, the total conductance of the system ðDÞ TðzÞ ¼ mn T mn ðzÞ where T mn ðzÞ can be calculated by Eq. (25). In the special case, tðDÞ ¼ t x y ¼ 0, the periodic dot is formed of the N x N y disconnected and parallel periodic polymer chains bridging the two simple cubic leads. Therefore, the conductance of a single chain is almost 1=ðN x N y Þ times of conductance of N x N y parallel polymer chains [20]. Figs. 8 and 9 show the conductance for a periodic chain such as ðCH ¼ CH Þn , which is attached to the same right and left leads (with square cross-sections) for arbitrary N x ¼ N y ¼ 1; 2; 5 and 10. Here, all on-site energies have been set to 0 in this system, and also, all hopping energies of the leads are set to 1. It is clear from Figs. 8 and 9 that by increasing of the cross-section the conductance is decreased. The reason for
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this effect in comparison to an ideal chain (Section 4.2) is related to the difference between dot and leads energy band. We have discussed the single gap dot; our results can be, however, easily generalized to other cases (as in Sections 4.1 and 4.2). This allows us to consider coherent transport in a more realistic setup. The effect of lead thickness, which modifies the dot self-energy due to leads, can also be analytically calculated in the GF formalism [20].
5. Conclusion We have studied the conductance of a general dot in the one-electron approximation. We have derived a general formula for the calculation of conductance in the tight-binding approach and assuming nearestneighbor interaction based on the transfer matrix method. Specially, the TC has been studied in much detail for a general periodic dot attached to two uniform leads. The equation describes the relation between the TC (conductance) and the TM for isolated cell in a periodic dot. It was shown that when the Fermi energy is within the periodic dot energy band, the TC has an oscillating behavior. At the band edge of the dot, the behavior is scaled as 1=n2cell . Outside the dot band, it decays exponentially with dot size. Next, we have derived a nonlinear equation which gives the bound state and surface state energies for the considered dot. Finally, as examples, we have obtained the TC for gapless, single and double gap dots in more details. Also, the effect of the cross-section of the leads has been considered in our study. The results may describe a realistic system such as a polymer chain coupled to leads with arbitrary cross-section. Generally, our results can also be used for molecular wires and periodic superlattices [20] attached to uniform leads.
Acknowledgements We thank Dr. A.T. Rezakhani for useful discussion and critical reading of this paper. References [1] P. Harrison, Quantum Wells, Wires and Dots, Wiley, New York, 2000. [2] D.K. Ferry, S.M. Goognick, Transport in Nanostructures, Cambridge University Press, Cambridge, 1997; G. Speyer, R. Akis, D.K. Ferry, Physica E 19 (2003) 145. [3] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [4] H.X. He, X.L. Li, N.J. Tao, Phys. Rev. B 68 (2003) 045302. [5] R. Landauer, Phys. Lett. 85 A (1986) 91. [6] C. Joachim, J.F. Vinuesa, Europhys. Lett. 33 (1996) 635. [7] D.W.L. Sprung, J.D. Sigetich, H. Wu, J. Martorell, Amer. J. Phys. 68 (2000) 715. [8] M.H. Tyc, W. Salejda, Physica A 303 (2002) 493. [9] C. Pacher, E. Gornik, Physica E 21 (2004) 783. [10] J. Heinrichs, J. Phys.: Condens. Mat. 12 (2000) 5565. [11] A.F. Sadreev, I. Rotter, J. Phys. A: Math. Gen. 36 (2003) 11413. [12] S. Chattopadhyah, A. Chakrabarti, J. Phys.: Condens. Mat. 16 (2004) 313. [13] C. Caroli, C.R. Combescot, P. Nozieres, D. Saint-James, J. Phys. C 4 (1971) 916. [14] D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1981) 6851. [15] P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47 (1981) 882. [16] H. Pastawski, Phys. Rev. B 44 (1991) 6329. [17] Y. Xue, M.A. Ratner, Phys. Rev. B 68 (2003) 115406 and references there in. [18] A. Opniko, Y. Klymenko, L. Malysheva, Phys. Rev. B 62 (2003) 10482. [19] G. Cuniberti, G. Fagas, K. Richter, Chem. Phys. 281 (2002) 465. [20] M. Mardaani, K. Esfarjani, Physica E 25 (2004) 119; M. Mardaani, K. Esfarjani, Physica E 27 (2005) 227.