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Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads
Accepted Manuscript Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads Bharat Bhushan Brogi, Shyam Chand, P.K. Ahluwalia PII:
S0749-6036(18)31661-6
DOI:
10.1016/j.spmi.2018.08.013
Reference:
YSPMI 5858
To appear in:
Superlattices and Microstructures
Please cite this article as: B.B. Brogi, S. Chand, P.K. Ahluwalia, Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads, Superlattices and Microstructures (2018), doi: 10.1016/j.spmi.2018.08.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads Bharat Bhushan Brogi a∗, Shyam Chandb , P. K. Ahluwaliaa
bUniversity
Department, Himachal Pradesh University, Shimla, Himachal Pradesh, India 171005
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aPhysics
Institute of Information Technology, Himachal Pradesh University, Shimla, Himachal Pradesh, India 171005
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Abstract
Electronic transport through asymmetrically parallel coupled quantum dot system hybridized between conventional
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superconducting leads has been investigated theoretically in the Coulomb blockade regime by using Non-Equilibrium Green Function formalism. For resonant Cooper pair tunneling, the Andreev transmission probability and Josephson supercurrent ( − characteristics) have been presented for each configuration of coupled quantum dot system and
for transition of coupled quantum dot system from series to symmetric parallel configuration. Superconducting order parameter regulates the Josephson Cooper pair tunneling in such a way that it initially enhances and then suppresses the resonant Cooper pair tunneling when system is tuned from series to symmetric parallel configuration.
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Furthermore, it has been found that the Fano effect, reported in asymmetric parallel configuration of CQD system, gets completely suppressed by Andreev tunneling for a particular value of superconducting order parameter and lead-dot coupling strength.
1. INTRODUCTION
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Keywords: Coupled quantum dot, Fano Effect, Josephson effect, Andreev reflection, supercurrent.
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Transport through non-superconducting nanostructures (such as quantum dots (QDs), nanowires, nanotubes, molecules etc.) hybridized between the normal-superconducting (N-S) leads, superconducting-ferromagnetic (S-F) leads and both the superconducting-superconducting (S-S) leads has been the subject of enormous research during past few years theoretically [1–5] and experimentally [6–10]. Electronic transport through these nanostructures is significantly affected when sandwiched between superconducting leads. In particular the transport properties of hybrid superconductor-quantum dot devices such as semiconductor QDs [3, 8, 11, 12], Carbon nanotubes QD [9, 13–15] and graphene QD [16] have been investigated during past few years. In these hybrid systems such as N-QDs-S, F-QDs-S, ∗Corresponding
Author: Research Fellow, Physics Department, Himachal Pradesh University, Shimla-171005, India, Ph. No. +91-9459749174 Email addresses: brogi−[email protected] ( Bharat Bhushan Brogi a), shyam−[email protected] (Shyam Chandb ), pk−[email protected] (P. K. Ahluwaliaa) Preprint submitted to Superlattice and Microstructures
January 31, 2017
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S-QDs-S, the tunneling is mediated by Andreev reflection process in which an electron is reflected at the interface as a hole and causes electrons near Fermi energy to bind into a Cooper pair. As a result a Cooper pair is injected into superconducting lead through discrete Andreev Bound States (ABS) near Fermi energy by virtue of proximity effect. The Cooper pair tunneling through hybrid-superconducting system known as Andreev tunneling constitutes
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current known as Josephson supercurrent. The current through these hybrid-superconducting systems is caused by both resonant single particle tunneling as well as by Josephson Cooper pair tunneling [7, 15, 17]. Contribution to the Josephson supercurrent by single particle tunneling or by Josephson Cooper pair tunneling through such a system can be tuned by varying the strength of Coulomb interaction, coupling strength, position of the dot levels and temperature
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[1, 12, 17]. The previous theoretical studies on such a hybrid system are mainly focused on Andreev tunneling through single-QD system [1, 2, 8, 12, 18, 19], double quantum dot (DQD) system such as parallel [20–23], T-shape DQD system [18, 24–29] and Aharonov Bohm (AB) interferometer [21, 30, 31] providing an interesting canvas to explore
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new physics in variety of ways.
Further, Fano effect, a manifestation of quantum interference phenomena, has been reported as an asymmetric line shape in the conductance spectrum of asymmetric parallel coupled quantum dot (CQD) system attached to normal [32–34] as well as N-S leads [20]. Fano effect in CQD system coupled to normal leads has also been explored during transition from series to symmetric parallel configuration [35–38] and by tuning the magnetic flux too [38–41]. Fano-type line shape in the conductance spectrum of DQD system has also been observed in T-shaped
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[5, 24, 25, 28] and parallel DQD [20, 42] configuration hybridized between N-S leads. But, most of the work done on hybrid superconductor-QD devices revolves around single and DQD systems hybridized as N-QDs-S, N-QDs-F and S-QDs-F and there are limited theoretical studies of CQD system coupled to S-S leads [23]. Recently Andreev tunneling through parallel DQD system symmetrically coupled to superconducting
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leads has been studied [23], with a focus on the role of inter-dot tunneling and superconducting order parameter on Andreev transmission probability. Further, transport through CQD systems in transition of configuration from series
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to symmetric parallel CQD system has also been studied for N-QDs-S system [20]. This particular transition helps to investigate Fano effect in a controlled manner. However, to the best of our knowledge, no theoretical study of Andreev transport with Fano effect have been reported in S-CQD-S hybrid system in transition from series to parallel configuration where an interesting interplay of Andreev tunneling and Fano effect is expected. Motivated by this, a theoretical investigation of Andreev transport and its effect on Fano resonance has been presented
for CQD system attached to superconducting leads (Fig.1), for different lead-dot coupling strength (Γ) and super-
conducting order parameter (∆). To model CQD system under investigation, two impurity Anderson Hamiltonian
has been generalized by replacing normal leads with superconducting leads. To calculate transmission probability and current, Keldysh Non-Equilibrium Green Function (NEGF) technique and Equation of Motion (EOM) method
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[36, 38] have been used. To truncate the endless chain of higher order Green Functions (GFs) appearing in EOM due to intra-dot Coulomb interaction, the simplest mean field approximation, valid above Kondo temperature, has been used.
Under these assumptions, transmission probability, − characteristics and Fano effect have been discussed for
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various configurations of CQD system during transition from series to symmetric parallel configuration both in the
absence and presence of ∆. Suppression of Fano effect appearing in asymmetric parallel CQD system has been
observed by tuning the values of ∆ and Γ.
Rest of the paper has been organized as follows: In section 2, model and theoretical formalism used has been
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introduced. In section 3, various formulae have been derived. Numerical results have been presented and discussed in Section 4. Finally, in Section 5, a brief summary of conclusions drawn from the present study are presented.
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2. Model and Formalism 2.1. Model Hamiltonian
The DQD system coupled to superconducting leads can be modeled by using two impurities Anderson model with an extra term of inter-dot tunneling corresponding to the electron hopping between the two dots. The dynamics of such a system can be governed by the Anderson impurity Hamiltonian as: = + +
(1)
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where, is the Hamiltonian for the superconducting leads, is the Hamiltonian for dots and is inter-dot
tunneling Hamiltonian. The first term in H can be described as:
= ∑ − ∑(Δ + . . )
(2)
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wherein, the first term is the kinetic energy of electrons and ( ) is electron creation (annihilation) operators in the Heisenberg picture, for left ( = ) and right ( = ) leads respectively. In the second term ∆ is
superconducting energy gap which represents the attractive interaction between electrons in superconducting leads
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responsible for the Cooper pair formation. The second term in is:
"! + ∑! #! $!↑ $!↓ = ∑! ! "!
(3)
where, ! is the energy of the discrete energy level on 'ℎ (' = 1, 2) dot. Here, only one energy level per dot is considered. The "! ("! ) are creation (annihilation) operators of electrons on the dots, #! is intra-dot Coulomb
interaction energy of electrons in 'ℎ dot for double occupancy with opposite spins and $! = "! "! is electron
occupation number operator. The third term in is:
= +∑! ! "! + ∑ , "- ". / + . .
(4)
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Here, the first term represents tunneling of electrons between DQD system and leads with ! as the coupling potentials
of left (right) barrier with the 'ℎ dot and the second term describes the inter-dot tunneling of electrons with coupling strength , .
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2.2. Theoretical Formalism The NEGF formalism [43] and EOM method have been used to study the electronic transport through CQD system coupled to superconducting leads. The EOM for Green Functions (GFs) generates an infinite chain of higher order GFs. To truncate these higher order GFs, the simplest Mean Field decoupling approximation [23] is adopted to break this hierarchy at some point and thereby following coupled equations have been obtained:
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+0 − -(.) −#-(.) < $-(.) >/ ≪ "-(.) , "- ≫ = 1(0) + , ≪ ".(-) , "- ≫ +-(.) ≪ , "- ≫
6 + -(.) ≪ 6 , "- ≫
(5)
(6) (6) +0 − (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ +≪ "- , "- ≫ +. ≪ ". , "- ≫
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(6)
(6) (6) +0 + (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ −≪ "- , "- ≫ −. ≪ ". , "- ≫
(7)
, "- ≫ = −, ≪ ".(-) , "- ≫ −-(.) ≪ , "- ≫ +0 + -(.) + #-(.) < $-(.) >/ ≪ "-(.) 6 −-(.) ≪ 6 , "- ≫
(8)
(6) (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ −≪ "- , "- ≫ −. ≪ ". , "- ≫ +0 + (6) / ≪ (6)
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(6) (6) +0 − (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ +≪ "- , "- ≫ +. ≪ ". , "- ≫
(9) (10)
where, the superconducting order parameter ∆ in the left (right) superconducting leads is ∆ (6) = Σ < (6) ,
(6) >.
Suppose,
= 6 = ,
∆ = ∆6 = ∆
and
using
the
notations
for
the
GFs
as
8--(..) = ≪ "-(.) , "-(.) ≫; , 8-.(.-) = ≪ "-(.) , ".(-) ≫; , the above GFs can be expressed as: 9 = 8--(..)
9(:)
<= <
9(:)
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9(:)
9 9 +>..(--) + '>.-(-.) /, 8-. = 8.= !6
A-. , @ = C + 'D, =
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where, @- = A-- +
.
. . C = (>-- >.. − >-. >.- − ?-+ ?-. ),
<= <
9(:)
(?-- − '?-. )
;EF ;GEF
, @H = I1 −
(11) ∆J
J) (;J EF
D = (>-- >.- + >.. >-. + 2?-- ?-. )
Here, >--(..) , >-.(.-) , ?11 and ?12 appearing in the Eq. (11) and Eq. (12) are >--(..) = IA-- L-(.) − >-.(.-) =
-
.
M=J N==(JJ) 6 O
+
M=P(JP) QJ
IA-- Γ--(..) + A-. L-(.) +
K and the factors X and Y are:
K
M=B N==(JJ) 6QJ
K 4
(12)
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?-- = IA-- , + ?-. =
-
.
O
−
MPP QJ
IA-- Γ-. − A-. , +
K M=B N=J 6QJ
K
with L-(.) = (0 − -(.) − #-(.) < $-(.) >). @ and various A R s in the above equations are
A-- = S- S. −
6 J M=B O
A-H = Γ-- Γ.. − Γ-. Γ.-
− ,. ,
A-. = S- Γ.. + S. Γ-- − , (Γ-. + Γ.- )
. . + S. Γ-− 2, Γ-- Γ-. , A-T = S- Γ-.
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@ = 2@H (0 + ) ,
. . A.T = S- Γ.. + S. Γ.− 2, Γ.- Γ..
ATT = S- Γ.. Γ-. + S. Γ-- Γ.- − , (Γ-- Γ.. + Γ-. Γ.- )
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where S-(.) = (0 + -(.) + #-(.) < $-(.) >).
6 Here, various lead-dot couplings in the above equations are defined as Γ!U = Γ!U + Γ!U , which are assumed spin
be expressed as Γ!U
(6)
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independent for present study. According to Wide-Band limit [44, 45] the coupling strength between lead and dot can (6) (6) (6) 9 = ∑(V) (V)!(U) (V)!(U) W(V) = 2X ∑ (6) (V) (V) (V) Y(0 − (V) ). (6)
(6)
(6)
As, CQD system can be tuned by adjusting the left-right asymmetry which transforms the system from the series to 6 symmetric parallel configuration, we introduce Z- and Z. as the coupling parameters defined as Z- = Γ-= Γ.. ,
6 6 6 Z. = Γ.. = Γ-and Γ-. = Γ-. = Γ.= Γ.= √Z- Z. , such that , for Z. = 0, QDs are in series configuration, for Z. ≠ 0, Z- , are
in asymmetric parallel configuration and for Z. = Z- , they are in symmetric parallel configuration. If Z. = ]Z- ; then, s
can be treated as configuration parameter, such that, change in the value of s from 0 to 1 allows one to tune the
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system from series configuration (s = 0) to symmetric parallel configuration (] = 1). For 0 < ] < 1, QDs are in
asymmetric parallel configuration of CQD system.
Now, Landaur-Buttiker current formula, valid in non-interacting case, can also be applied for interacting case by
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taking care of Coulomb correlations via Mean Field approximation and neglecting spin flip process, inelastic processes and multiple electron scatterings [38]. Further, making use of NEGF formalism and Meir-Wingreen ^
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techniques, the expression for current can be derived to a well-known Landaur-Buttiker current formula, written as: ^
= ∑ `(a − a6 )bc(8 : Γ 6 8 9 Γ )d0 = ∑ `(a − a6 )b(0)d0 (13) _ _ . where, b(0) = bc(8 : Γ 6 8 9 Γ ) is called transmission coefficient and a (6) are Fermi distribution functions for left 8 (right) leads. By using matrices 8 9(:) = e -9(:) 8.-
9(:)
8-.
Γ f and Γ (6) = e -9(:) (6) 8.. Γ.9(:)
(6)
Γ-.
(6)
Γ..
(6)
f, the final form of b(0) can be
written as: b(0) =
|<= |J |<|J
. . . . ) . . [Z- Z. (>-+ >.. + >-. + >.+ (Z- + Z. ). (?-+ ?-. ) + 2(Z- + Z. )√Z- Z. {(>-- + >.. )?-- −
(>-. + >.- )?-. )} + 2Z- Z. (>-- >.. + >-. >.- )]
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ACCEPTED MANUSCRIPT 3. Results and Discussion For the numerical calculation of Andreev transmission probability (b (0)) and Josephson supercurrent ( −
characteristics) we take identical quantum dots such that, - = . = = 0.4U, #1 = #2 = # and the average
value of occupation number operator i.e. < $! > =< $!, > = 0.5. The values of all other parameters are scaled in term of U. The transport properties such as Andreev transmission probability and Josephson supercurrent have been
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calculated for the transition of the system from series configuration to symmetrical parallel configuration. The
numerical results for b (0) and − characteristics have been presented with the variation of ∆ as well as
configuration parameter s. 3.1. Effect of different values of ∆ for various configurations.
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For various configurations of CQD system such as series, asymmetric and symmetric parallel configuration the
transport properties have been presented for different values of superconducting order parameter, ∆. For ∆ = 0 (Normal case), Fig. 2 presents the transmission probability spectrum and I − V characteristics. We obtained single
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resonant peak for b (0) spectrum and stair-case like structure with single current step for − characteristics for
series configuration shown in insets of Fig. 2. Single resonant peak in b (0) spectrum and stair-case like structure
with single current step for − characteristics sustain for all other configurations of CQD system [Fig. 2] except
the shape of peak in b (0) spectrum which depends upon the geometry of CQD system. For asymmetric parallel
configuration we obtained the Fano type peak around 0 ≃ 2 and of Lorentzian type for symmetric parallel configuration [Fig. 2(a)]. Similar results for different configurations of CQD system attached to normal leads have
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also been reported in previous theoretical work [36, 38]. For ∆ = 0.5#, a resonant two peak structure evolves around
the Fermi energy level in b (0) n] 0 curve [Fig. 3(a)]. The similar observations were obtained in some of the previous works [20, 23] for parallel and T shaped DQD system coupled to superconducting leads. Appearance of
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two peak structure around Fermi energy level is clear signatures of tunneling of Cooper pair through Andreev Bound States by virtue of proximity effect, a manifestation of Josephson effect. Besides two resonant peaks around Fermi
energy level in b (0) spectrum of CQD system, a single resonant peak around 0 ≃ 2 is also observed for
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∆ = 0.5# having Fano type peak structure for 0 < ] < 1 as shown in inset II of Fig. 3(a). Hence our theoretical
investigation reveals both Josephson effect and Fano effect in asymmetric parallel configuration of CQD system hybridized between superconducting leads. The Cooper pair tunneling through hybrid-superconducting system known as Andreev tunneling constitutes supercurrent. In addition to Cooper pair tunneling, single electron tunneling also contributes for the current. The −
characteristics of system show stair-case like structure having two steps one very close to Fermi energy level at ≃ ∆ [Fig. 3(b) (Inset II)] due to Cooper pair tunneling and second step arises around + # due to single particle tunneling [Fig. 3(b)]. Here, this two peak structure around the Fermi energy level and stair case like structure with two steps exists for all configurations of CQD system [Fig. 3]. 6
ACCEPTED MANUSCRIPT Now, for further increase in ∆ i.e. ∆ = # [Fig. 4], two peak structure shifts farther away from the Fermi energy
level with decrease in height of the peaks in b (0) spectrum as well − steps compared to ∆ = 0 and 0.5# for each configuration of CQD system. Decrease in height of the peak in transmission probability spectrum and the step height in − curve signifies the reduction of Cooper pair tunneling probability and hence the supercurrent
because of increase in energy difference between Cooper pair and dot levels ( ).
Hence for ∆ ≠ 0, position as well as height of resonant peaks around Fermi energy level appearing for each
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configuration of CQD system depends upon superconducting order parameter ∆. With increase in ∆ resonant peaks around Fermi energy level shifts farther away from Fermi energy level and height of the peaks initially increase and
then decrease. Change in the height of the peaks signify that superconducting order parameter regulates the Josephson Cooper pair tunneling in such a way that it initially enhances and then suppresses the resonant Cooper pair
tunneling. Besides the reduction of the Cooper pair tunneling, increase in ∆ suppresses the single particle tunneling
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completely such that single peak appearing around 0 ≃ 2 in transmission probability spectrum disappears completely [Fig.4(a)] and hence there is only a single step in − curve [Fig. 4(b)] for ∆ = #.
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Furthermore, variation of s from 0 to 1 tunes configurations of CQD system from series to symmetric parallel configuration. The effect of variation of configuration parameter s for CQD system at ∆ = 0 (normal case) on b(0)
and − characteristics have also been presented in Fig. 2. In the absence of superconducting leads (∆ = 0), change
in configuration of CQD system from series to symmetric parallel shows Lorentzian type peaks for
b (0) n] 0 curve [Fig. 2(a)] and stair case like structure for − characteristics [Fig. 2(b)]. Clearly there is
successive increase in height of peaks and current steps with change in s from 0 to 1. This is similar to the results
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reported in previous theoretical works [36, 38]. But, for finite value of superconducting order parameter (∆ =
0.5#), a resonant two peak structure evolving around the Fermi energy level [Fig. 3(a)], exists for all
configurations of CQD system and for increase in s from 0 to 1, peaks continue to move farther away from Fermi
energy level accompanied by successive increase in width and height of the peak [Fig. 3(a)]. While for ∆ = #, for
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change in s from 0 to 1 peaks do not move farther away from the Fermi energy level but superimpose over each other with successive increase in width and height of the peak .
As we know that in series case (] = 0) there is only one channel for electron transmission and current is minimum.
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But for ] > 0, the number of transmission channels increases and hence the − step-height increases such that current is minimum in series and maximum for symmetric parallel configuration for all values of ∆ including ∆ = 0
[Fig. 2(b) - Fig. 4(b)]. So presence of ∆ does not make any difference quantitatively on the role of the configuration parameter s which enhances the height of the peak in b (0) and step heights of − characteristics.
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For 0 < ] < 1, appearance of asymmetric line shape in b (0) spectrum is the clear signature of Fano effect in
asymmetric parallel CQD system, which has been reported in many of the previous works on CQD system attached
to normal leads [32–34]. In the present work, this corresponds to the case for ∆ = 0 where b (0) peaks clearly
depict the existence of Fano effect [Fig. 2(a)]. Furthermore, present study also shows that the Fano effect survives
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for asymmetric parallel CQD system with superconducting leads, a new theoretical finding [Fig. 3(a)]. Hence it will be interesting to investigate an interplay between Fano effect and Andreev tunneling in case of asymmetric parallel
CQD system with superconducting leads by tuning ∆ and Γ. We consider an asymmetric parallel configuration with ] = 0.5, and note that b (0) peaks correspond to single particle tunneling with Fano resonance [Fig. 5]. Fig. 5 also
shows an interesting influence of ∆ on Fano effect. Clearly Fano peaks show successive decrease in height with
increase in ∆ upto ∆ = 0.7#, after which Fano peaks disappear completely and become Lorentzian type only [Fig.
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5 (Inset (a))] which also disappears completely for further increase in ∆ i.e. for ∆ = # [Fig. 5 (Inset (b))].
Further consider a case when ∆ = 0.2#, where both Cooper pair and single particle tunneling exist in b (0)
spectrum [Fig. 6]. The peak around 0 ≃ 2 is of Fano type for small values of Γ [Fig. 6 (Inset (a))]. If we increase
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the value of Γ, thereby increasing the tunneling rate, then it is found that Fano peaks disappear for Γ = 0.3# [Fig. 6
(Inset (a))]. This signifies that only Cooper pair tunneling is dominating and there is complete suppression of Fano
effect. The variation of Γ shows that Fano peak sustains only upto Γ = 0.3# and thereafter Fano peaks disappears
completely for higher value of Γ [Fig. 6 (Inset (b-e))]. So both ∆ and Γ shows there influence over Fano effect which
4. Conclusions
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gets suppressed with increase in ∆ and Γ.
Using NEGF formalism and EOM method, Andreev transmission probability and Josephson supercurrent through hybrid superconductor CQD system have been studied while carrying a system from series to symmetric parallel configuration. Present work showed that in such a hybrid system the electron transport is mediated by Cooper pair
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tunneling, a manifestation of Josephson effect showing resonant two peak structure around Fermi energy level in b (0) spectrum and also by single particle tunneling for which b (0) n] 0 peaks show Fano resonance in asymmetric
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parallel configuration. The conclusions of the study are summarized below: • In the absence of superconducting leads (∆ = 0), transport properties of CQD system during transition of system from series to symmetric parallel configuration reveals the similar results reported in previous theoretical work.
• In the presence of ∆, the pair of resonant peaks around Fermi energy level in transmission probability
spectrum and stair-case like structure for − characteristics remain intact for each configuration during transition of the configuration of CQD system from series to symmetric parallel configuration.
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• Superconducting order parameter (∆), regulates the transport in such a way that it initially enhances and then suppresses both, the resonant Cooper pair tunneling as well as the single particle tunneling when system is tuned from series to symmetric parallel configuration. • Presence of ∆ does not intervene to the role of configuration parameter s which enhances the height of the peak
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in b (0) and step heights of − characteristics when tuned from ] = 0 to ] = 1 and are therefore independent of each other.
• Fano effect reported in asymmetric parallel configuration of S-CQD-S system is being regulated by Andreev tunneling. Andreev tunneling shows its dominance over Fano effect in such a way that it suppresses the Fano
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effect which ultimately gets completely suppressed.
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J. Baranski and T. Domanski, Acta Physica Polonica A 121 (2012) 1213. J. Baranski and T. Domanski, Phys. Rev. B 85 (2012) 205451. [25] A. M. Calle, M. Pacheco, P. A. Orellana, Physics Letters A 377 (2013) 1474-1478. [26] Yu-Xian Li, Han-Yong Choi, Hyun-Woo Lee and Jian-Jun Liu, Journal of Applied Physics 101 (2007) 103918. [27] Shu-guang Cheng and Qing-feng Sun, J. Phys.: Condens. Matter 20 (2008) 505202. [28] Gao Wen-Zhu, Gong Wei Jiang, Zheng Yi-Song, Liu Yu and Lu Tian Quan, Commun.Theor. Phys. 49 (2008) 771-776. [29] Long Bai, Yang Jin Wu and Baigeng Wang, Phys. Status Solidi B 247 (2010) 335-341. [30] Ju Peng, Baigeng Wang and D. Y. Xing, Phys. Rev. B 71 (2005) 214523. [31] Anatoly Golub and Yshai Avishai, Phys. Rev. B 69 (2004) 165325. [32] K. Kobayashi, H. Aikawa, S. Katusmoto, and Y. Iye, Phys. Rev. Lett. 88 (2002) 256802. [33] Kensuke Kobayashi, Hisashi Aikawa, Shingo Katsumoto, and Yasuhiro Iye, Phys. Rev. B 68 (2003) 235304.
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[34] A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. c. Gossard, Phys. Rev. Lett. 93 (2004) 106803. [35] M. L. Ladron de Guevara, F. Carlo, P. A. Orellana, Phys. Rev. B 67 (2003) 195335. [36] Shyam Chand, R. K. Moudgil, P. K. Ahluwalia, Physica B 405 (2010) 239-246. [37] Gagan Rajput, S. Chand, P. K. Ahluwalia, K. C. Sharma, Physica B 405 (2010) 4323-4329.
[39] D. Sztenkiel and R. Swirkowicz, J. Phys.: Condens Matter 19 (2007) 176202. [40] Haizhou Lu, Rong Lu, and Bang-Fen Zhu, Phys. Rev. B 71 (2005) 235320. Haizhou Lu, Rong Lu, and Bang-Fen Zhu, J. Phys: Condens. Matter 18 (2006) 8961-8972.
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[38] Bharat B. Brogi, Shyam Chand and P. K. Ahluwalia, Physica B 461 (2015) 110-117.
[41] Haizhou Lu, Zuo-Zi Chen, Rong Lu, and Bang-Fen Zhu, Phys. stat. sol. (c) 4 No. 2, (2007) 574-577. [42] Yoichi Tanaka, Norio Kawakami and Akira Oguri, Phys. Rev. B 78 (2008) 035444.
[44] A. P. Jouho, N. S. Wingreen, Y. Meir, Phys. Rev. B 50 (1994) 5528. [45] Y. Meir, N. S. Wingreen, Phys. Rev. Lett. 68 (1992) 2512. N. S. Wingreen, Y. Meir, Phys. Rev. B 49 (1994) 11040.
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[46] Piotr Trocha and Jozef Barnas, Phys. Rev. B 76 (2007) 165432.
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[43] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515. L. V. Keldysh, Zh. Soviet. Phys. JETP 20 (1965) 1018.
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Figure 1: Schematic setup of tunneling asymmetric parallel CQD system coupled to conventional superconducting leads. Here d is inter-dot
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coupling strength, Z-(.) are coupling strength of dots with the leads.
Figure 2: For ∆ = 0, effect of variation of configuration parameter s on b n] 0 and − characteristics with parameters = 0.4#, d = 0.1#, Γ = 0.1# and Ab = 0.025#. Inset showing the variation of b n] 0 and − characteristics for series configuration (] = 0).
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ERROR: ioerror OFFENDING COMMAND: image
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STACK:
S0749-6036(18)31661-6
DOI:
10.1016/j.spmi.2018.08.013
Reference:
YSPMI 5858
To appear in:
Superlattices and Microstructures
Please cite this article as: B.B. Brogi, S. Chand, P.K. Ahluwalia, Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads, Superlattices and Microstructures (2018), doi: 10.1016/j.spmi.2018.08.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Andreev tunneling assisted suppression of Fano resonance in asymmetric parallel quantum dot system coupled to superconducting leads Bharat Bhushan Brogi a∗, Shyam Chandb , P. K. Ahluwaliaa
bUniversity
Department, Himachal Pradesh University, Shimla, Himachal Pradesh, India 171005
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aPhysics
Institute of Information Technology, Himachal Pradesh University, Shimla, Himachal Pradesh, India 171005
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Abstract
Electronic transport through asymmetrically parallel coupled quantum dot system hybridized between conventional
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superconducting leads has been investigated theoretically in the Coulomb blockade regime by using Non-Equilibrium Green Function formalism. For resonant Cooper pair tunneling, the Andreev transmission probability and Josephson supercurrent ( − characteristics) have been presented for each configuration of coupled quantum dot system and
for transition of coupled quantum dot system from series to symmetric parallel configuration. Superconducting order parameter regulates the Josephson Cooper pair tunneling in such a way that it initially enhances and then suppresses the resonant Cooper pair tunneling when system is tuned from series to symmetric parallel configuration.
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Furthermore, it has been found that the Fano effect, reported in asymmetric parallel configuration of CQD system, gets completely suppressed by Andreev tunneling for a particular value of superconducting order parameter and lead-dot coupling strength.
1. INTRODUCTION
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Keywords: Coupled quantum dot, Fano Effect, Josephson effect, Andreev reflection, supercurrent.
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Transport through non-superconducting nanostructures (such as quantum dots (QDs), nanowires, nanotubes, molecules etc.) hybridized between the normal-superconducting (N-S) leads, superconducting-ferromagnetic (S-F) leads and both the superconducting-superconducting (S-S) leads has been the subject of enormous research during past few years theoretically [1–5] and experimentally [6–10]. Electronic transport through these nanostructures is significantly affected when sandwiched between superconducting leads. In particular the transport properties of hybrid superconductor-quantum dot devices such as semiconductor QDs [3, 8, 11, 12], Carbon nanotubes QD [9, 13–15] and graphene QD [16] have been investigated during past few years. In these hybrid systems such as N-QDs-S, F-QDs-S, ∗Corresponding
Author: Research Fellow, Physics Department, Himachal Pradesh University, Shimla-171005, India, Ph. No. +91-9459749174 Email addresses: brogi−[email protected] ( Bharat Bhushan Brogi a), shyam−[email protected] (Shyam Chandb ), pk−[email protected] (P. K. Ahluwaliaa) Preprint submitted to Superlattice and Microstructures
January 31, 2017
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S-QDs-S, the tunneling is mediated by Andreev reflection process in which an electron is reflected at the interface as a hole and causes electrons near Fermi energy to bind into a Cooper pair. As a result a Cooper pair is injected into superconducting lead through discrete Andreev Bound States (ABS) near Fermi energy by virtue of proximity effect. The Cooper pair tunneling through hybrid-superconducting system known as Andreev tunneling constitutes
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current known as Josephson supercurrent. The current through these hybrid-superconducting systems is caused by both resonant single particle tunneling as well as by Josephson Cooper pair tunneling [7, 15, 17]. Contribution to the Josephson supercurrent by single particle tunneling or by Josephson Cooper pair tunneling through such a system can be tuned by varying the strength of Coulomb interaction, coupling strength, position of the dot levels and temperature
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[1, 12, 17]. The previous theoretical studies on such a hybrid system are mainly focused on Andreev tunneling through single-QD system [1, 2, 8, 12, 18, 19], double quantum dot (DQD) system such as parallel [20–23], T-shape DQD system [18, 24–29] and Aharonov Bohm (AB) interferometer [21, 30, 31] providing an interesting canvas to explore
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new physics in variety of ways.
Further, Fano effect, a manifestation of quantum interference phenomena, has been reported as an asymmetric line shape in the conductance spectrum of asymmetric parallel coupled quantum dot (CQD) system attached to normal [32–34] as well as N-S leads [20]. Fano effect in CQD system coupled to normal leads has also been explored during transition from series to symmetric parallel configuration [35–38] and by tuning the magnetic flux too [38–41]. Fano-type line shape in the conductance spectrum of DQD system has also been observed in T-shaped
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[5, 24, 25, 28] and parallel DQD [20, 42] configuration hybridized between N-S leads. But, most of the work done on hybrid superconductor-QD devices revolves around single and DQD systems hybridized as N-QDs-S, N-QDs-F and S-QDs-F and there are limited theoretical studies of CQD system coupled to S-S leads [23]. Recently Andreev tunneling through parallel DQD system symmetrically coupled to superconducting
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leads has been studied [23], with a focus on the role of inter-dot tunneling and superconducting order parameter on Andreev transmission probability. Further, transport through CQD systems in transition of configuration from series
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to symmetric parallel CQD system has also been studied for N-QDs-S system [20]. This particular transition helps to investigate Fano effect in a controlled manner. However, to the best of our knowledge, no theoretical study of Andreev transport with Fano effect have been reported in S-CQD-S hybrid system in transition from series to parallel configuration where an interesting interplay of Andreev tunneling and Fano effect is expected. Motivated by this, a theoretical investigation of Andreev transport and its effect on Fano resonance has been presented
for CQD system attached to superconducting leads (Fig.1), for different lead-dot coupling strength (Γ) and super-
conducting order parameter (∆). To model CQD system under investigation, two impurity Anderson Hamiltonian
has been generalized by replacing normal leads with superconducting leads. To calculate transmission probability and current, Keldysh Non-Equilibrium Green Function (NEGF) technique and Equation of Motion (EOM) method
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[36, 38] have been used. To truncate the endless chain of higher order Green Functions (GFs) appearing in EOM due to intra-dot Coulomb interaction, the simplest mean field approximation, valid above Kondo temperature, has been used.
Under these assumptions, transmission probability, − characteristics and Fano effect have been discussed for
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various configurations of CQD system during transition from series to symmetric parallel configuration both in the
absence and presence of ∆. Suppression of Fano effect appearing in asymmetric parallel CQD system has been
observed by tuning the values of ∆ and Γ.
Rest of the paper has been organized as follows: In section 2, model and theoretical formalism used has been
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introduced. In section 3, various formulae have been derived. Numerical results have been presented and discussed in Section 4. Finally, in Section 5, a brief summary of conclusions drawn from the present study are presented.
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2. Model and Formalism 2.1. Model Hamiltonian
The DQD system coupled to superconducting leads can be modeled by using two impurities Anderson model with an extra term of inter-dot tunneling corresponding to the electron hopping between the two dots. The dynamics of such a system can be governed by the Anderson impurity Hamiltonian as: = + +
(1)
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where, is the Hamiltonian for the superconducting leads, is the Hamiltonian for dots and is inter-dot
tunneling Hamiltonian. The first term in H can be described as:
= ∑ − ∑(Δ + . . )
(2)
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wherein, the first term is the kinetic energy of electrons and ( ) is electron creation (annihilation) operators in the Heisenberg picture, for left ( = ) and right ( = ) leads respectively. In the second term ∆ is
superconducting energy gap which represents the attractive interaction between electrons in superconducting leads
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responsible for the Cooper pair formation. The second term in is:
"! + ∑! #! $!↑ $!↓ = ∑! ! "!
(3)
where, ! is the energy of the discrete energy level on 'ℎ (' = 1, 2) dot. Here, only one energy level per dot is considered. The "! ("! ) are creation (annihilation) operators of electrons on the dots, #! is intra-dot Coulomb
interaction energy of electrons in 'ℎ dot for double occupancy with opposite spins and $! = "! "! is electron
occupation number operator. The third term in is:
= +∑! ! "! + ∑ , "- ". / + . .
(4)
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Here, the first term represents tunneling of electrons between DQD system and leads with ! as the coupling potentials
of left (right) barrier with the 'ℎ dot and the second term describes the inter-dot tunneling of electrons with coupling strength , .
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2.2. Theoretical Formalism The NEGF formalism [43] and EOM method have been used to study the electronic transport through CQD system coupled to superconducting leads. The EOM for Green Functions (GFs) generates an infinite chain of higher order GFs. To truncate these higher order GFs, the simplest Mean Field decoupling approximation [23] is adopted to break this hierarchy at some point and thereby following coupled equations have been obtained:
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+0 − -(.) −#-(.) < $-(.) >/ ≪ "-(.) , "- ≫ = 1(0) + , ≪ ".(-) , "- ≫ +-(.) ≪ , "- ≫
6 + -(.) ≪ 6 , "- ≫
(5)
(6) (6) +0 − (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ +≪ "- , "- ≫ +. ≪ ". , "- ≫
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(6)
(6) (6) +0 + (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ −≪ "- , "- ≫ −. ≪ ". , "- ≫
(7)
, "- ≫ = −, ≪ ".(-) , "- ≫ −-(.) ≪ , "- ≫ +0 + -(.) + #-(.) < $-(.) >/ ≪ "-(.) 6 −-(.) ≪ 6 , "- ≫
(8)
(6) (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ −≪ "- , "- ≫ −. ≪ ". , "- ≫ +0 + (6) / ≪ (6)
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(6) (6) +0 − (6) / ≪ (6) , "- ≫ = −∆ (6) ≪ (6) , "- ≫ +≪ "- , "- ≫ +. ≪ ". , "- ≫
(9) (10)
where, the superconducting order parameter ∆ in the left (right) superconducting leads is ∆ (6) = Σ < (6) ,
(6) >.
Suppose,
= 6 = ,
∆ = ∆6 = ∆
and
using
the
notations
for
the
GFs
as
8--(..) = ≪ "-(.) , "-(.) ≫; , 8-.(.-) = ≪ "-(.) , ".(-) ≫; , the above GFs can be expressed as: 9 = 8--(..)
9(:)
<= <
9(:)
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9(:)
9 9 +>..(--) + '>.-(-.) /, 8-. = 8.= !6
A-. , @ = C + 'D, =
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where, @- = A-- +
.
. . C = (>-- >.. − >-. >.- − ?-+ ?-. ),
<= <
9(:)
(?-- − '?-. )
;EF ;GEF
, @H = I1 −
(11) ∆J
J) (;J EF
D = (>-- >.- + >.. >-. + 2?-- ?-. )
Here, >--(..) , >-.(.-) , ?11 and ?12 appearing in the Eq. (11) and Eq. (12) are >--(..) = IA-- L-(.) − >-.(.-) =
-
.
M=J N==(JJ) 6 O
+
M=P(JP) QJ
IA-- Γ--(..) + A-. L-(.) +
K and the factors X and Y are:
K
M=B N==(JJ) 6QJ
K 4
(12)
ACCEPTED MANUSCRIPT M=J N=J 6
?-- = IA-- , + ?-. =
-
.
O
−
MPP QJ
IA-- Γ-. − A-. , +
K M=B N=J 6QJ
K
with L-(.) = (0 − -(.) − #-(.) < $-(.) >). @ and various A R s in the above equations are
A-- = S- S. −
6 J M=B O
A-H = Γ-- Γ.. − Γ-. Γ.-
− ,. ,
A-. = S- Γ.. + S. Γ-- − , (Γ-. + Γ.- )
. . + S. Γ-− 2, Γ-- Γ-. , A-T = S- Γ-.
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@ = 2@H (0 + ) ,
. . A.T = S- Γ.. + S. Γ.− 2, Γ.- Γ..
ATT = S- Γ.. Γ-. + S. Γ-- Γ.- − , (Γ-- Γ.. + Γ-. Γ.- )
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where S-(.) = (0 + -(.) + #-(.) < $-(.) >).
6 Here, various lead-dot couplings in the above equations are defined as Γ!U = Γ!U + Γ!U , which are assumed spin
be expressed as Γ!U
(6)
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independent for present study. According to Wide-Band limit [44, 45] the coupling strength between lead and dot can (6) (6) (6) 9 = ∑(V) (V)!(U) (V)!(U) W(V) = 2X ∑ (6) (V) (V) (V) Y(0 − (V) ). (6)
(6)
(6)
As, CQD system can be tuned by adjusting the left-right asymmetry which transforms the system from the series to 6 symmetric parallel configuration, we introduce Z- and Z. as the coupling parameters defined as Z- = Γ-= Γ.. ,
6 6 6 Z. = Γ.. = Γ-and Γ-. = Γ-. = Γ.= Γ.= √Z- Z. , such that , for Z. = 0, QDs are in series configuration, for Z. ≠ 0, Z- , are
in asymmetric parallel configuration and for Z. = Z- , they are in symmetric parallel configuration. If Z. = ]Z- ; then, s
can be treated as configuration parameter, such that, change in the value of s from 0 to 1 allows one to tune the
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system from series configuration (s = 0) to symmetric parallel configuration (] = 1). For 0 < ] < 1, QDs are in
asymmetric parallel configuration of CQD system.
Now, Landaur-Buttiker current formula, valid in non-interacting case, can also be applied for interacting case by
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taking care of Coulomb correlations via Mean Field approximation and neglecting spin flip process, inelastic processes and multiple electron scatterings [38]. Further, making use of NEGF formalism and Meir-Wingreen ^
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techniques, the expression for current can be derived to a well-known Landaur-Buttiker current formula, written as: ^
= ∑ `(a − a6 )bc(8 : Γ 6 8 9 Γ )d0 = ∑ `(a − a6 )b(0)d0 (13) _ _ . where, b(0) = bc(8 : Γ 6 8 9 Γ ) is called transmission coefficient and a (6) are Fermi distribution functions for left 8 (right) leads. By using matrices 8 9(:) = e -9(:) 8.-
9(:)
8-.
Γ f and Γ (6) = e -9(:) (6) 8.. Γ.9(:)
(6)
Γ-.
(6)
Γ..
(6)
f, the final form of b(0) can be
written as: b(0) =
|<= |J |<|J
. . . . ) . . [Z- Z. (>-+ >.. + >-. + >.+ (Z- + Z. ). (?-+ ?-. ) + 2(Z- + Z. )√Z- Z. {(>-- + >.. )?-- −
(>-. + >.- )?-. )} + 2Z- Z. (>-- >.. + >-. >.- )]
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ACCEPTED MANUSCRIPT 3. Results and Discussion For the numerical calculation of Andreev transmission probability (b (0)) and Josephson supercurrent ( −
characteristics) we take identical quantum dots such that, - = . = = 0.4U, #1 = #2 = # and the average
value of occupation number operator i.e. < $! > =< $!, > = 0.5. The values of all other parameters are scaled in term of U. The transport properties such as Andreev transmission probability and Josephson supercurrent have been
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calculated for the transition of the system from series configuration to symmetrical parallel configuration. The
numerical results for b (0) and − characteristics have been presented with the variation of ∆ as well as
configuration parameter s. 3.1. Effect of different values of ∆ for various configurations.
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For various configurations of CQD system such as series, asymmetric and symmetric parallel configuration the
transport properties have been presented for different values of superconducting order parameter, ∆. For ∆ = 0 (Normal case), Fig. 2 presents the transmission probability spectrum and I − V characteristics. We obtained single
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resonant peak for b (0) spectrum and stair-case like structure with single current step for − characteristics for
series configuration shown in insets of Fig. 2. Single resonant peak in b (0) spectrum and stair-case like structure
with single current step for − characteristics sustain for all other configurations of CQD system [Fig. 2] except
the shape of peak in b (0) spectrum which depends upon the geometry of CQD system. For asymmetric parallel
configuration we obtained the Fano type peak around 0 ≃ 2 and of Lorentzian type for symmetric parallel configuration [Fig. 2(a)]. Similar results for different configurations of CQD system attached to normal leads have
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also been reported in previous theoretical work [36, 38]. For ∆ = 0.5#, a resonant two peak structure evolves around
the Fermi energy level in b (0) n] 0 curve [Fig. 3(a)]. The similar observations were obtained in some of the previous works [20, 23] for parallel and T shaped DQD system coupled to superconducting leads. Appearance of
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two peak structure around Fermi energy level is clear signatures of tunneling of Cooper pair through Andreev Bound States by virtue of proximity effect, a manifestation of Josephson effect. Besides two resonant peaks around Fermi
energy level in b (0) spectrum of CQD system, a single resonant peak around 0 ≃ 2 is also observed for
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∆ = 0.5# having Fano type peak structure for 0 < ] < 1 as shown in inset II of Fig. 3(a). Hence our theoretical
investigation reveals both Josephson effect and Fano effect in asymmetric parallel configuration of CQD system hybridized between superconducting leads. The Cooper pair tunneling through hybrid-superconducting system known as Andreev tunneling constitutes supercurrent. In addition to Cooper pair tunneling, single electron tunneling also contributes for the current. The −
characteristics of system show stair-case like structure having two steps one very close to Fermi energy level at ≃ ∆ [Fig. 3(b) (Inset II)] due to Cooper pair tunneling and second step arises around + # due to single particle tunneling [Fig. 3(b)]. Here, this two peak structure around the Fermi energy level and stair case like structure with two steps exists for all configurations of CQD system [Fig. 3]. 6
ACCEPTED MANUSCRIPT Now, for further increase in ∆ i.e. ∆ = # [Fig. 4], two peak structure shifts farther away from the Fermi energy
level with decrease in height of the peaks in b (0) spectrum as well − steps compared to ∆ = 0 and 0.5# for each configuration of CQD system. Decrease in height of the peak in transmission probability spectrum and the step height in − curve signifies the reduction of Cooper pair tunneling probability and hence the supercurrent
because of increase in energy difference between Cooper pair and dot levels ( ).
Hence for ∆ ≠ 0, position as well as height of resonant peaks around Fermi energy level appearing for each
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configuration of CQD system depends upon superconducting order parameter ∆. With increase in ∆ resonant peaks around Fermi energy level shifts farther away from Fermi energy level and height of the peaks initially increase and
then decrease. Change in the height of the peaks signify that superconducting order parameter regulates the Josephson Cooper pair tunneling in such a way that it initially enhances and then suppresses the resonant Cooper pair
tunneling. Besides the reduction of the Cooper pair tunneling, increase in ∆ suppresses the single particle tunneling
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completely such that single peak appearing around 0 ≃ 2 in transmission probability spectrum disappears completely [Fig.4(a)] and hence there is only a single step in − curve [Fig. 4(b)] for ∆ = #.
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Furthermore, variation of s from 0 to 1 tunes configurations of CQD system from series to symmetric parallel configuration. The effect of variation of configuration parameter s for CQD system at ∆ = 0 (normal case) on b(0)
and − characteristics have also been presented in Fig. 2. In the absence of superconducting leads (∆ = 0), change
in configuration of CQD system from series to symmetric parallel shows Lorentzian type peaks for
b (0) n] 0 curve [Fig. 2(a)] and stair case like structure for − characteristics [Fig. 2(b)]. Clearly there is
successive increase in height of peaks and current steps with change in s from 0 to 1. This is similar to the results
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reported in previous theoretical works [36, 38]. But, for finite value of superconducting order parameter (∆ =
0.5#), a resonant two peak structure evolving around the Fermi energy level [Fig. 3(a)], exists for all
configurations of CQD system and for increase in s from 0 to 1, peaks continue to move farther away from Fermi
energy level accompanied by successive increase in width and height of the peak [Fig. 3(a)]. While for ∆ = #, for
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change in s from 0 to 1 peaks do not move farther away from the Fermi energy level but superimpose over each other with successive increase in width and height of the peak .
As we know that in series case (] = 0) there is only one channel for electron transmission and current is minimum.
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But for ] > 0, the number of transmission channels increases and hence the − step-height increases such that current is minimum in series and maximum for symmetric parallel configuration for all values of ∆ including ∆ = 0
[Fig. 2(b) - Fig. 4(b)]. So presence of ∆ does not make any difference quantitatively on the role of the configuration parameter s which enhances the height of the peak in b (0) and step heights of − characteristics.
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For 0 < ] < 1, appearance of asymmetric line shape in b (0) spectrum is the clear signature of Fano effect in
asymmetric parallel CQD system, which has been reported in many of the previous works on CQD system attached
to normal leads [32–34]. In the present work, this corresponds to the case for ∆ = 0 where b (0) peaks clearly
depict the existence of Fano effect [Fig. 2(a)]. Furthermore, present study also shows that the Fano effect survives
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for asymmetric parallel CQD system with superconducting leads, a new theoretical finding [Fig. 3(a)]. Hence it will be interesting to investigate an interplay between Fano effect and Andreev tunneling in case of asymmetric parallel
CQD system with superconducting leads by tuning ∆ and Γ. We consider an asymmetric parallel configuration with ] = 0.5, and note that b (0) peaks correspond to single particle tunneling with Fano resonance [Fig. 5]. Fig. 5 also
shows an interesting influence of ∆ on Fano effect. Clearly Fano peaks show successive decrease in height with
increase in ∆ upto ∆ = 0.7#, after which Fano peaks disappear completely and become Lorentzian type only [Fig.
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5 (Inset (a))] which also disappears completely for further increase in ∆ i.e. for ∆ = # [Fig. 5 (Inset (b))].
Further consider a case when ∆ = 0.2#, where both Cooper pair and single particle tunneling exist in b (0)
spectrum [Fig. 6]. The peak around 0 ≃ 2 is of Fano type for small values of Γ [Fig. 6 (Inset (a))]. If we increase
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the value of Γ, thereby increasing the tunneling rate, then it is found that Fano peaks disappear for Γ = 0.3# [Fig. 6
(Inset (a))]. This signifies that only Cooper pair tunneling is dominating and there is complete suppression of Fano
effect. The variation of Γ shows that Fano peak sustains only upto Γ = 0.3# and thereafter Fano peaks disappears
completely for higher value of Γ [Fig. 6 (Inset (b-e))]. So both ∆ and Γ shows there influence over Fano effect which
4. Conclusions
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gets suppressed with increase in ∆ and Γ.
Using NEGF formalism and EOM method, Andreev transmission probability and Josephson supercurrent through hybrid superconductor CQD system have been studied while carrying a system from series to symmetric parallel configuration. Present work showed that in such a hybrid system the electron transport is mediated by Cooper pair
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tunneling, a manifestation of Josephson effect showing resonant two peak structure around Fermi energy level in b (0) spectrum and also by single particle tunneling for which b (0) n] 0 peaks show Fano resonance in asymmetric
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parallel configuration. The conclusions of the study are summarized below: • In the absence of superconducting leads (∆ = 0), transport properties of CQD system during transition of system from series to symmetric parallel configuration reveals the similar results reported in previous theoretical work.
• In the presence of ∆, the pair of resonant peaks around Fermi energy level in transmission probability
spectrum and stair-case like structure for − characteristics remain intact for each configuration during transition of the configuration of CQD system from series to symmetric parallel configuration.
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• Superconducting order parameter (∆), regulates the transport in such a way that it initially enhances and then suppresses both, the resonant Cooper pair tunneling as well as the single particle tunneling when system is tuned from series to symmetric parallel configuration. • Presence of ∆ does not intervene to the role of configuration parameter s which enhances the height of the peak
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in b (0) and step heights of − characteristics when tuned from ] = 0 to ] = 1 and are therefore independent of each other.
• Fano effect reported in asymmetric parallel configuration of S-CQD-S system is being regulated by Andreev tunneling. Andreev tunneling shows its dominance over Fano effect in such a way that it suppresses the Fano
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effect which ultimately gets completely suppressed.
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Figure 1: Schematic setup of tunneling asymmetric parallel CQD system coupled to conventional superconducting leads. Here d is inter-dot
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coupling strength, Z-(.) are coupling strength of dots with the leads.
Figure 2: For ∆ = 0, effect of variation of configuration parameter s on b n] 0 and − characteristics with parameters = 0.4#, d = 0.1#, Γ = 0.1# and Ab = 0.025#. Inset showing the variation of b n] 0 and − characteristics for series configuration (] = 0).
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