- Email: [email protected]
Tunneling transport in d-wave superconductor-silicene junction
Accepted Manuscript Tunneling transport in d-wave superconductor-silicene junction Y. Hajati, S. Vosoughi nia, G. Rashedi
PII:
S0749-6036(16)31324-6
DOI:
10.1016/j.spmi.2016.11.067
Reference:
YSPMI 4702
To appear in:
Superlattices and Microstructures
Received Date: 25 October 2016 Revised Date:
29 November 2016
Accepted Date: 29 November 2016
Please cite this article as: Y. Hajati, S. Vosoughi nia, G. Rashedi, Tunneling transport in dwave superconductor-silicene junction, Superlattices and Microstructures (2017), doi: 10.1016/ j.spmi.2016.11.067. 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.
ACCEPTED MANUSCRIPT
Tunneling transport in d-wave superconductor-silicene junction Y. hajatib, S. Vosoughi niaa, G. Rashedia a
Department of Physics, Faculty of Science, University of Isfahan, Isfahan 81746-73441, Iran Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
RI PT
b
Abstract: We theoretically study the tunneling conductance of a normal/d-wave superconductor silicene junction using Blonder-Tinkham-Klapwijk (BTK) formalism. We discuss in detail how the conductances spectra are affected by inducing d-wave superconducting pairing symmetry in the
SC
buckled silicene. It is obtained that the amplitude of the spin/valley-dependent Andreev reflection and subgap conductance of the junction can be strongly modulated by the orientation angle of tune the transport properties of the junction through changing and EZ. We demonstrate that the
M AN U
superconductive gap () and perpendicular electric field (EZ), suggesting that one may experimentally
subgap conductance exhibits an oscillatory behavior as a function of the orientation angle of
superconductive gap () with a period of and by increasing the insulating gap of silicene, the charge
conductance oscillations suppress. Remarkably, due to the buckled structure of silicene at the maximum orientation angle of the d-wave superconducting = , we found a very distinct behavior
from the graphene-based NS junction where the charge conductance is insensitive to the bias energy.
electric field.
TE D
In addition, the Andreev reflection and subgap conductance can be switched on and off by applying
AC C
EP
Keywords: Silicene; Tunneling conductance; Superconducting pairing symmetry; Andreev reflection
1
ACCEPTED MANUSCRIPT
1. Introduction: Andreev reflection (AR) is a fundamental process in understanding quantum transport in a normal metal/superconductor (NS) junction.1 Based on Andreev theory, an incident electron with the energy smaller than the superconducting energy gap cannot enter into the superconductor. In the conventional
RI PT
materials an electron in the normal metal is retro-reflected at the interface of NS as a hole by transferring a cooper pair to the superconductor. Pure graphene2 is not supercoductor, but superconductivity can be induced in graphene layer by proximity effect.3-8 Beenakker obtained that in a graphene-based normal metal/superconductor (NS) junction when the Fermi energy in the normal region is much smaller than that of the superconducting energy gap, the specular Andreev reflection
SC
occurs which is different from the usual retro-reflection in a conventional NS junction.3,9
It is known that an anisotropic superconducting order parameter (unconventional superconductor) induced in graphene has a strong effect on quantum transport in graphene-based NS junction.7 In
M AN U
contrast to the s-wave superconductor cases, at the interface of a dx2-y2-wave superconductor, zero-bias conductance peaks (ZBCP) are formed due to the sign change of the pair potential.10-13 It is found that in graphene-based N/dx2-y2-wave superconductor the variation of rotation angle of the superconductive
gap has led to a progressive shift of the peak in the conductance without any formation of the
ZBCP except for = /4.7
Silicene is a single layer of silicon atoms arranged in a two-dimentsional honeycomb lattice with a
TE D
buckled sublattice structure which has been successfully synthesis on variety of substrates.14-17 Its low-energy band structure in the K and K' valleys is described by the Dirac theory as in graphene.18,19 Silicene has a large spin-orbit coupling compared to graphene and due to the low-buckled geometry, its energy gap can be further tuned by an external electric field Ez perpendicular to the silicene sheet.
EP
Tunabilty of the silicene energy gap leads to change its topological phases which is very important in the modern condensed matter physics.20-23 Recently, a possible conductive gap, about 35 meV, is observed in monolayer silicene epitaxially
AC C
grown on Ag(111) surface at temperature as high as 30-40 K via scanning tunneling microscope.24 Observation of Andreev reflection strongly indicates that this gap is a superconducting gap. It is also found that the pairing symmetry of superconductivity in silicene can be tuned by applying external electric field.25,26 To date, there is no experimental report about proximity induced superconductivity in silicene. However, currently Linder et al. studied the superconducting proximity effect silicene and calculated conductance across NS and NSN junctions.27 It is obtained that the local and non local Andreev reflection can be tuned by an external electric field owing to the buckled structure of silicene. In this letter, we study the proximity-induced unconventional superconductivity in the buckled structure silicene. Here, we have mainly focused how the anisotropic superconducting order parameter can affect the transmission coefficient and spin-valley conductance in the NS silicene 2
ACCEPTED MANUSCRIPT
junction. We model our NS setup within the scattering matrix theory. Similar setup has been studied in graphene, however the transport properties of silicene-based unconventional superconductivity is still unexplored to the best of our knowledge. 2. Model and Hamiltonian
RI PT
We consider a two-dimensional silicene-based normal/superconductor (NS) junction, as shown in Fig 1. The NS silicene junction can be described by the Dirac-Bogoliubov-de Gennes (DBdG) equation27
where,
= ℏ − − (!"#$ − %&' ) ' − &( ()), () = ∆(.)+() (2)
&( () = &(* +(−) + &(# +() ,
SC
() ∗ ψ = εψ, (1) () −
M AN U
Here = 5.5 × 104 m/s represents the Fermi velocity in silicene and ! = +1 (↑) or ! = −1 (↓) denotes the spin indices in the silicene and superconductor regions. = 1(−1) corresponds to the K
(K') valley point and , and ' are 2 × 2 Pauli matrices in sublattice pseudospin. "#$ = 3.9 9:; is
respond differently to an externally applied electric field &' . So, %&' is the on-site potential difference
the spin-orbit coupling term. Due to the buckled structure of the silicene, the atoms in two sublattices between A and B sublattices which can be tuned by an external electric field.20 For the dx2-y2-wave
pairing, the order parameter in graphene induced by a high-Tc cuprate ∆(.) = ∆ cos2(. − ) ,
TE D
where is the orientation angle of the superconductive gap with respect to normal as illustrated in Fig
1. The DBdG equation is derived within the mean-field approximation.3,9 So, the chemical potential in
the superconductor region, &(# , should be very large compared to all other energy scales in our
EP
calculations: &(# ≫ ∆ , "@A .
the normal silicene (N) region, the wavefunction ψB (x) for incident electron, reflected electron, and
AC C
By solving the DBdG equation (Eq.1), we obtain the wavefunctions in each region of the junction. In
Andreev reflected hole is given by
H : IJKL − H : OIJKL N H F G F M + G M ψ* ( = 0) = 0 0 D2E F D2E F 0 0 1
+
NP
0 0
G OIJ .ℎ M , (3) D2E O P : O
3
ACCEPTED MANUSCRIPT H P where NJ,R and NJ,R are normal and Andreev reflection amplitudes, .H is incident angle of the electron
and .P is the Andreev reflection angle of the hole in the N region with respect to the normal axis of
the interface. Hereafter, for brevity of notation we set ℏ = = 1. By setting the chemical potential
&(* = 0 (undoped silicene), wave vectors of electron and hole become identical; H = P = =
RI PT
DE − (%&' − !"#$ ) . We have also defined: ± = E ± (!"#$ − %&' ). From the Hamiltonian (Eq.
band gap (insulating gap) is &T = 2|%&' − !"#$ |=29JR , thus we have transport only if |E| > &T /2.
1) it can be seen that there are four bands in the energy spectrum. At the valley points (K and K') the As Ez increases, the band gap decreases linearly and finally closes at EX = EY = 17 meV/A. Note
also that the K' valley with spin up orientation ( = −1, ! = 1) and K valley with spin down
SC
orientation bands ( = 1, ! = −1) do not contribute to the subgap transport in our studied energy
range, since they are far away from Fermi level.27,28 It is worth mentioning that the magnitude of spin-
electric field EZ.
M AN U
orbit interaction in silicene is constant, so the only way to tune the insulating gap is changing the In the dx2-y2-wave superconductive silicene (S) region, the wave function for the transmission of DBdG quasiparticles with excitation energy ε is given by
−:−l.@ f(. O ):lg m _H c k _P c k f (. O ):lg ψ# (x = 0) = , (4) b: IJKd f(. F ): OIgh j + b −l. O @ −: e(. ) j √2 √2 F OIgh e(. O ) a f(. ): i a i : IJKd e(. F ) e(. F )
m
TE D
where _:,! and _ℎ,! are the transmission amplitudes of electron-like (ELQ) and hole-like (HLQ)
quasiparticles. .# is the incident angle of ELQ in the superconductive region and we have defined the parameters . F (. O ) = .# ( − .# ) and : Ig = ∆(. ± )/|∆(. ± )| . Because of the translational ±
EP
invariance in the y-direction, all transverse wave vectors during the scattering process are conserved
and the relationship among the incident angles in each region is obtained through the following component conservation relation,
AC C
H sin .H = P sin .P , H sin .H = p sin .p . (5)
Since we consider heavy doped superconductor ( &(# ≫ ℏ ), we set .# → 0 in the following calculations. The coherent factors read
DE − |∆(θ)| 1 1 DE − |∆(θ)| e(.) = r s1 + t , f(.) = r (1 − ) . (6) 2 E 2 E
4
ACCEPTED MANUSCRIPT
Andreev reflection Nℎ,! and normal reflection N:,! can be calculated by, ψv ( = 0) = ψ@ ( = 0), as Using the continuity of wavefunctions at the boundaries, The amplitude of spin/valley-dependent
follows =
[e(.) + f(.) :l(x
−
z
−x+ )
][:2l. − − + ]
with z = 2 {e(.) − f(.) :l(x
−
−x+ )
, Nℎ,!
=
4 e(.)f(.)cos.: z
:
l. OIg h
, (7)
RI PT
N:,!
|cos. + 2E{e(.) + f(.) :l(x
−
−x+ )
|. Using the Blonder-
Tinkham-Klapwijk (BTK) formalism,29 we define the normalized charge conductance (} ):
SC
/2 1 = . .(1 + − ), (8) * 4 ! O/
}
~
where v is the ballistic conductance of silicene. Here, R= N:,! and RA= Nℎ,! are normal and
M AN U
2
2
chosen to normalize all our energy scales by ∆ , so that all our results are in terms of dimensionless Andreev reflection probabilities, respectively. Note that to carry out our numerical analysis, we have
quantities. Also for spin-orbit coupling we set 3. Results and discussions:
d ∆
= 5 thoroughout the paper.
TE D
In this section, we present our numerical results for NS silicene junction for different parameter of the Dirac electrons on the incident angle θ for different values of insulating gap (9/∆ ). The regimes. Figure 2 shows the dependence of the amplitude of normal (R) and Andreev reflection (AR)
orientation angle of d-wave superconductor in Figs. 2 (a) and (b) is β =0 and in Figs. 2 (c) and (d) is
β = . From Fig. 2(a) (Fig. 2 (b)), it is seen that in the absence of insulating gap, the amplitude of AR
EP
(R) for normal incidence is unity (zero) and by increasing incident angle, AR (R) decreases (increases). By increasing insulating gap, the amplitude of AR (R) decreases (increases) and by
AC C
approaching 9/∆ towards
= 0.7, the AR (R) reaches zero (unity), as seen from gray curves in
Figs. 2(a) and (b). From Fig. 2 (c), we observe that for the maximum superconducting orientation
angle β = , the amplitude of AR vanishes independent of mass gap and incidence angle values (in
this case, the coherent factor in Eq. 6 is f(.) = 0, so the amplitude of Nℎ,! in Eq. 7 will be zero).
While the amplitude of R depends on . and mass gap variations and it will be zero only at normal
incidence (. = 0) for ∆ = 0, as can be seen from Fig. 2 (d).
bias energy eV for different superconducting gap orientation angles () for normal incidence . = 0,
Figs 3. (a) and (b) show the amplitude of Andreev reflection (AR) and normal reflection (R) versus
5
ACCEPTED MANUSCRIPT ∆
respectively. In these figures, the silicene insulating gap (Eg) is zero (electric field
= 5). It is seen
that the amplitude of AR has been strongly modulated by increasing . In the case of = 0, the AR
is unity for subgap energies and then for energies higher than gap ∆ , AR amplitude decreases. For a
determined value of energy :;/∆ by increasing the superconducting orientation angle, the amplitude of Andreev reflection decreases. This scenario is best understood as follow: at .p = 0, by increasing
RI PT
from 0 to the order parameter in superconducting region decreases and thus Andreev reflection will be suppressed. Actually by varying the subgap energy decreases and consequently AR is unity for lower energies. In the case of =
the AR is zero. We should mention that at = , the coherent
factor f(.) = 0, so Nℎ,! in Eq. 7 will be zero for any bias energy as seen in figure 3(a). In fact, for
d ∆
= 5), the silicene is similar to gapless monolayer
SC
case of zero insulating gap &T = 0 ( ∆ = silicene d-wave superconductor.7
M AN U
graphene3 and the suppression of AR is due to the tunneling into the nodes on the Fermi surface of
the R is zero and it is insensitive to any increasing of (see figure 3(b)). This is due to the fact that at In the absence of insulating gap of silicene, the amplitude of AR dominates normal reflection R, so
normal incidence angle . = 0, in the absence of insulating gap (
T ∆
= 0), the + = − = E. Hence, in
this case the AR amplitude N:,! in Eq. 7 will be zero. In the Fig. 3(c) we have plotted the normalized
charge conductance (G/GN) versus bias energy :;/∆ for different superconducting gap orientations
TE D
in the absence of insulating gap. It is seen that by increasing from 0 to = , the conductance
alters drastically and for =
the conductance decreases to constant value and it is not sensitive to
the bias energies. This latter case was obtained for the case of gapless monolayer graphene.7 Actually
EP
for undoped silicene EB =0 and in the absence of insulating gap, the reduction of G/GN is due to the tunneling into the nodes of the superconducting gap. Hence, the AR which significantly contributes to
the conductance is suppressed in the d-wave superconductor as compared to s-wave case. This behavior is very similar to the graphene-based NS junction.7 Also, it is remarkable that for undoped
AC C
silicene (EB = 0), the condition for formation of zero-energy state (ZES) is not satisfied at =
and we do not obtain zero-bias conductance peak (ZBCP) in the silicene-based d-wave superconductor junction.
Now, we study the effect of the insulating gap on the transport properties of the junction by applying external electric field. In Fig. 4 we present Andreev reflection (AR), normal reflection (R) and G/GN H
versus bias energy for different superconducting gap orientations () in the presence of insulating
gap ∆ = 0.2 (electric field
∆
= 5.2). As it is expected from the wave vectors equation of electron
and hole in the normal silicene region in Eq. (3), there are no transport properties for sub-insulating 6
ACCEPTED MANUSCRIPT
energy gap values. In Fig. 4(b) we see that the amplitude of normal reflection R is sensitive to bias
energy for any superconducting rotation angle even for = . Interestingly, it can be seen that in the
presence of insulating gap ( ∆ = 0.2) the normalized charge conductance G/GN depends on bias
energy for =
(see Fig. 4(c)). This behavior is essentially different from that in graphene-based
RI PT
NS junction7 and this is the most distinguished character of the present silicene-based d-wave
superconductor junction. In this case due to the presence of the insulating gap, electrons cannot tunnel into the nodes of the superconductor, so the amplitude of normal reflection R is not zero (see the brown curve in figure 4(b)). It is worth mentioning that the amplitude of normal reflection curves R starts from 1 at
H
= 0.2 and by increasing bias energy the amplitude of R decreases for any values of
SC
in contrary to the AR curves in Fig. 4(a) where they have started from zero at = 0.2 and have H
increased with bias energy for any values of the superconducting rotation angle except = . Note
M AN U
that in the presence of the insulating gap (for a fixed value of external electric field), the energy spectrum of Dirac electrons in proposed silicene junction is similar to energy spectrum of gapped graphene-based N/d-wave superconductor junction.30-32 More precisely the probability amplitudes of Andreev reflection (AR) and normal reflection (R) for = 0 in Fig. 4(a) and Fig. 4(b) respectively,
with &(* = 0.5∆ . We should emphasize from Figs. 3 and 4 that owing to the buckled structure of
are similar to the amplitude of RA and R for gapped graphene N/d-wave junction in Fig. 2 in Ref. 33
TE D
silicene, the tunneling conductance of the junction strongly depends on the rotation angle of d-wave superconductor () and perpendicular electric field (EZ).
To show more clear the effect of superconducting rotation angle on the transport properties of the silicene-based NS junction, in Fig. 5, we have plotted the Andreev reflection (AR), normal reflection (R) and G/GN versus for different values of bias energy in the absence of insulating gap (
∆
= 0).
EP
The periodicity of the charge conductance and AR with respect to (by a period of ⁄2) for any
AC C
values of bias energy (except for zero-bias energy E ⁄∆ = 0 ) is seen in Fig. 5(a) and 5(c), respectively. Note from Figs. 5(a) and 5(c) that the amplitude of AR and consequently G/GN is
minimum at = and ± . These features can be understood from the following discussion: at the
maximum orientation angle of d-wave superconductor = , the coherent factor in Eq. 6 is f(.) =
0, so the amplitude of Nℎ,! in Eq. 7 will be zero. Similar to Fig. 3(b), the amplitude of R does not
insensitive to and by increasing the bias energy the amplitude of G/GN curves suppresses. Note that
depend on the bias energy in Fig. 5(b). In the case of zero-bias energy, the charge conductance G(0) is
increasing the bias energy does not have any effect on the periodicity of AR and G/GN oscillations.
7
ACCEPTED MANUSCRIPT
6, we have plotted Andreev reflection (AR), normal reflection (R) and G/GN versus for different
To see the effect of silicene insulating gap on the transport properties of silicene NS junction, in Fig.
values of bias energy for the case of
∆
= 0.2. One can see that we cannot see reflections and
conductance for bias energies smaller than insulating gap ( ∆ < ∆ ). This can be understood as follow: H
RI PT
for bias energies lower than insulating gap, the wave vector in Eq. 3 will be imaginary which is due to the unique buckling structure of silicene. Similar to Fig. 4(b) the amplitude of R depends on the bias energy in Fig. 6(b). As a result, this junction can be considered as an energy filter (switch) for charge
conductance in silicene-based structures.
In Fig. 7 we have demonstrated the effect of insulating gap ∆ on the AR, R and G/GN versus β for a
fixed energy = 0.5. It can be seen that by increasing the insulating gap ∆ (or perpendicular electric
SC
the amplitudes of AR and G/GN oscillations decrease. Also it should be emphasized that, the
M AN U
field
), ∆
presence of insulating gap in silicene enhances Fermi wave vector mismatches and suppresses the AR and G/GN oscillations. Note that we have AR, R and G/GN for insulating gap smaller than bias energy (∆ <
H ) ∆
and at ∆ = = 0.5, the AR and G/GN are zero. However, we obtained that by increasing
H
the perpendicular electric field the amplitude of R enhances and finally reaches unity for
∆
= 5.5
(∆ = 0.5). These features can be explained from the wave vector component for electrons and holes
TE D
H = P = = DE − (%&' − !"#$ ) as follow: by increasing electric field and consequently
insulating gap, the wave vector for electrons and holes diminishes and for ∆ =
∆
−
d ∆
= = 0.5 H
P case, the wave vector is zero, so the amplitude of NJ,R in Eq. 7 is zero. Furthermore, in this case as O
EP
H and k in Eq. 7 are zero, consequently the amplitude of NJ,R is unity.
Also, in Fig. 8 we have plotted the Andreev reflection (AR), normal reflection (R) and G/GN versus for different values of orientation angle of d-wave superconductor () for a fixed energy = 1. It
AC C
∆
H
satisfying condition−E + "p ≤ %&' ≤ E + "p . Therefore, beyond this criterion the wave vector will
is seen that we obtained transmission and charge conductance for perpendicular electric field be imaginary and we do not have transport. It is found that by varying , the amplitude of AR
strongly suppresses and finally it will be zero for = that by varying from 0 to
as seen in Fig. 8(a). Form Fig. 8(b), it is seen
there is a little distinction between different superconducting gap
orientation curves. We should point out that at
∆
= 5, where the insulating gap is zero, the amplitude
of AR and consequently G/GN is maximum and by increasing both decrease. 8
∆
and , the AR and G/GN
ACCEPTED MANUSCRIPT
4. Conclusion In summary, we have theoretically studied the transport properties through a NS junction in buckled silicene, where the pairing potential of the superconductor is anisotropic d-wave. We adopted the BTK formalism to carry out our calculations. The most striking aspect of the present work is that the
RI PT
transport properties of the junction can be effectively tuned by the orientation angle of superconductive gap () and perpendicular electric field (EZ) which are due to the buckled structure of silicene. Remarkably, it is obtained that the Andreev reflection (AR) and charge conductance exhibit an oscillatory behavior with the orientation angle of superconductive gap () with a period of
and by increasing the insulating gap of silicene, the amplitudes of AR and charge conductance
SC
oscillations strongly suppress. Moreover, the insulating gap of silicene can be used as a switch to turn on or off the Andreev reflection and charge conductance. It is expected that the obtained theoretical results offer a potential strategy for improving the performance of buckled silicene superconductor
M AN U
junctions.
References 1
A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science 306, 666 (2004). 3 C. W. J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006). 4 S. Bhattacharjee, M. Maiti and K. Sengupta, Phys. Rev. B 76, 184514 (2007). 5 S. Bhattacharjee and K. Sengupta, Phys. Rev. Lett. 97, 217001 (2006). 6 H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen and A. F. Morpurgo, Nature (London) 446, 56 (2007). 7 J. Linder and A. Sudbø, Phys. Rev. Lett. 99, 147001 (2007). 8 M. Maiti and K. Sengupta, Phys. Rev. B 76, 054513 (2007). 9 C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008). 10 G. Deutscher, Rev. Mod. Phys. 77, 109 (2005). 11 Y. Hajati, AIP Adv. 5, 047112 (2015). 12 S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). 13 J. Zou and G. Jin, EPL 87, 27008 (2009). 14 B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. b. Vizzini, B. n. Ealet and B. Aufray, Appl. Phys. Lett. 97, 223109 (2010). 15 P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet and G. Le Lay, Phys. Rev. Lett. 108, 155501 (2012). 16 D. Chiappe, E. Scalise, E. Cinquanta, C. Grazianetti, B. van den Broek, M. Fanciulli, M. Houssa and A. Molle, Adv. Mater. 26, 2096 (2014). 17 A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y. Wang and Y. Yamada-Takamura, Phys. Rev. Lett. 108, 245501 (2012). 18 C.-C. Liu, W. Feng and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). 19 C.-C. Liu, H. Jiang and Y. Yao, Phys. Rev. B 84, 195430 (2011). 20 M. Ezawa, New J. Phys. 14, 033003 (2012). 21 M. Ezawa, Phys. Rev. Lett. 110, 026603 (2013). 22 M. Ezawa, Phys. Rev. B 87, 155415 (2013). 23 A. Yamakage, M. Ezawa, Y. Tanaka and N. Nagaosa, Phys. Rev. B 88, 085322 (2013).
AC C
EP
TE D
2
9
ACCEPTED MANUSCRIPT
24
L. Chen, B. Feng and K. Wu, Appl. Phys. Lett. 102, 081602 (2013). F. Liu, C.-C. Liu, K. Wu, F. Yang and Y. Yao, Phys. Rev. Lett. 111, 066804 (2013). 26 L.-D. Zhang, F. Yang and Y. Yao, Sci. Rep. 5, 8203 (2015). 27 J. Linder and T. Yokoyama, Phys. Rev. B 89, 020504 (2014). 28 H. Li, Phys. Rev. B 94, 075428 (2016). 29 G. E. Blonder, M. Tinkham and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982). 25
S. Y. Zhou, G. H. Gweon, A. V. Fedorov, P. N. First, W. A. de Heer, D. H. Lee, F. Guinea, A. H.
Castro Neto and A. Lanzara, Nature Mater. 6, 770 (2007).
RI PT
30
B. Soodchomshom, Physica B 405, 1383-1387 (2010).
32
B. Soodchomshom, I. M. Tang, and R. Hoonsawat, Physica E, 41(8), 1475-1478 (2009).
33
H. Goudarzi, M. Khezerlou, and T. Dezhaloud, Physica C 502, 36-40 (2014).
AC C
EP
TE D
M AN U
SC
31
10
ACCEPTED MANUSCRIPT
Figure captions Figure 1. Schematic view of a normal/d-wave superconductor silicene junction in the xy plane. Figure 2. RA and R versus incidence angle θ for different values of insulating gap ∆ at = 0.7. The
orientation angle of d-wave superconductor in (a) and (b) is β =0 and in (c) and (d) is β = .
RI PT
Figure 3. RA, R and G/GN versus bias energy eV/∆0 for different values of orientation angle of d-wave
superconducting gap (β) in the absence of insulating gap (∆ = 0). In figure (a) and (b) the incidence T
(E ≫ ∆ , λSO ).
SC
angle is θ = 0 and η = σ = 1. We will consider in all cases a heavy doped superconducting region
Figure 4. RA, R and G/GN versus bias energy eV/∆0 for different values of orientation angle of d-wave
superconductor (β) in the presence of insulating gap (∆ = 0.2 with 9 = |%&' − "#$ |). In figure (a)
M AN U
and (b) the incidence angle is θ = 0.
Figure 5. RA, R and G/GN versus β for different values of bias energy (E ⁄∆ ) in the absence of
insulating gap (∆ = 0). In figure (a) and (b) the incidence angle is θ = 0.
Figure 6. RA, R and G/GN versus β for different values of bias energy (E ⁄∆ ) in the presence of insulating gap ∆ = 0.2 ( ∆ ¥ = 5.2). In figure (a) and (b) the incidence angle is θ = 0.
¤
TE D
Figure 7. RA, R and G/GN versus β for different values of insulating gap ∆ at = 0.5. In figure (a)
EP
and (b) the incidence angle is θ = 0.
Figure 8. RA, R and G/GN versus
¤¥ ∆
AC C
for different values of orientation angle of d-wave
superconductor (β) at = 1. In figure (a) and (b) the incidence angle is θ = 0.
11
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 1
V
AC C
EP
¦
§¨© Oª© - Superconductor
Normal silicene
12
ACCEPTED MANUSCRIPT
Figure 2
=0
(a)
0.8
1.0
= /4
RI PT
1.0
(c)
0.8
RA 0.4
0.2
0.2
m / ∆ 0 = 0.6
SC
RA 0.4
0.4
0.2
0.0
0.2
1.0
M AN U
0.0
0.0
m / ∆ 0 = 0.2
m / ∆ 0 = 0.4
0.6
0.6
m / ∆ 0 = 0.0
0.4
0.4
0.2
0.0
m / ∆ 0 = 0.699
0.2
0.4
0.2
= /4 0.4
1.0 0.8
0.8
0.6
R
R
0.6
0.4
0.2 0.0
(b) 0.2
0.0
0.2
=0 0.4
AC C
EP
0.4
TE D
0.4
13
0.2 0.0
(d) 0.4
0.2
0.0
ACCEPTED MANUSCRIPT
Figure 3 1.0
(a)
0.6 0.4 0.2 0.2
0.4
0.6 0.8 eV 0
1.2
1.4
M AN U
1.0 (b)
1.0
SC
0.0 0.0
RI PT
RA
0.8
0.8 R
0.6 0.4 0.2
TE D
0.0 0.0
0.2
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
2.0
1.0
AC C
G GN
EP
1.5
0.5
0.0 0.0
(c)
0.2
14
β β β β β β
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π /4
ACCEPTED MANUSCRIPT
Figure 4 1.0 (a)
0.6 0.4 0.2 0.2
0.4
0.6 0.8 eV 0
0.8 R
0.6 0.4 0.2
TE D
0.0 0.0 2.0
1.2
1.4
M AN U
1.0 (b)
1.0
SC
0.0 0.0
RI PT
RA
0.8
0.2
β β β β β β
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
(c)
1.0
AC C
G GN
EP
1.5
0.5
0.0 0.0
0.2
15
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π /4
ACCEPTED MANUSCRIPT
Figure 5 1.0
0.6
RI PT
RA
0.8
0.4 0.2 (a)
0.2
0.6
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.8
1.0
1.2
1.4
M AN U
1.0 (b)
0.4
SC
0.0 0.0
0.2
ε / ∆0 ε / ∆0 ε / ∆0 ε / ∆0
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.6
0.8
1.0
1.2
1.4
EP
2.0
G GN
AC C
1.5 1.0 0.5
(c)
0.0 0.0
0.2
16
= 0.0 = 0.1 = 0.5 = 1.0
ACCEPTED MANUSCRIPT
Figure 6 1.0 (a)
0.6 0.4 0.2
(b)
0.4
0.6
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.8
1.0
1.2
1.4
M AN U
1.0
0.2
SC
0.0 0.0
RI PT
RA
0.8
0.2
ε / ∆0 ε / ∆0 ε / ∆0 ε / ∆0
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.6
0.8
1.0
1.2
1.4
EP
2.0 (c)
G GN
AC C
1.5 1.0 0.5
0.0 0.0
0.2
17
= 0.0 = 0.1 = 0.5 = 1.0
ACCEPTED MANUSCRIPT
Figure 7 1.0 (a)
0.6 0.4 0.2
(b)
0.2
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.4
0.5
0.1
0.2
0.3
0.4
m / ∆0 = 0.0 m / ∆0 = 0.2 m / ∆0 = 0.4 m / ∆0 = 0.45 m / ∆0 = 0.5
0.5
(c)
EP
2.0
0.3
M AN U
1.0
0.1
SC
0.0 0.0
RI PT
RA
0.8
G GN
AC C
1.5 1.0 0.5
0.0 0.0
0.2
0.4
0.6
18
0.8
1.0
1.2
1.4
ACCEPTED MANUSCRIPT
Figure 8 1.0
(a)
0.6 0.4 0.2 3
4
5 lEz
(b)
0.8 R
0.6 0.4
TE D
0.2 0.0 2
3
4
5
lEz
8
β β β β β β
6
7
8
6
7
8
0
(c)
EP
2.0
0
7
M AN U
1.0
6
SC
0.0 2
RI PT
RA
0.8
G GN
AC C
1.5 1.0 0.5
0.0 2
3
4
5 lEz
19
0
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π / 4
ACCEPTED MANUSCRIPT
. *We study the tunneling conductance of a normal/d-wave superconductor silicene junction. *We investigate effect of d-wave superconducting pairing symmetry on the conductances
RI PT
spectra of silicene. *We study the influence of insulating gap of silicene on the transport properties of the
AC C
EP
TE D
M AN U
SC
system.
PII:
S0749-6036(16)31324-6
DOI:
10.1016/j.spmi.2016.11.067
Reference:
YSPMI 4702
To appear in:
Superlattices and Microstructures
Received Date: 25 October 2016 Revised Date:
29 November 2016
Accepted Date: 29 November 2016
Please cite this article as: Y. Hajati, S. Vosoughi nia, G. Rashedi, Tunneling transport in dwave superconductor-silicene junction, Superlattices and Microstructures (2017), doi: 10.1016/ j.spmi.2016.11.067. 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.
ACCEPTED MANUSCRIPT
Tunneling transport in d-wave superconductor-silicene junction Y. hajatib, S. Vosoughi niaa, G. Rashedia a
Department of Physics, Faculty of Science, University of Isfahan, Isfahan 81746-73441, Iran Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
RI PT
b
Abstract: We theoretically study the tunneling conductance of a normal/d-wave superconductor silicene junction using Blonder-Tinkham-Klapwijk (BTK) formalism. We discuss in detail how the conductances spectra are affected by inducing d-wave superconducting pairing symmetry in the
SC
buckled silicene. It is obtained that the amplitude of the spin/valley-dependent Andreev reflection and subgap conductance of the junction can be strongly modulated by the orientation angle of tune the transport properties of the junction through changing and EZ. We demonstrate that the
M AN U
superconductive gap () and perpendicular electric field (EZ), suggesting that one may experimentally
subgap conductance exhibits an oscillatory behavior as a function of the orientation angle of
superconductive gap () with a period of and by increasing the insulating gap of silicene, the charge
conductance oscillations suppress. Remarkably, due to the buckled structure of silicene at the maximum orientation angle of the d-wave superconducting = , we found a very distinct behavior
from the graphene-based NS junction where the charge conductance is insensitive to the bias energy.
electric field.
TE D
In addition, the Andreev reflection and subgap conductance can be switched on and off by applying
AC C
EP
Keywords: Silicene; Tunneling conductance; Superconducting pairing symmetry; Andreev reflection
1
ACCEPTED MANUSCRIPT
1. Introduction: Andreev reflection (AR) is a fundamental process in understanding quantum transport in a normal metal/superconductor (NS) junction.1 Based on Andreev theory, an incident electron with the energy smaller than the superconducting energy gap cannot enter into the superconductor. In the conventional
RI PT
materials an electron in the normal metal is retro-reflected at the interface of NS as a hole by transferring a cooper pair to the superconductor. Pure graphene2 is not supercoductor, but superconductivity can be induced in graphene layer by proximity effect.3-8 Beenakker obtained that in a graphene-based normal metal/superconductor (NS) junction when the Fermi energy in the normal region is much smaller than that of the superconducting energy gap, the specular Andreev reflection
SC
occurs which is different from the usual retro-reflection in a conventional NS junction.3,9
It is known that an anisotropic superconducting order parameter (unconventional superconductor) induced in graphene has a strong effect on quantum transport in graphene-based NS junction.7 In
M AN U
contrast to the s-wave superconductor cases, at the interface of a dx2-y2-wave superconductor, zero-bias conductance peaks (ZBCP) are formed due to the sign change of the pair potential.10-13 It is found that in graphene-based N/dx2-y2-wave superconductor the variation of rotation angle of the superconductive
gap has led to a progressive shift of the peak in the conductance without any formation of the
ZBCP except for = /4.7
Silicene is a single layer of silicon atoms arranged in a two-dimentsional honeycomb lattice with a
TE D
buckled sublattice structure which has been successfully synthesis on variety of substrates.14-17 Its low-energy band structure in the K and K' valleys is described by the Dirac theory as in graphene.18,19 Silicene has a large spin-orbit coupling compared to graphene and due to the low-buckled geometry, its energy gap can be further tuned by an external electric field Ez perpendicular to the silicene sheet.
EP
Tunabilty of the silicene energy gap leads to change its topological phases which is very important in the modern condensed matter physics.20-23 Recently, a possible conductive gap, about 35 meV, is observed in monolayer silicene epitaxially
AC C
grown on Ag(111) surface at temperature as high as 30-40 K via scanning tunneling microscope.24 Observation of Andreev reflection strongly indicates that this gap is a superconducting gap. It is also found that the pairing symmetry of superconductivity in silicene can be tuned by applying external electric field.25,26 To date, there is no experimental report about proximity induced superconductivity in silicene. However, currently Linder et al. studied the superconducting proximity effect silicene and calculated conductance across NS and NSN junctions.27 It is obtained that the local and non local Andreev reflection can be tuned by an external electric field owing to the buckled structure of silicene. In this letter, we study the proximity-induced unconventional superconductivity in the buckled structure silicene. Here, we have mainly focused how the anisotropic superconducting order parameter can affect the transmission coefficient and spin-valley conductance in the NS silicene 2
ACCEPTED MANUSCRIPT
junction. We model our NS setup within the scattering matrix theory. Similar setup has been studied in graphene, however the transport properties of silicene-based unconventional superconductivity is still unexplored to the best of our knowledge. 2. Model and Hamiltonian
RI PT
We consider a two-dimensional silicene-based normal/superconductor (NS) junction, as shown in Fig 1. The NS silicene junction can be described by the Dirac-Bogoliubov-de Gennes (DBdG) equation27
where,
= ℏ − − (!"#$ − %&' ) ' − &( ()), () = ∆(.)+() (2)
&( () = &(* +(−) + &(# +() ,
SC
() ∗ ψ = εψ, (1) () −
M AN U
Here = 5.5 × 104 m/s represents the Fermi velocity in silicene and ! = +1 (↑) or ! = −1 (↓) denotes the spin indices in the silicene and superconductor regions. = 1(−1) corresponds to the K
(K') valley point and , and ' are 2 × 2 Pauli matrices in sublattice pseudospin. "#$ = 3.9 9:; is
respond differently to an externally applied electric field &' . So, %&' is the on-site potential difference
the spin-orbit coupling term. Due to the buckled structure of the silicene, the atoms in two sublattices between A and B sublattices which can be tuned by an external electric field.20 For the dx2-y2-wave
pairing, the order parameter in graphene induced by a high-Tc cuprate ∆(.) = ∆ cos2(. − ) ,
TE D
where is the orientation angle of the superconductive gap with respect to normal as illustrated in Fig
1. The DBdG equation is derived within the mean-field approximation.3,9 So, the chemical potential in
the superconductor region, &(# , should be very large compared to all other energy scales in our
EP
calculations: &(# ≫ ∆ , "@A .
the normal silicene (N) region, the wavefunction ψB (x) for incident electron, reflected electron, and
AC C
By solving the DBdG equation (Eq.1), we obtain the wavefunctions in each region of the junction. In
Andreev reflected hole is given by
H : IJKL − H : OIJKL N H F G F M + G M ψ* ( = 0) = 0 0 D2E F D2E F 0 0 1
+
NP
0 0
G OIJ .ℎ M , (3) D2E O P : O
3
ACCEPTED MANUSCRIPT H P where NJ,R and NJ,R are normal and Andreev reflection amplitudes, .H is incident angle of the electron
and .P is the Andreev reflection angle of the hole in the N region with respect to the normal axis of
the interface. Hereafter, for brevity of notation we set ℏ = = 1. By setting the chemical potential
&(* = 0 (undoped silicene), wave vectors of electron and hole become identical; H = P = =
RI PT
DE − (%&' − !"#$ ) . We have also defined: ± = E ± (!"#$ − %&' ). From the Hamiltonian (Eq.
band gap (insulating gap) is &T = 2|%&' − !"#$ |=29JR , thus we have transport only if |E| > &T /2.
1) it can be seen that there are four bands in the energy spectrum. At the valley points (K and K') the As Ez increases, the band gap decreases linearly and finally closes at EX = EY = 17 meV/A. Note
also that the K' valley with spin up orientation ( = −1, ! = 1) and K valley with spin down
SC
orientation bands ( = 1, ! = −1) do not contribute to the subgap transport in our studied energy
range, since they are far away from Fermi level.27,28 It is worth mentioning that the magnitude of spin-
electric field EZ.
M AN U
orbit interaction in silicene is constant, so the only way to tune the insulating gap is changing the In the dx2-y2-wave superconductive silicene (S) region, the wave function for the transmission of DBdG quasiparticles with excitation energy ε is given by
−:−l.@ f(. O ):lg m _H c k _P c k f (. O ):lg ψ# (x = 0) = , (4) b: IJKd f(. F ): OIgh j + b −l. O @ −: e(. ) j √2 √2 F OIgh e(. O ) a f(. ): i a i : IJKd e(. F ) e(. F )
m
TE D
where _:,! and _ℎ,! are the transmission amplitudes of electron-like (ELQ) and hole-like (HLQ)
quasiparticles. .# is the incident angle of ELQ in the superconductive region and we have defined the parameters . F (. O ) = .# ( − .# ) and : Ig = ∆(. ± )/|∆(. ± )| . Because of the translational ±
EP
invariance in the y-direction, all transverse wave vectors during the scattering process are conserved
and the relationship among the incident angles in each region is obtained through the following component conservation relation,
AC C
H sin .H = P sin .P , H sin .H = p sin .p . (5)
Since we consider heavy doped superconductor ( &(# ≫ ℏ ), we set .# → 0 in the following calculations. The coherent factors read
DE − |∆(θ)| 1 1 DE − |∆(θ)| e(.) = r s1 + t , f(.) = r (1 − ) . (6) 2 E 2 E
4
ACCEPTED MANUSCRIPT
Andreev reflection Nℎ,! and normal reflection N:,! can be calculated by, ψv ( = 0) = ψ@ ( = 0), as Using the continuity of wavefunctions at the boundaries, The amplitude of spin/valley-dependent
follows =
[e(.) + f(.) :l(x
−
z
−x+ )
][:2l. − − + ]
with z = 2 {e(.) − f(.) :l(x
−
−x+ )
, Nℎ,!
=
4 e(.)f(.)cos.: z
:
l. OIg h
, (7)
RI PT
N:,!
|cos. + 2E{e(.) + f(.) :l(x
−
−x+ )
|. Using the Blonder-
Tinkham-Klapwijk (BTK) formalism,29 we define the normalized charge conductance (} ):
SC
/2 1 = . .(1 + − ), (8) * 4 ! O/
}
~
where v is the ballistic conductance of silicene. Here, R= N:,! and RA= Nℎ,! are normal and
M AN U
2
2
chosen to normalize all our energy scales by ∆ , so that all our results are in terms of dimensionless Andreev reflection probabilities, respectively. Note that to carry out our numerical analysis, we have
quantities. Also for spin-orbit coupling we set 3. Results and discussions:
d ∆
= 5 thoroughout the paper.
TE D
In this section, we present our numerical results for NS silicene junction for different parameter of the Dirac electrons on the incident angle θ for different values of insulating gap (9/∆ ). The regimes. Figure 2 shows the dependence of the amplitude of normal (R) and Andreev reflection (AR)
orientation angle of d-wave superconductor in Figs. 2 (a) and (b) is β =0 and in Figs. 2 (c) and (d) is
β = . From Fig. 2(a) (Fig. 2 (b)), it is seen that in the absence of insulating gap, the amplitude of AR
EP
(R) for normal incidence is unity (zero) and by increasing incident angle, AR (R) decreases (increases). By increasing insulating gap, the amplitude of AR (R) decreases (increases) and by
AC C
approaching 9/∆ towards
= 0.7, the AR (R) reaches zero (unity), as seen from gray curves in
Figs. 2(a) and (b). From Fig. 2 (c), we observe that for the maximum superconducting orientation
angle β = , the amplitude of AR vanishes independent of mass gap and incidence angle values (in
this case, the coherent factor in Eq. 6 is f(.) = 0, so the amplitude of Nℎ,! in Eq. 7 will be zero).
While the amplitude of R depends on . and mass gap variations and it will be zero only at normal
incidence (. = 0) for ∆ = 0, as can be seen from Fig. 2 (d).
bias energy eV for different superconducting gap orientation angles () for normal incidence . = 0,
Figs 3. (a) and (b) show the amplitude of Andreev reflection (AR) and normal reflection (R) versus
5
ACCEPTED MANUSCRIPT ∆
respectively. In these figures, the silicene insulating gap (Eg) is zero (electric field
= 5). It is seen
that the amplitude of AR has been strongly modulated by increasing . In the case of = 0, the AR
is unity for subgap energies and then for energies higher than gap ∆ , AR amplitude decreases. For a
determined value of energy :;/∆ by increasing the superconducting orientation angle, the amplitude of Andreev reflection decreases. This scenario is best understood as follow: at .p = 0, by increasing
RI PT
from 0 to the order parameter in superconducting region decreases and thus Andreev reflection will be suppressed. Actually by varying the subgap energy decreases and consequently AR is unity for lower energies. In the case of =
the AR is zero. We should mention that at = , the coherent
factor f(.) = 0, so Nℎ,! in Eq. 7 will be zero for any bias energy as seen in figure 3(a). In fact, for
d ∆
= 5), the silicene is similar to gapless monolayer
SC
case of zero insulating gap &T = 0 ( ∆ = silicene d-wave superconductor.7
M AN U
graphene3 and the suppression of AR is due to the tunneling into the nodes on the Fermi surface of
the R is zero and it is insensitive to any increasing of (see figure 3(b)). This is due to the fact that at In the absence of insulating gap of silicene, the amplitude of AR dominates normal reflection R, so
normal incidence angle . = 0, in the absence of insulating gap (
T ∆
= 0), the + = − = E. Hence, in
this case the AR amplitude N:,! in Eq. 7 will be zero. In the Fig. 3(c) we have plotted the normalized
charge conductance (G/GN) versus bias energy :;/∆ for different superconducting gap orientations
TE D
in the absence of insulating gap. It is seen that by increasing from 0 to = , the conductance
alters drastically and for =
the conductance decreases to constant value and it is not sensitive to
the bias energies. This latter case was obtained for the case of gapless monolayer graphene.7 Actually
EP
for undoped silicene EB =0 and in the absence of insulating gap, the reduction of G/GN is due to the tunneling into the nodes of the superconducting gap. Hence, the AR which significantly contributes to
the conductance is suppressed in the d-wave superconductor as compared to s-wave case. This behavior is very similar to the graphene-based NS junction.7 Also, it is remarkable that for undoped
AC C
silicene (EB = 0), the condition for formation of zero-energy state (ZES) is not satisfied at =
and we do not obtain zero-bias conductance peak (ZBCP) in the silicene-based d-wave superconductor junction.
Now, we study the effect of the insulating gap on the transport properties of the junction by applying external electric field. In Fig. 4 we present Andreev reflection (AR), normal reflection (R) and G/GN H
versus bias energy for different superconducting gap orientations () in the presence of insulating
gap ∆ = 0.2 (electric field
∆
= 5.2). As it is expected from the wave vectors equation of electron
and hole in the normal silicene region in Eq. (3), there are no transport properties for sub-insulating 6
ACCEPTED MANUSCRIPT
energy gap values. In Fig. 4(b) we see that the amplitude of normal reflection R is sensitive to bias
energy for any superconducting rotation angle even for = . Interestingly, it can be seen that in the
presence of insulating gap ( ∆ = 0.2) the normalized charge conductance G/GN depends on bias
energy for =
(see Fig. 4(c)). This behavior is essentially different from that in graphene-based
RI PT
NS junction7 and this is the most distinguished character of the present silicene-based d-wave
superconductor junction. In this case due to the presence of the insulating gap, electrons cannot tunnel into the nodes of the superconductor, so the amplitude of normal reflection R is not zero (see the brown curve in figure 4(b)). It is worth mentioning that the amplitude of normal reflection curves R starts from 1 at
H
= 0.2 and by increasing bias energy the amplitude of R decreases for any values of
SC
in contrary to the AR curves in Fig. 4(a) where they have started from zero at = 0.2 and have H
increased with bias energy for any values of the superconducting rotation angle except = . Note
M AN U
that in the presence of the insulating gap (for a fixed value of external electric field), the energy spectrum of Dirac electrons in proposed silicene junction is similar to energy spectrum of gapped graphene-based N/d-wave superconductor junction.30-32 More precisely the probability amplitudes of Andreev reflection (AR) and normal reflection (R) for = 0 in Fig. 4(a) and Fig. 4(b) respectively,
with &(* = 0.5∆ . We should emphasize from Figs. 3 and 4 that owing to the buckled structure of
are similar to the amplitude of RA and R for gapped graphene N/d-wave junction in Fig. 2 in Ref. 33
TE D
silicene, the tunneling conductance of the junction strongly depends on the rotation angle of d-wave superconductor () and perpendicular electric field (EZ).
To show more clear the effect of superconducting rotation angle on the transport properties of the silicene-based NS junction, in Fig. 5, we have plotted the Andreev reflection (AR), normal reflection (R) and G/GN versus for different values of bias energy in the absence of insulating gap (
∆
= 0).
EP
The periodicity of the charge conductance and AR with respect to (by a period of ⁄2) for any
AC C
values of bias energy (except for zero-bias energy E ⁄∆ = 0 ) is seen in Fig. 5(a) and 5(c), respectively. Note from Figs. 5(a) and 5(c) that the amplitude of AR and consequently G/GN is
minimum at = and ± . These features can be understood from the following discussion: at the
maximum orientation angle of d-wave superconductor = , the coherent factor in Eq. 6 is f(.) =
0, so the amplitude of Nℎ,! in Eq. 7 will be zero. Similar to Fig. 3(b), the amplitude of R does not
insensitive to and by increasing the bias energy the amplitude of G/GN curves suppresses. Note that
depend on the bias energy in Fig. 5(b). In the case of zero-bias energy, the charge conductance G(0) is
increasing the bias energy does not have any effect on the periodicity of AR and G/GN oscillations.
7
ACCEPTED MANUSCRIPT
6, we have plotted Andreev reflection (AR), normal reflection (R) and G/GN versus for different
To see the effect of silicene insulating gap on the transport properties of silicene NS junction, in Fig.
values of bias energy for the case of
∆
= 0.2. One can see that we cannot see reflections and
conductance for bias energies smaller than insulating gap ( ∆ < ∆ ). This can be understood as follow: H
RI PT
for bias energies lower than insulating gap, the wave vector in Eq. 3 will be imaginary which is due to the unique buckling structure of silicene. Similar to Fig. 4(b) the amplitude of R depends on the bias energy in Fig. 6(b). As a result, this junction can be considered as an energy filter (switch) for charge
conductance in silicene-based structures.
In Fig. 7 we have demonstrated the effect of insulating gap ∆ on the AR, R and G/GN versus β for a
fixed energy = 0.5. It can be seen that by increasing the insulating gap ∆ (or perpendicular electric
SC
the amplitudes of AR and G/GN oscillations decrease. Also it should be emphasized that, the
M AN U
field
), ∆
presence of insulating gap in silicene enhances Fermi wave vector mismatches and suppresses the AR and G/GN oscillations. Note that we have AR, R and G/GN for insulating gap smaller than bias energy (∆ <
H ) ∆
and at ∆ = = 0.5, the AR and G/GN are zero. However, we obtained that by increasing
H
the perpendicular electric field the amplitude of R enhances and finally reaches unity for
∆
= 5.5
(∆ = 0.5). These features can be explained from the wave vector component for electrons and holes
TE D
H = P = = DE − (%&' − !"#$ ) as follow: by increasing electric field and consequently
insulating gap, the wave vector for electrons and holes diminishes and for ∆ =
∆
−
d ∆
= = 0.5 H
P case, the wave vector is zero, so the amplitude of NJ,R in Eq. 7 is zero. Furthermore, in this case as O
EP
H and k in Eq. 7 are zero, consequently the amplitude of NJ,R is unity.
Also, in Fig. 8 we have plotted the Andreev reflection (AR), normal reflection (R) and G/GN versus for different values of orientation angle of d-wave superconductor () for a fixed energy = 1. It
AC C
∆
H
satisfying condition−E + "p ≤ %&' ≤ E + "p . Therefore, beyond this criterion the wave vector will
is seen that we obtained transmission and charge conductance for perpendicular electric field be imaginary and we do not have transport. It is found that by varying , the amplitude of AR
strongly suppresses and finally it will be zero for = that by varying from 0 to
as seen in Fig. 8(a). Form Fig. 8(b), it is seen
there is a little distinction between different superconducting gap
orientation curves. We should point out that at
∆
= 5, where the insulating gap is zero, the amplitude
of AR and consequently G/GN is maximum and by increasing both decrease. 8
∆
and , the AR and G/GN
ACCEPTED MANUSCRIPT
4. Conclusion In summary, we have theoretically studied the transport properties through a NS junction in buckled silicene, where the pairing potential of the superconductor is anisotropic d-wave. We adopted the BTK formalism to carry out our calculations. The most striking aspect of the present work is that the
RI PT
transport properties of the junction can be effectively tuned by the orientation angle of superconductive gap () and perpendicular electric field (EZ) which are due to the buckled structure of silicene. Remarkably, it is obtained that the Andreev reflection (AR) and charge conductance exhibit an oscillatory behavior with the orientation angle of superconductive gap () with a period of
and by increasing the insulating gap of silicene, the amplitudes of AR and charge conductance
SC
oscillations strongly suppress. Moreover, the insulating gap of silicene can be used as a switch to turn on or off the Andreev reflection and charge conductance. It is expected that the obtained theoretical results offer a potential strategy for improving the performance of buckled silicene superconductor
M AN U
junctions.
References 1
A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science 306, 666 (2004). 3 C. W. J. Beenakker, Phys. Rev. Lett. 97, 067007 (2006). 4 S. Bhattacharjee, M. Maiti and K. Sengupta, Phys. Rev. B 76, 184514 (2007). 5 S. Bhattacharjee and K. Sengupta, Phys. Rev. Lett. 97, 217001 (2006). 6 H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen and A. F. Morpurgo, Nature (London) 446, 56 (2007). 7 J. Linder and A. Sudbø, Phys. Rev. Lett. 99, 147001 (2007). 8 M. Maiti and K. Sengupta, Phys. Rev. B 76, 054513 (2007). 9 C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008). 10 G. Deutscher, Rev. Mod. Phys. 77, 109 (2005). 11 Y. Hajati, AIP Adv. 5, 047112 (2015). 12 S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). 13 J. Zou and G. Jin, EPL 87, 27008 (2009). 14 B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. b. Vizzini, B. n. Ealet and B. Aufray, Appl. Phys. Lett. 97, 223109 (2010). 15 P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet and G. Le Lay, Phys. Rev. Lett. 108, 155501 (2012). 16 D. Chiappe, E. Scalise, E. Cinquanta, C. Grazianetti, B. van den Broek, M. Fanciulli, M. Houssa and A. Molle, Adv. Mater. 26, 2096 (2014). 17 A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y. Wang and Y. Yamada-Takamura, Phys. Rev. Lett. 108, 245501 (2012). 18 C.-C. Liu, W. Feng and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). 19 C.-C. Liu, H. Jiang and Y. Yao, Phys. Rev. B 84, 195430 (2011). 20 M. Ezawa, New J. Phys. 14, 033003 (2012). 21 M. Ezawa, Phys. Rev. Lett. 110, 026603 (2013). 22 M. Ezawa, Phys. Rev. B 87, 155415 (2013). 23 A. Yamakage, M. Ezawa, Y. Tanaka and N. Nagaosa, Phys. Rev. B 88, 085322 (2013).
AC C
EP
TE D
2
9
ACCEPTED MANUSCRIPT
24
L. Chen, B. Feng and K. Wu, Appl. Phys. Lett. 102, 081602 (2013). F. Liu, C.-C. Liu, K. Wu, F. Yang and Y. Yao, Phys. Rev. Lett. 111, 066804 (2013). 26 L.-D. Zhang, F. Yang and Y. Yao, Sci. Rep. 5, 8203 (2015). 27 J. Linder and T. Yokoyama, Phys. Rev. B 89, 020504 (2014). 28 H. Li, Phys. Rev. B 94, 075428 (2016). 29 G. E. Blonder, M. Tinkham and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982). 25
S. Y. Zhou, G. H. Gweon, A. V. Fedorov, P. N. First, W. A. de Heer, D. H. Lee, F. Guinea, A. H.
Castro Neto and A. Lanzara, Nature Mater. 6, 770 (2007).
RI PT
30
B. Soodchomshom, Physica B 405, 1383-1387 (2010).
32
B. Soodchomshom, I. M. Tang, and R. Hoonsawat, Physica E, 41(8), 1475-1478 (2009).
33
H. Goudarzi, M. Khezerlou, and T. Dezhaloud, Physica C 502, 36-40 (2014).
AC C
EP
TE D
M AN U
SC
31
10
ACCEPTED MANUSCRIPT
Figure captions Figure 1. Schematic view of a normal/d-wave superconductor silicene junction in the xy plane. Figure 2. RA and R versus incidence angle θ for different values of insulating gap ∆ at = 0.7. The
orientation angle of d-wave superconductor in (a) and (b) is β =0 and in (c) and (d) is β = .
RI PT
Figure 3. RA, R and G/GN versus bias energy eV/∆0 for different values of orientation angle of d-wave
superconducting gap (β) in the absence of insulating gap (∆ = 0). In figure (a) and (b) the incidence T
(E ≫ ∆ , λSO ).
SC
angle is θ = 0 and η = σ = 1. We will consider in all cases a heavy doped superconducting region
Figure 4. RA, R and G/GN versus bias energy eV/∆0 for different values of orientation angle of d-wave
superconductor (β) in the presence of insulating gap (∆ = 0.2 with 9 = |%&' − "#$ |). In figure (a)
M AN U
and (b) the incidence angle is θ = 0.
Figure 5. RA, R and G/GN versus β for different values of bias energy (E ⁄∆ ) in the absence of
insulating gap (∆ = 0). In figure (a) and (b) the incidence angle is θ = 0.
Figure 6. RA, R and G/GN versus β for different values of bias energy (E ⁄∆ ) in the presence of insulating gap ∆ = 0.2 ( ∆ ¥ = 5.2). In figure (a) and (b) the incidence angle is θ = 0.
¤
TE D
Figure 7. RA, R and G/GN versus β for different values of insulating gap ∆ at = 0.5. In figure (a)
EP
and (b) the incidence angle is θ = 0.
Figure 8. RA, R and G/GN versus
¤¥ ∆
AC C
for different values of orientation angle of d-wave
superconductor (β) at = 1. In figure (a) and (b) the incidence angle is θ = 0.
11
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 1
V
AC C
EP
¦
§¨© Oª© - Superconductor
Normal silicene
12
ACCEPTED MANUSCRIPT
Figure 2
=0
(a)
0.8
1.0
= /4
RI PT
1.0
(c)
0.8
RA 0.4
0.2
0.2
m / ∆ 0 = 0.6
SC
RA 0.4
0.4
0.2
0.0
0.2
1.0
M AN U
0.0
0.0
m / ∆ 0 = 0.2
m / ∆ 0 = 0.4
0.6
0.6
m / ∆ 0 = 0.0
0.4
0.4
0.2
0.0
m / ∆ 0 = 0.699
0.2
0.4
0.2
= /4 0.4
1.0 0.8
0.8
0.6
R
R
0.6
0.4
0.2 0.0
(b) 0.2
0.0
0.2
=0 0.4
AC C
EP
0.4
TE D
0.4
13
0.2 0.0
(d) 0.4
0.2
0.0
ACCEPTED MANUSCRIPT
Figure 3 1.0
(a)
0.6 0.4 0.2 0.2
0.4
0.6 0.8 eV 0
1.2
1.4
M AN U
1.0 (b)
1.0
SC
0.0 0.0
RI PT
RA
0.8
0.8 R
0.6 0.4 0.2
TE D
0.0 0.0
0.2
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
2.0
1.0
AC C
G GN
EP
1.5
0.5
0.0 0.0
(c)
0.2
14
β β β β β β
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π /4
ACCEPTED MANUSCRIPT
Figure 4 1.0 (a)
0.6 0.4 0.2 0.2
0.4
0.6 0.8 eV 0
0.8 R
0.6 0.4 0.2
TE D
0.0 0.0 2.0
1.2
1.4
M AN U
1.0 (b)
1.0
SC
0.0 0.0
RI PT
RA
0.8
0.2
β β β β β β
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
0.4
0.6 0.8 eV 0
1.0
1.2
1.4
(c)
1.0
AC C
G GN
EP
1.5
0.5
0.0 0.0
0.2
15
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π /4
ACCEPTED MANUSCRIPT
Figure 5 1.0
0.6
RI PT
RA
0.8
0.4 0.2 (a)
0.2
0.6
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.8
1.0
1.2
1.4
M AN U
1.0 (b)
0.4
SC
0.0 0.0
0.2
ε / ∆0 ε / ∆0 ε / ∆0 ε / ∆0
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.6
0.8
1.0
1.2
1.4
EP
2.0
G GN
AC C
1.5 1.0 0.5
(c)
0.0 0.0
0.2
16
= 0.0 = 0.1 = 0.5 = 1.0
ACCEPTED MANUSCRIPT
Figure 6 1.0 (a)
0.6 0.4 0.2
(b)
0.4
0.6
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.8
1.0
1.2
1.4
M AN U
1.0
0.2
SC
0.0 0.0
RI PT
RA
0.8
0.2
ε / ∆0 ε / ∆0 ε / ∆0 ε / ∆0
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.6
0.8
1.0
1.2
1.4
EP
2.0 (c)
G GN
AC C
1.5 1.0 0.5
0.0 0.0
0.2
17
= 0.0 = 0.1 = 0.5 = 1.0
ACCEPTED MANUSCRIPT
Figure 7 1.0 (a)
0.6 0.4 0.2
(b)
0.2
0.8 R
0.6 0.4
TE D
0.2 0.0 0.0
0.4
0.5
0.1
0.2
0.3
0.4
m / ∆0 = 0.0 m / ∆0 = 0.2 m / ∆0 = 0.4 m / ∆0 = 0.45 m / ∆0 = 0.5
0.5
(c)
EP
2.0
0.3
M AN U
1.0
0.1
SC
0.0 0.0
RI PT
RA
0.8
G GN
AC C
1.5 1.0 0.5
0.0 0.0
0.2
0.4
0.6
18
0.8
1.0
1.2
1.4
ACCEPTED MANUSCRIPT
Figure 8 1.0
(a)
0.6 0.4 0.2 3
4
5 lEz
(b)
0.8 R
0.6 0.4
TE D
0.2 0.0 2
3
4
5
lEz
8
β β β β β β
6
7
8
6
7
8
0
(c)
EP
2.0
0
7
M AN U
1.0
6
SC
0.0 2
RI PT
RA
0.8
G GN
AC C
1.5 1.0 0.5
0.0 2
3
4
5 lEz
19
0
= 0.0 = π / 20 = π /10 = 3π / 20 =π /5 =π / 4
ACCEPTED MANUSCRIPT
. *We study the tunneling conductance of a normal/d-wave superconductor silicene junction. *We investigate effect of d-wave superconducting pairing symmetry on the conductances
RI PT
spectra of silicene. *We study the influence of insulating gap of silicene on the transport properties of the
AC C
EP
TE D
M AN U
SC
system.