and spin-entangled states in a MoS2-based bipolar transistor

PHYSICAL REVIEW B 90, 195445 (2014)

Pure valley- and spin-entangled states in a MoS2 -based bipolar transistor Chunxu Bai,1,2 Yonglian Zou,1 Wen-Kai Lou,1 and Kai Chang1,* 1

SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, Peoples Republic of China 2 School of Physics, Anyang Normal University, Anyang 455000, Peoples Republic of China (Received 6 May 2014; revised manuscript received 12 October 2014; published 25 November 2014)

In this study, we show that the local Andreev reflection not only can be tuned largely by the type of the normal metal electrode, it also is related to the electrostatic potential in the superconductor region in a MoS2 -based n(p)-type metal/superconductor junction. In a MoS2 -based n-type metal/n(p)-type superconductor/p-type metal (nSp) transistor, nonlocal pure valley- and spin-entangled current can be tuned by the length and local gate voltage of a superconductor region. In particular, switching the quasiparticle type in both structures results in a series of intriguing features. Such an effect is not attainable in a graphene-based junction where the electron-hole symmetry enables the symmetry results to be observed. Besides, we have shown that the crossed Andreev reflection exhibits a maximum around ξ/2 instead of the exponential decay behavior in conventional superconductors and a maximum around ξ in the graphene material. The proposed straightforward experimental design and pure valley- and spin-entangled state can pave the way for a wider use in the entanglement based on material group-VI dichalcogenides. DOI: 10.1103/PhysRevB.90.195445

PACS number(s): 74.78.Na, 73.63.−b, 74.45.+c, 72.25.−b

I. INTRODUCTION

The recent proposal of valleytronics in a graphene sheet with a zigzag edge constriction [1] has made the valley-based electronics a realistic possibility. Valleytronics devices employ the valley degree of freedom, and operate in the same way as the charge or spin is used in electronics or spintronics [2]. The first key problem of valleytronics is how to produce a valley polarization, i.e., a valley current. Following the pioneering work of Rycerz, several alternative ways in graphene are driven by both the novel physics involved and a crucial step towards the device reality. Aside from the initial proposal based on a nanoribbon, valley current has been proposed in the bulk graphene with the trigonal warping [3], staggered sublattice potentials [4], the strain [5], and the line defects [6]. Furthermore, Martin et al. [7] have shown that bilayers still offer a promising platform for valley filtering. However, due to the physical and technological reasons, it seems that there is a long way to realize valleytronics in graphene. Fortunately, transition-metal dichalcogenides with a considerably large gap at the valleys become a wonderful platform for the valleytronics [8–10]. In particular, the dynamic valley polarization has been demonstrated experimentally by the circularly polarized light in the MoS2 monolayer [11]. In nature, MoS2 with a layered crystal structure is a semiconductor and not a superconductor. However, recent exciting developments in this material are the theoretical and experimental studies of possible superconductivity phenomena [12–15]. From an experimental point of view, there are two routes toward superconductivity, i.e., intrinsic superconductivity and proximity-induced superconductivity. Indeed, similar to the case of graphene [16,17], there is experimental evidence for the achievement of proximity-induced superconductivity in MoS2 by superconducting gates at the presence of solid and liquid gating [15]. From a theoretical point of view, two possible categories of pairing phase have been revealed previ-

*

[email protected]

1098-0121/2014/90(19)/195445(9)

ously, i.e., conventional s-wave pairing phase (induced by the electron-phonon interaction) [12] and unconventional pairing phase (induced by the electron-electron and electron-phonon interactions) [13]. As we show in the following, without the consideration of electron-electron interaction, we assume the s-wave superconducting phases are the only possible superconducting phases induced by an s-wave superconducting lead. This fundamental property therefore offers a unique possibility to a combination of valleytronics and superconductive electronics in the MoS2 monolayer. It paves the way toward MoS2 -based superconductor quantum device applications and gives a test bed for the interaction between the valley degree of freedom and superconductivity in a real material. One of the allures of superconductive electronics stems from the hunting of the entangled electronic states which may be found very useful in quantum information applications [18]. In solid-state physics, the conventional singlet superconductor is regarded as a natural source of entangled electronic state. In general, the quantum correlations can be provided by the crossed Andreev reflection (CAR) process through a normal metal–superconductor–normal metal (NSN) three-terminal transistor where the entangled Cooper pairs from the superconductor split into the two different normal electrodes [19,20]. However, the main drawback of such NSN transistor is attributed to the unavoidable processes including the local Andreev reflection (LAR) and the elastic cotunneling (EC). Furthermore, to detect the entangled state, shot-noise measurement must be demanded due to the crossover between CAR and EC in the nonlocal electrode [21]. Yet few proofs have been demonstrated experimentally [22,23], there has already been a remarkable progress. Recently, a theoretical study reported a pure CAR transport had been observed directly in a bipolar graphene transistor [24]. However, the pure CAR process is achieved only at one precise bias so that the nonlocal current is not completely carried by CAR. To get out of this dilemma, a very useful route as shown in experiment and theory by employing the energy to discriminate CAR has been suggested [25]. Note that although an ideal nonlocal Cooper pair splitting can be observed, the direct application of it is in general a very difficult task because the efficiency is yet small.

195445-1

©2014 American Physical Society

CHUNXU BAI, YONGLIAN ZOU, WEN-KAI LOU, AND KAI CHANG

In this paper, we mainly propose a setup for pure valley- and spin-entangled states, with no contributions from AR and EC, in a MoS2 -based n-type metal/n(p)-type superconductor/ptype metal (nSp) transistor. The valley-resolved transport in a metal/superconductor interface was first revealed in the graphene [26]. As proclaimed by Beenakker, the subgap current through the graphene-based n-type metal/superconductor (nS) junction stems from the combination of an electron in one valley and another in the other valley. The two different valley electrons couple with each other and transfer into superconductor as a Cooper pair. The valley degree of freedom consequently plays an important role in a superconducting contact in graphene. Due to the similarity and difference between graphene and MoS2 , we expect that those novel characteristics will facilitate the observation of pure valley- and spin-entangled states in MoS2 , thus, opening the way for their wider use in superconductor quantum device applications [27]. The rest of this paper is organized as follows. In Sec. II, we provide the detailed results of the single MoS2 -based n-type metal/superconductor (nS) and p-type metal/superconductor (pS) junction. In Sec. III, the results of calculations and discussions are presented for the MoS2 -based nSp transistor. In Sec. IV, a summary of the results obtained is given. In this paper, we confine ourselves to zero temperature.

single layer MoS2 x

e(h)+ e(h)− + r n(p) n(p) + rA n(p) , L = n(p)

A. Theory and model

We first consider a single nS or pS junction in a monolayer MoS2 sheet occupying the xy plane; the schematic illustration of the model is shown in Fig. 1. The growth direction is taken along the x axis. Here, we assume the S region is in the n-type superconductor. The n or p region occupies x < 0, while the S region occupies 0 < x. The conductive properties of left MoS2 lead on SiO2 in n or p type may be engineered by depositing intentionally adsorbates over the substrate surface [28–31]. The S region can be produced by depositing an n(p)

S 0

(b)

s-wave superconductor on the surface of MoS2 layer [26]. In the present junction, we assume the Fermi energy in the left lead is described by EF = μ and −μ corresponding to the n- and p-type leads, respectively. In the right, the s-wave pairing superconducting symmetry is assumed to be induced and is taken in the form (x) = s (x), where (x) is the Heaviside step function and s is the superconducting gap. We also assume that the potential U (x) is only nonzero in S, U (x) = −U (x). Furthermore, we suppose that the transversal width w is large enough, thus the details of the microscopic description of the strip edges become irrelevant. For monolayer MoS2 , in contrast with the monolayer graphene, the inversion symmetry is explicitly broken. However, the time-reversal symmetry is preserved even if the spin-orbit interaction lifts the spin degeneracy in valence band. To describe the low-energy excitation quasiparticles transport characteristics in the present junction, we employ the following Bogoliubov–de Gennes equation commonly used in graphene [8,9]: Hγ s − EF (x) = E, (1) ∗ (x) EF − Hγ s where Hγ s = at(γ kx σx + ky σy ) + N /2σz − λϑs γ (σz − 1)/ 2 + U (x), the four-dimensional spinor = (us , vs¯ ) contains us for the spin s electronlike quasiparticle and vs¯ for the spin s¯ holelike quasiparticle, s(¯s ) is physical spin [if s denotes up (down) spin, s¯ denotes down (up) spin], and E is the quasiparticle energy measured from EF . In Hγ s , γ = ±1 is the valley index corresponding to K and K , a is lattice constant, t is effective hopping integral, N is energy gap, 2λ is spin splitting in valence band, σ is pseudospin, and ϑs = ±1 denotes spin up and down, respectively. Note that the two spin subbands decouple in Eq. (1). The reason is that a hole seems to be an antiparticle of a removed electron. In the Andreev reflection process, a hole therefore has the same spin as an incident electron. In addition, two valleys of the band structure also decouple due to the time-reversal-invariant symmetry. For a particle of energy ε and transverse momentum q incidents from the left n (p) lead, the corresponding wave functions in the two regions can be written as

II. A SINGLE nS AND pS JUNCTION

(a)

n(p)

R = f Se+ + gSh+ ,

(c)

EF

K K

n(p)

EF

s N

K K K

nS junction

K


K pS junction

K

FIG. 1. (Color online) Schematic diagram of a n(p)S junction. The yellow and the blue blocks denote a local gate voltage electrode and an s-wave superconductor electrode which are deposited on the surface of the single-layer MoS2 , respectively. By the proximity effect, a superconducting gap s is induced in the single-layer MoS2 . Schematic representation of energy bands for nS and pS junctions corresponds to left and right figures in the lower panel, respectively. Black (blue) curves denote spin-up (-down) bands. Black dashed line represents the Fermi energy EF measured from the middle of the gap s .

n(p)−

(2)

where r n(p) and rA are the amplitudes of normal and Andreev reflections in the left n- (p-) type lead, respectively, and f and g are the corresponding transmission amplitudes of electronlike and holelike quasiparticles in right S. The wave functions in Eq. (2) can be obtained directly by solving Eq. (1). In the left n lead, the wave functions are given by e 2iatq ∓ kns e , 1, 0, 0 e±ikns x+iqy , ne± = N − 2ε − 2μ (3) h h 2iatq ± kns nh± = 0, 0, , 1 e∓ikns x+iqy , N + 2ε − 2μ e(h) = −4a 2 t 2 q 2 − [N − 2(±ε + μ)]χ± /2at with where kns χ± = [N + 2(±ε − λ + μ)]. ne± and nh± are the wave functions traveling along the ±x directions for electron and

195445-2

PURE VALLEY- AND SPIN-ENTANGLED STATES IN A . . .


−π/2

where G0 = e N (eV)/ h is the ballistic conductance of metallic MoS2 , θ is the incident angle, V is the bias voltage, and N (eV) = (EF + eV)w/(π vF ) denotes the number of available channels for a MoS2 sample of width w. Note that we have set ε = eV. 2

B. Numerical results and discussion

We are now ready to present the results of calculations of the zero-temperature tunneling conductance using Eq. (6). Experimentally, both n-type [28,29] and p-type [30,31] current polarity has been reported in ultrathin MoS2 flake deposited on SiO2 . Recently, ab initio calculation [33] suggests that the conductive properties of MoS2 are mainly determined by the interaction between the MoS2 layer and the substrate. Moreover, doping the substrate can be regarded as a viable strategy for engineering MoS2 -based devices. Thus, we present our numerical results for nS structure with n-doped normal metal and pS structure with p-doped normal metal. In effect, the low-energy electronic structure of monolayer MoS2 exhibits two degenerate valleys (K,K ) locating at the corners of the hexagonal Brillouin zone. As in graphene, those two inequivalent and degenerate valleys K and K in the Brillouin zone are related to each other by time-reversal symmetry. As a result, only the K valley is taken into account in the calculation and the other valley can be added in a straight manner. The parameters used in the calculations are N = 1.66 eV, t = 1.1 eV, λ = 75 meV, a = 0.3193 nm, V = 0, and s = 1 meV. It is worthwhile to note that although the parameters follow from the material MoS2 , the results would be also applicable to the other valley- and spin-coupled systems, i.e., the other group-VI dichalcogenides. Figures 2(a) and 2(b) show the

2

(a)

(b) in n type

G/G0

hole, respectively. Similarly, if the left lead is in p type, we have e 2iatq ± kps e pe± = , 1, 0, 0 e∓ikps x+iqy , N − 2ε + 2μ (4) h 2iatq ∓ kps h ±ikps x+iqy e ph± = 0, 0, , ,1 N + 2ε + 2μ e(h) where kps = −4a 2 t 2 q 2 − [N − 2(±ε − μ)]δ± /2at with δ± = [N + 2(±ε − λ − μ)]. In the S lead, the wave functions can be expressed in the following form: e Se± = χ1e , χ2e , χ3e , χ4e e±ikSs x+iqy , (5) h Sh± = χ1h , χ2h , χ3h , χ4h e∓ikSs x+iqy , e(h) where kSs = −4q 2 − [( − 42 ) ∓ ]/a 2 t 2 /2 with = (N − 2μ)(N − 2ϑs λ + 2μ) and = 4 (ϑs λ − 2μ)2 . The coherent factors in the wave functions are shown in the Appendix. Applying the continuity boundary conditions of the wave functions at the boundary L (0) = R (0), the amplitudes r n(p) n(p) and rA can be obtained directly. By means of the BlonderTinkham-Klapwijk formula, the zero-temperature tunneling conductance thus is calculated straightforwardly [32]: π/2 n(p) 2 2 Gn(p) (eV) = G0 1 − |r n(p) | + rA cos θ dθ, (6)

1.0

in p type

0.5 1 spin up spin down

spin up spin down

U=1eV

U=0eV 0.0 0.50

0.75

EF/

1.00 N

0.50

0.75

EF/

1.0

N

FIG. 2. (Color online) Dependence of the Andreev conductance on the magnitude of the chemical potential for nS and pS is plotted in (a) and (b), respectively. Here, N = 1.66 eV, t = 1.1 eV, λ = 75 meV, a = 0.3193 nm, V = 0, and s = 1 meV. The other parameters are shown in the figure. Solid lines correspond to spin-up electron and dashed lines to spin-down electron.

zero-bias tunneling conductance as a function of the Fermi energy for nS and pS junctions, respectively. Solid lines correspond to spin-up and dashed lines to spin-down electrons. It is seen clearly from the figures that the different features appear for two kinds of structures. First, as shown in Fig. 2(a), zero-bias tunneling conductances of the nS junction for spin up increase rapidly with increasing Fermi energy, while, in the pS junction, the spin-up conductances show a much slowly increasing feature as a function of Fermi energy. Second, the striking distinctness is shown in the case of spin down for the two kinds of structures. The spin-down conductance of nS junction is identical to the spin-up case, which is in sharp contrast to the phenomenon in the case of pS junction, in which the spin-down conductance remains nearly constant (zero) for a range of Fermi energy and shows a slowly increasing function. Third, an excess conductance value 2 can be reached in the nS junction and is almost double the value in the pS junction. Note that those intriguing phenomena are not attainable in a graphene-based junction, where the low-energy excitation quasiparticles are invariant under time reversal and keep the electron-hole symmetry [26]. The physical origin of those phenomena can be given as follows. As shown in Fig. 1, due to a loss of the inversion symmetry in the monolayer case, spin-orbit coupling induces spin splitting of up to 150 meV at the valence band. Time-reversal symmetry leads to spinand valley-coupled effects and requires that the spin splitting at two valleys K and K must be opposite. As well known, the Andreev reflection peculiar to inhomogeneous superconductors can be described, using pure electron language, as the two electrons (with opposite spin) going into S and eventually joining the condensate [32]. As in graphene, an electron in one valley is coupled to another electron in the other valley due to the time-reversal operator which interchanges the valleys in the single-layer MoS2 . Because of the valley-contrasting spin splitting in the valence band, the valley- and spin-dependent Andreev reflection becomes spin related different in the spin-splitting energy region, as illustrated in Fig. 2(b). In the spin-splitting energy region, a spin-up (-down) electron

195445-3


2.0

(a)

G/G0

spin up (down) 1.5 in n type EF=0.55 N

U=10eV U=1eV U=0.5eV U=0eV

PHYSICAL REVIEW B 90, 195445 (2014) 1.5

2 U=10eV U=1eV U=0.5eV U=0eV

(b) spin up in p type EF=0.55

N

(c)

U=10eV U=1eV U=0.5eV U=0eV

spin down in p type 1.0 EF=0.55 N

1 1.0 0.5 0.5

0.0

0.5

1.0

eV/

1.5

0 2.0 0.0

0.5

s

1.0

eV/

1.5 s

0.0 2.0 0.0

0.5

1.0

eV/

1.5

2.0

s

FIG. 3. Andreev conductance of a n(p)S junction as a function of incident energy eV for nS and pS is plotted in (a) and (b), respectively. The parameters are similar to Fig. 2 and the differences are shown in the figure.

from K (K ) valley must combine a spin-down (-up) electron from K (K) valley and jumps into S. Considering an electron incident from K valley, a spin-down electron is forbidden in the spin-splitting energy region while a spin-up Andreev reflection can happen. Beyond the spin-splitting energy region, both spin-resolved Andreev reflections are permitted. On the other hand, in the conduction band, as shown in Fig. 1, the dominant contribution of the lowest-energy state cancels the spin-orbit splitting, leading to an overlooked value. This means that the strength of the Andreev reflections for the spin-up and the spin-down electrons will equate to each other, as shown in Fig. 2(a). As for the different slopes of the increasing function, they can be elucidated by the nonzero potential barrier strength in the case of pS junction where Andreev reflection is suppressed at any angle of incidence. Meanwhile, the nonzero potential results in a half-excess value of zero-bias tunneling conductance as compared to the case of the nS junction. Those are why we can find the distinct features between the two kinds of junctions in Figs. 2(a) and 2(b). Next, we briefly explore the effect of electrostatic potential in S on the tunneling conductance. Figure 3(a) plots tunneling conductance of a nS junction as a function of bias voltage for different electrostatic potential strengths. Figures 3(b) and 3(c) represent the results of a pS junction for spin-up and -down electrons, respectively. The parameters are similar to those in Fig. 2 and the differences are shown in the figure. In Fig. 3, we find that the tunneling conductance at the gap edge reaches a maximum 2 independent of the electrostatic potential strength and structure of the junction (nS or pS). It is also seen that tunneling conductance changes significantly by varying the electrostatic potential strength. This indicates that the tunneling conductance can be manipulated electrically by a gate voltage. Moreover, a noteworthy different feature can be found that the electrostatic potential has an inverse effect on the tunneling conductance between the nS and pS junctions, which is attributed to quasiparticles transport through the junction involving only the conduction band or both conduction band and valence band. Besides, the different subgap tunneling conductances between nS and pS junctions can be understood in a similar way as that in Fig. 2.

III. A nSp TRANSISTOR A. Theory and model

In the above, the issue we tackle is the local tunneling conductance in nS and pS junctions. Next, we calculate the nonlocal conductance in a nSp transistor, which is the main focus of our work. In general, LAR and CAR can occur through a NSN three-terminal transistor. In particular, the time reversal of CAR process corresponds to the splitting of a Cooper pair from the superconductor into two entangled electrons tunneling into the two different normal electrodes. The spatially separated electrons without destroying the entanglement are ideally useful for quantum information purposes. However, the main drawback of such NSN transistor is the unavoided process EC in the right nonlocal electrode. Therefore, to detect the entangled state, shot-noise measurement must be demanded due to the crossover between CAR and EC in the nonlocal electrode. More recently, theoretically feasible realization of a pure CAR by a bipolar graphene-based transistor has been proposed. On the other hand, a very useful route in experiment and theory by employing the energy to discriminate CAR has been suggested. In this work, we investigate what happens to CAR in a nSp transistor when a single layer of MoS2 replaces the normal material or graphene layer. Our motivation comes from the following fact. As in graphene, the potential strength of MoS2 fleet can be very efficiently and independently tuned by the application of a gate voltage which is very amenable to experiments. In contrast to a pristine graphene, the band-gap problem can be overcome by using the single layer of MoS2 . It is therefore suggested to be used for optoelectronic applications and regarded as one of the most promising candidates for post-silicon electronics. Besides, it also has intriguing features relating to the coupled spin and valley freedom, for example, the possibility of controlling the spin and valley freedom in a single material. This indicates that the valley freedom can be manipulated magnetically, leading to a combination of valleytronics and spintronics. We expect that those characteristics of MoS2 will facilitate

195445-4

PURE VALLEY- AND SPIN-ENTANGLED STATES IN A . . . (a)

n

p

S

single layer MoS2 x

l

0 (b) n-type S nSp junction


(c) p-type S nSp junction

EF

EF

N

K K K K K K

K K

KK K K

FIG. 4. (Color online) Schematic diagram of a nSp transistor in the top panel. In the lower panel, left and right figures correspond to the schematic representation of energy bands for nSp with n-type and p-type S electrodes, respectively. By applying gate voltages, the bands in each electrode can be tuned with respect to the Fermi energy. In a nSp junction, it is possible to achieve a perfect nonlocal entangled states splitting.

the observation of pure valley- and spin-entangled states, thus, paving the way for their wider use in valleytronics and spintronics. In Fig. 4, we sketch our proposed MoS2 -based nSp transistor. The n and p regions occupy x < 0 and x > l, respectively, while the S region occupies 0 < x < l. The conductive properties of different regions can be achieved by the same route as the above single junction. The potential profile U (x) in the three regions may be adjusted independently by a gate voltage or doping, which is taken as ⎧ ⎨0, x < 0 U (x) = −U, 0 < x < l ⎩V , x > l. p In order to solve the transport problem in our nSp transistor (sketched in Fig. 4), we assume that a spin-up K valley electron with energy ε and transverse momentum q is incident from the left n lead. Taking into account both Andreev and normal reflection processes in the n lead (x < 0) and both elastic cotunneling and Andreev transmission processes in the p lead (x > l), the wave functions in the three regions can then be written as 1 = ne+ + r n ne− + rAn nn− , 2 = eSe+ + f Se− + gSh+ + hSh− ,

(7)

3 = tpe+ + t ph+ , where r and rA are the amplitudes of normal reflection and LAR, respectively; t and t are the amplitudes of EC and CAR in the p regions; e, f , g, and h are the amplitudes of electronlike and holelike quasiparticles in the S region. The wave functions are the same as those in Sec. II. All the amplitudes in Eq. (7) can be obtained by the following boundary conditions: 1 (0) = 2 (0), 2 (l) = 3 (l).

(8)

If we take l ξ (ξ = v F /s = at/s is the superconductor coherence length), we recover the results of the transmission and reflection amplitudes in the above nS junction. Note that

ξ ≈ 350 nm is for a typical superconductor electrode with s ≈ 1 meV. Since the gate potentials can be adjusted independently in the present transistor, we assume the energy spectrum in p-type lead can be tuned by Vp and does not equate to that in the left lead. By Vp , we can adjust the position of the Fermi energy in p-type lead always set on the top of the valence edge which results in a zero contribution of EC to the nonlocal conductivity and pure valley- and spin-entangled conductivity. Furthermore, n-type and p-type S can be realized by tuning the magnitude of the chemical potential of S through U . Experimentally, n-type MoS2 -based and p-type MoS2 -based transistors were recently demonstrated [28–31]. It indicates that the settings of both nSp transistor with n-type S and with p-type S envisaged here are experimentally accessible. After all the amplitudes in Eq. (7) are obtained, the nonlocal conductivity in this study is given by the following formula, i.e., only come from the crossed Andreev conductivity (GCAR ) [24] π/2 2 |t | cos(θ )dθ. (9) GCAR = G0 −π/2

Using Eq. (1), conductivity GCAR for our three-terminal transistor can be obtained easily by the numerical calculations. B. Numerical results and discussion

Indeed, an ideal setup for pure valley- and spin-entangled states without AR and EC can be fulfilled. Making use of energy gap, it is possible to cancel both AR and EC individually as shown in Fig. 4. The essential thing here is that quasiparticle wave functions of EC and AR decay exponentially in the forbidden energy gap. This means that pure CAR without AR and EC can be obtained for all positive bias voltages in the superconductor gap. In practice, due to the vanishing density of states at the bottom of conduction band edge, we slightly lift the Fermi energy in the left n electrode, whereas the doping level of the right p electrode always perfectly aligns with the top of valence band edge. EC is therefore prohibited, and a pure CAR current is possible. In this section, we always set EF in the left electrode as 0.502N and EF in the right electrode as 0.5N − λ. The parameters are the same as those in Sec. II. In Fig. 5, we show results for the GCAR as a function of eV and potential strength U for l/ξ = 1. Figure 5(a) shows the bias-voltage dependence of the GCAR for a n-type superconductor nSp transistor with various values of the potential strength U . We should note that GCAR begins to oscillate with U right after the incident energy eV not equating to the zero. In particular, the generated oscillation of GCAR becomes more strenuous with the increase of eV and U . Physically, we can regard the oscillating behavior as a consequence of interference effects in the S between electronlike and holelike quasiparticles. The oscillation period is detere h mined by the resonant condition: (kSs − kSs )l cos(θ ) = 2nπ e(h) with kSs the wave vector in the S region and n an integer. Thus, the oscillation is not only determined by the incident energy eV, also related to the values U and EF in the S interlayer. For a very small value of eV in the S region, the amplitudes of electronlike and holelike quasiparticles are negligible so that the oscillation is not pronounced. With the increasing U in

195445-5


(a) 0.1 0.000 0.001250 0.002500 0.003750

U(eV)

0.005000 0.006250 0.007500 0.008750 0.01000

0.08 0

eV/

1.2 s

(b) - 1.688

0.000 0.01500 0.03000 0.04500

U(eV)

0.06000 0.07500 0.09000 0.1050 0.1200

- 1.668 0

eV/

s

1.2

FIG. 5. (Color online) Crossed Andreev conductivity as a function of eV and potential strength U for l/ξ = 1. (a) and (b) correspond to n-type and p-type superconductors, respectively.

the S region, the mismatch of Fermi surface which acts as an effective barrier for the quasiparticles tunneling at the interface becomes bigger and it makes oscillation much stronger. As a result, the oscillating behavior shows an increasing function of eV and U . However, the situation is different in a p-type superconductor nSp transistor which gives rise to two explicitly different Fermi velocities and momenta for the Cooper pairs in the p-type superconductor region. Making an assumption that in S electrode residents in the spin-splitting energy gap as shown in Fig. 4(c), only one type of Cooper pairs can generate in the S electrode. Beyond the spin-splitting energy gap, the present p-type superconductor region has properties similar to the n-type superconductor region. The situation becomes very intriguing by including the intervalley scattering. In this situation, the present p-type superconductor region (the EF in S electrode beyond the spin-splitting energy gap) has properties similar to biaxial crystals: a quasiparticle state incident from the left n electrode splits at the nS interface and the two resulting quasiparticle states propagate in the S region with different momenta. This effect, similar to the optical double refraction, produces an extra interference pattern when the quasiparticle state transfers into the right p electrode. On another hand, both spin-flip and spin-conserving intervalley


scattering are permitted for quasiparticle state at the Fermi level in the left n lead [34], while only a spin-flip intervalley scattering process is possible in the right p lead due to intrinsic spin splitting in the valence band [34]. Opening intervalley and spin-flip scattering channel reduces the entanglement of the valley- and spin-entangled states. For a gradual increase of intervalley and spin-flip scattering, it may drive the system out of the valley- and spin-entangled states. However, intervalley and spin-flip scattering is suppressed in pure samples as revealed in graphene [35]. We therefore study the relevant case without the intervalley and spin-flip scattering. As obviously expected, it is clearly seen from Fig. 5(b) that GCAR exhibits a similar interference pattern as a function of eV. However, the noteworthy thing is that the oscillation amplitudes for a p-type superconductor nSp transistor are 12 times larger than that for a n-type superconductor nSp transistor. Such feature can be elucidated by the competition between LAR and CAR in the structure. Here, we adjust the position of the Fermi energy in p-type lead as always set on the top of valence edge which results in a zero contribution of EC to the nonlocal conductivity. Therefore, for an electron incident from the left n lead, it may combine another electron from the same lead (a LAR process) transferred into a superconducting condensate as a Cooper pair or may involve another electron from the other spatially separated p lead (a CAR process) to the same charge transfer as in a LAR process to a Cooper pair in the superconductor. In this way, CAR occurs in competition with LAR. On the other hand, as shown in Sec. II, the value of LAR in a n-type superconductor nSp transistor must be bigger than that in a p-type superconductor nSp transistor because the quasiparticle states contributing to the transport in the two cases are different. In a n-type superconductor nSp transistor, the quasiparticles (both electronlike and holelike) in the n lead all transmit in the conduction band. Meanwhile, the quasiparticles in the n-type superconductor originate from the same conduction band. As a result, the quasiparticle’s conversion between n (Andreev bound states) and S (Cooper pairs) can be served as a classical motion, at least from a point of view of the transmission, while in a p-type superconductor nSp transistor, in contrast, those supercurrent carrying quasiparticles in S come from the valence band. Thus, the quasiparticle’s conversion between n (in the conduction band) and S (in the valence band) originates form the different bands amounting to a Klein tunneling. Therefore, the different quasiparticle types in S result in distinct strengths of the LAR. As a consequence, the difference of LAR processes between the two transistors then allows 12 times different oscillation amplitudes of GCAR as depicted in Fig. 5. The GCAR is not only an oscillation function of U , as seen in Fig. 6, but also quite sensitive to the precise value of l at a fixed U . Figures 6(a) and 6(b) show GCAR as a function of eV and l for a n-type superconductor nSp transistor and p-type superconductor nSp transistor, respectively. The parameters which are different from Fig. 5 are shown in Fig. 6. It is shown that GCAR can be modulated largely by l with a big bias voltage eV and the resonant peaks can be yielded. Although there are similar oscillation phenomena of GCAR for the two structures, a difference can be also found. It is shown that the maximum of oscillation amplitude of a p-type superconductor

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PHYSICAL REVIEW B 90, 195445 (2014) 0.02

(a)

0.02

U=0.4eV eV=0.1 s

(a)

l/

GCAR/G0

1 0.01

1

GCAR/G0

eV/

1 .2 s

1

3

4

0.00 5 0.20

1

2

1

2

3

4

5

4

5

(d)

eV=0.1 s EF=0.502 N EFL=0.5 N-

0.1

0.0 0

(b)

2

s

0.01

U=-2eV

(c)

0

eV=0.7

EF=0.502 N EFL=0.5 N-

0.00 0.20

0 .9 5

(b)

3

eV=0.7

s

0.1

4

5

0.0 0

l/

1

2

3

l/

l/

FIG. 7. Crossed Andreev conductivity as a function of the superconducting length with U = 0.4 eV and U ≈ −2 eV corresponding to n-type and p-type superconductors, respectively. The other parameters are the same as Fig. 6.

0 .9 5 0

eV/

s

1 .2

FIG. 6. (Color online) (a) and (b) show the crossed Andreev conductivity as a function of eV and l for cases of n-type (U = 0.4 eV) and p-type (U = −2 eV) superconductors nSp transistor, respectively.

nSp transistor is almost six times larger than the case of a n-type superconductor. This oscillation phenomenon can be interpreted by the coherence stemming from the change of l (similar to the variety of the potential strength U ). Besides, the distinct strength of the oscillation amplitude is similar to Fig. 5 and can also be ascribed to the same or different band of the quasiparticle involved in the transport process. In the following, we want to show how the fate of the GCAR depends on the length of the superconducting region l in a large range. The parameters are the same as in Fig. 6. As seen in Fig. 7, the upper panels correspond to the n-type superconductor nSp transistor for a small potential strength (e.g., U = 0.4) while the lower panels correspond to the p-type superconductor nSp transistor for a small potential strength [e.g., U ≈ −2 corresponds to −(0.5N − λ) − 0.4]. In both cases, GCAR varies strongly when considering different l/ξ due to the fast oscillations which correspond to the resonant transmission levels in S. Also, as a function of the length l we find that both GCAR vanish for small and large values. Upon increasing eV, the resonant peaks of GCAR are shown by a large enhancement and the maximum of GCAR can be tuned to l ≈ ξ/2. The remarkable feature, clearly seen in Fig. 6, is in sharp contrast to the conventional superconductor junction where GCAR always exhibits an exponential decay behavior as a function of l/ξ [36]. On the other hand, compared to the graphene case (the GCAR current is maximum around ξ ) [37], the novel aspect of the present junction is the resonant peaks maximum central around ξ/2. The novel behavior of GCAR

may be understood as follows. Nonrelativistic quasiparticles with parabolic dispersions which obey the Schr¨odinger equation play a major role in the studies about the conventional superconductor junction. Instead, relativistic quasiparticles with linear dispersions which obey the Dirac equation have a main contribution to the transport properties in a graphene superconductor junction. However, the intriguing characteristics feature can be found in the single-layer MoS2 . In general, a parabolic dispersion is given around the conduction and valence band edges, similar to conventional nonrelativistic quasiparticle cases. However, the quasiparticle excitations also host chirality, similar to relativistic quasiparticles in graphene. At a certain point, the single-layer MoS2 amounts to a bilayer graphene without the energy gap between the conduction and valence bands. Thus, in the single-layer MoS2 , quasiparticle transport properties are subject to the competitions between the parabolic dispersions and chirality of the quasiparticle excitations. In particular, the chirality manifests itself in CAR as the novel length dependence character as shown in Fig. 6. Moreover, it also possesses a similar feature as Fig. 6 and the enhancement of the oscillating amplitude in a p-type superconductor nSp transistor can be understood as above. IV. SUMMARY

To conclude, we have investigated LAR and CAR in the n(p)S single junction and n(p)-type superconductor nSp transistor, respectively. We have shown that LAR not only can be tuned largely by the type of the normal metal electrode, it also is related to electrostatic potential in S. In a nSp transistor, it is found that GCAR is pure valley- and spin-entangled conductivity due to the coupling between the valley and the spin degrees of freedom, when the Fermi energy aligns to the top of the valence band. In particular, an ideal nonlocal pure valley- and spin-entangled splitting without LAR and EC can be obtained based on the energy difference between the electron state and the AR hole state. Moreover, the pure valleyand spin-entangled GCAR can be tuned by the length l and

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the local application of a gate voltage. In particular, we have shown that CAR exhibits a maximum around ξ/2 instead of the exponential decay behavior in conventional superconductors and a maximum around ξ in graphene material. These effects suggest the potential of combining spintronic with valleytronic applications. Also, we can expect that our findings could give a reference to prepare a pure valley- and spin-entangled state which can be regarded as a building block for solid-state Bellinequality experiments, quantum computation, and quantum teleportation.

e(h) = where the wave vectors kSs 2 2 2 2 −4q − [( − 4 ) ∓ ]/a t /2 with

= (N − 2μ)(N − 2ϑs λ + 2μ) and = 4 (ϑs λ − 2μ)2 , χ2e± = (ε − (ϑs λ − 2μ)2 /(ϑs λ − 2μ))/S ,

χ1e± = ±[(ϑs λ − 2μ)S (−4ρ + β)]/

e , × at[(N + 2ε − 2μ)ρ + γ ] ±2iq + kSs

e± 2 χ3 = ± ρ η + 4S + (ϑs λ − 2μ) × εη + 4(ε + ϑs λ − 2μ)2S

e × at[(N + 2ε − 2μ)ρ + γ ] ±2iq + kSs ,

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grants No. U1204110, No. 11304306, and No. 61290303). This project was also supported by China Postdoctoral Science Foundation (Grant No. 2013M540126). C.B. also acknowledges partial support from Program of Young Core Teachers in Higher Education Institutions of Henan Province, China (Grant No. 2013GGJS-148).

χ4e± = 1. The parameters ρ, β, , and γ are defined by γ = (ϑs λ − 2μ) ε(N + 2ε − 2μ) − 22S , β = (N − 2μ)(N − 2ϑs λ + 2μ) − 42 , ρ = (ϑs λ − 2μ)2 , = ε2 − 2S , η = [N − 2(ε + ϑs λ − μ)](N + 2ε − 2μ), χ2h± = (ε + (ϑs λ − 2μ)2 /(ϑs λ − 2μ))/S ,

APPENDIX: CALCULATION OF THE STATES IN S

In the superconducting region, the low-energy excitation quasiparticles are described by the following Bogoliubov–de Gennes equation: Hγ s − EF (x) = E, ∗ (x) EF − Hγ s

χ1h± = ±[(ϑs λ − 2μ)(4ρ + β)]/

h × at[(N + 2ε − 2μ)ρ − γ ] ∓2iq + kSs ,

χ3h± = ∓ ρ η + 42S − (ϑs λ − 2μ) × εη + 4(ε + ϑs λ − 2μ)2S

h , × at[(N + 2ε − 2μ)ρ − γ ] ∓2iq + kSs

where Hγ s = at(γ kx σx + ky σy ) + N /2σz − λρs γ (σz − 1)/ 2 + U (x). For a particle of energy ε and transverse momentum q, the solutions of the above equation take the form e Se± = χ1e , χ2e , χ3e , χ4e e±ikSs x+iqy , h Sh± = χ1h , χ2h , χ3h , χ4h e∓ikSs x+iqy ,

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