Transmission in bilayer graphene through time-periodic potential

Physica B 456 (2015) 167–170

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Transmission in bilayer graphene through time-periodic potential HaiYang Chen School of Physics, Beijing Institute of Technology, Beijing 100081, China

art ic l e i nf o

a b s t r a c t

Article history: Received 21 July 2014 Accepted 2 September 2014 Available online 16 September 2014

The transport property of Dirac electrons through bilayer graphene under a time periodic field is investigated. The effect of external field strength and the system parameters on the transmission probabilities is investigated. The results show that the applied time-periodic potential provides sidebands for Dirac electrons to tunnel through the bilayer graphene, and transport property of system exhibits various kinds of behaviors with the change of external field strength and structure parameters. We also find that, for normal and close to normal incidence, the perfect reflection that was observed for the static barrier is found to persist for the oscillating barrier. & 2014 Elsevier B.V. All rights reserved.

Keywords: Bilayer graphene Time-periodic potential Transmission probabilities

1. Introduction Following the successful fabrication experiment by Novoselov et al. [1], the study of single-layer and bilayer graphene has been intensified. Monolayer graphene is a layer of carbon atoms with hexagonal lattice of sp2-bonded carbon atoms, its low-energy dynamics of electrons are described by the Dirac-like equation. Near the Brillouin zone the graphene is the linear energy spectrum. For bilayer graphene, composed of two layers of graphene, the electronic structure and the spectrum were essentially different from that of a monolayer [2]. The charge carriers in bilayer graphene have parabolic energy spectrum, which means that they are massive quasiparticle, similar to the conventional nonrelativistic fermions. Graphene exhibits a number of unusual electronic and transport properties, for example, half-integer and unconventional Hall effect [2–5]. In the past few years, electronic structure [6–11], optical excitations [12–16], many-particle interactions [17– 20], and transport properties [21–24] have been investigated and achieved many interesting results. Photon-assisted tunneling (PAT) has been attracting much research interest. In the presence of an additional time-periodic potential, V cos ωt , an electron of energy E without microwave fields will generate many sidebands which have energies E 7 lℏω ðl ¼ 0; 7 1; 7 2; …Þ due to the exchange of energy between the electrons and the time-periodic potential, and the wave function will contain the Bessel function. Dayem and Martin reported evidences of absorption or emission of photons by a single tunneling electron according to the experiments on the tunneling between superconducting films in the presence of microwave fields [25]. Tien and Gordon explained qualitatively the

E-mail address: [email protected] http://dx.doi.org/10.1016/j.physb.2014.09.002 0921-4526/& 2014 Elsevier B.V. All rights reserved.

multiphoton-assisted electron tunneling current in superconducting diodes by calculation [26]. Zeb et al. studied the transport of Dirac electrons in monolayer graphene through a single barrier with a time-periodic potential, they find that at normal and close to normal incidence the system shows perfect transmission (Klein tunnelings) because of chiral nature of the particles [27]. In a series of papers, Wagner studied photon-assisted transport through quantum barriers and wells with driving V cos ωt based on a transfer-matrix formalism [28–30]. In this paper, the transport property of charge carriers through bilayer graphene with a time periodic field is investigated. We study the transmission probabilities of central band and sidebands as external field strength, incident angle, barrier width and height are changed. Our numerical results demonstrate that the applied time oscillating field provides sidebands for charge carriers to tunnel through the bilayer graphene, and transport property of system exhibits various kinds of behavior with the change of external field strength and structure parameters of the system. We also find the perfect transmission for normal incidence and close to normal incidence that has been reported for a static barrier that persists for the oscillating barrier.

2. Theory and model In the present study we consider bilayer graphene sheet in the xy plane. The square potential barrier is set up in the x-direction while carriers are free in the y-direction. The width of the barrier is D and the height of the barrier is oscillating sinusoidally around V with amplitude V1 and frequency ω. Charge carriers with energy E are incident from one side of the barrier in graphene with an angle ϕ0 with the x-axis. the Hamiltonian H of the system is as

168

H. Chen / Physica B 456 (2015) 167–170

follows: H ¼ H0 þ H1

ð1Þ

among them H0 ¼ 

ðkx  iky Þ2

0 ℏ2 2m ðkx þ iky Þ2

!

ð12Þ þV

0

ð2Þ

H 1 ¼ V 1 cos ðωtÞ

ð3Þ

V and V1 are the static square potential barrier and the amplitude of the oscillating potential, respectively. Here the effective mass m is 0.035 me, where me is the bare electron mass. By solving the equation in the absence of the oscillating potential [31], the solution to Eq. (2) can be expressed as h

1

ψ 1 ðx; y; tÞ ¼ eiky y ∑

l ¼ 1



 δi;2

i ai;l eiki;l x þ bi;l e  iki;l x þ ci;l eiκi;l x þ di;l e  iκi;l x e  iðE þ lℏωÞt=ℏ 

1

∑

m ¼ 1

1

Jm

  V1 e  imℏωt=ℏ þ ð1  δi;2 Þ ℏω



ψ 2 ðx; y; tÞ ¼ eiky y ∑ si;l ai;l eiki;l x þ 2iϕi;l þbi;l e  iki;l x  2iϕi;l ci;l hi;l eiκi;l x þ l ¼ 1

e  iðE þ lℏωÞt=ℏ



δi;2

1

∑

m ¼ 1

  1 V1 s1;n ðδn;0 ie2iϕ1;n k1;n  ie  2iϕ1;n b1;n k1;n  c1;n h1;l κ 1;l Þ ¼ ∑ J n  m ℏω m ¼ 1   d2;m κ 2;m  s2;m e2iϕ2;m a2;m ik2;m  e  2iϕ2;m b2;m ik2;m  c2;m h2;m κ 2;m þ h2;m

ð4Þ di;l  iκi;l x e hi;l

  V1 e  imℏωt=ℏ þð1 δi;2 Þ ℏω

 Jm



ð5Þ

where J m ðV 1 =ℏωÞ is the mth-order Bessel function, δi;2 is the Dirichlet function. i ¼ 1; 2; 3, representing different regions x o 0; 0 o x o a; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 4 a, si;l ¼ sgnðV i  E lℏωÞ; ky ¼ E sin ϕ0 , ℏki;l ¼ 2mjE V i þ lℏωj cos ϕi;l , q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ℏky ¼ 2mjE V i j sin ϕi;l , κ ¼ ki;l þ 2ky , ϕi;l ¼ tan  1 ðky =ki;l Þ and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hi;l ¼ ð 1 þ sin 2 ϕi;l  sin ϕi;l Þ2 . Applying the continuity of the wave function and their derivatives at the boundaries x ¼ 0; D, i.e., ψ 1 ð0; y; tÞ ¼ ψ 2 ð0; y; tÞ; ψ 2 ðD; y; tÞ ¼ ψ 3 ðD; y; tÞ, we obtain the coupled equations in form of infinite series:   1 V δn;0 þb1;n þ c1;n ¼ ∑ J n  m 1 ða2;m þ b2;m þ c2;l þ d2;m Þ ð6Þ ℏω m ¼ 1 ik1;n δn;0  ib1;n k1;n þc1;n κ 1;n   1 V1 ¼ ∑ Jn  m ðia2;m k2;m  ib2;m k2;m þ c2;m κ 2;m  d2;m κ 2;m Þ ℏω m ¼ 1 ð7Þ s1;n ðe2iϕ1;n δl;0 þ e  2iϕ1;n b1;n  c1;n h1;n Þ     1 V1 d2;m s2;m e2iϕ2;m a2;m þ e  2iϕ2;m b2m  c2;m h2;m  ¼ ∑ Jn  m ℏω h2;m m ¼ 1

  e  Dk1;n d3;n k1;n s3;n ieiDk1;n þ 2iϕ1;n a3;n k1;n þ h3;n    1 V1 ¼ ∑ Jn  m s2;m e2iϕ2;m þ Dik2;m a2;m ik2;m ℏ ω m ¼ 1  e  2iϕ2;m  Dik2;m b2;m ik2;m  eDκ2;m c2;m h2;m κ 2;m þ

e  Dκ 2;m d2;m κ 2;m h2;m

 ð13Þ

Usually, these coupled equations can be truncated into finite number of terms starting from  N up to NðN 4V 1 =ℏωÞ [27], the transmission probability for the n-th sideband Tn be obtained numerically.

3. Numerical results and discussion In this study, we have determined the numerical results for the Dirac electrons in graphene, corresponding to the incident electron wavelength of λ ¼ 50 nm, barrier width of a ¼ 5 nm, barrier height of V¼50 mev and oscillation frequency of ω ¼ 5  1012 Hz. Fig. 1 shows transmission probabilities of the central band and first sidebands as a function of αðα ¼ V 1 =ℏωÞ for the incident angle π =8 . Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. It can be observed that the transmission probability of central band T0 is dominated for smaller values of α. This is because the barrier can be seen as a static barrier when the values of α are lower. With the increase of α, the transmission probabilities of various sidebands become significant, indicating the contribution of time-periodic potential on the sidebands of electron in the graphene. This is similar to the typical characteristics of single barrier under a time-periodic potential [27]. Fig. 2 presents transmission probabilities of the central band and first sidebands as a function of the incident angle. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. We find, for normal and close to normal incidence, perfect reflection that was observed for the static barrier is found to persist for the oscillating barrier and spread over the central band and sidebands. Analytical expressions for the transmission 1.0

ð8Þ s3;n

 e  Dk1;n d3;n e a3;n  h3;n    1 V1 s2;m e2iϕ2;m þ Dik2;m a2;m þ e  2iϕ2;m  Dik2;m b2;m ¼ ∑ Jn  m ℏ ω m ¼ 1  e  Dκ 2;m d2;m eDκ 2;m c2;m h2;m  ð9Þ h2;m

0.8

iDk1;n þ 2iϕ1;n

ik1;n δn;0  ib1;n k1;n þc1;n κ 1;n   1 V1 ðia2;m k2;m  ib2;m k2;m þ c2;m κ 2;m  d2;m κ 2;n Þ ¼ ∑ Jn  m ℏω m ¼ 1 ð10Þ ieiDk1;n a3;n k1;n  e  Dk1;n d3;n k1;n   1 V1 ¼ ∑ Jn  m ðeDik2;m a2;m ik2;m ie  iDk2;m b2;m k2;m ℏω m ¼ 1 eDκ 2;m c2;m κ 1;m  e  Dκ1;m d2;m κ 2;m Þ

ð11Þ

0.6 T



0.4 0.2 0.0

0

2

4

6

8

10

V1 Fig. 1. Transmission probabilities of the central band and first sidebands as a function of αðα ¼ V 1 =ℏωÞ for the incident angle π=8. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

H. Chen / Physica B 456 (2015) 167–170

1.0

0.8

0.8

0.6

0.6 T

T

1.0

169

0.4

0.4

0.2

0.2

0.0

50

0

0.0

50

0

20

40

60

80

100

V mev Fig. 2. Transmission probabilities of the central band and first sidebands as a function of the incident angle. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 4. Transmission probabilities of the central band and first sidebands as a function of the barrier height for the incident angle π=8. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

find that applied time-dependent fields provide more transmission sidebands for electron to tunnel through the barrier. Furthermore, as evidenced from Figs. 1–4, one could control the transmission probability of the system by adjusting the external field strength αðα ¼ V 1 =ℏωÞ, incident angle, barrier width and barrier height.

1.0 0.8

T

0.6

4. Conclusion

0.4 0.2 0.0

0

10

20

30

40

50

D nm Fig. 3. Transmission probabilities of the central band and first sidebands as a function of the barrier width for the incident angle π=8. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

coefficient under the time oscillating field are complicated. But we can understand how they occur, according to the static case, transmission coefficient t ¼ 4ik1 k2 =ððk2 þ ik1 Þ2 e  k2 D  ðk2  ik1 Þ2 ek2 D Þ, it shows that the transmission coefficient should be perfect reflection for normally incident, and this contrasts to single-layer graphene, where electrons are always perfectly transmitted [31]. Fig. 3 we present the transmission probabilities of the central band and first sidebands as a function of the barrier width for the incident angle π =8. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. As shown in the figure, the central band T0 is initially dominant when the barrier width is small, and the higher sidebands rise as the barrier width increases. This occurs as the increase in the barrier width allows effective interaction of electrons with more phonons. Fig. 4 displays the transmission probabilities as a function of the barrier height for the incident angle π =8. Green, blue, red, purple, orange lines correspond to T  2 , T  1 , T0, T1, T2, respectively. As is seen, the central band T0 shows oscillatory but damped behavior as the barrier height increases. This is plausible according to static case, for the limit of high barriers V b E, the transmission probability is T ¼ ðE=VÞ sin 2 ð2ϕÞ [31]. That is to say, the amplitude of transmission decreases with increasing barrier height. We also

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