The influence of the optical Stark effect on chiral tunneling in graphene

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by  Jiang-Tao Liu

The influence of the optical Stark effect on chiral tunneling in graphene
Jiang-Tao Liu,1,∗ Fu-Hai Su,2 Hai Wang,3 and Xin-Hua Deng1
1Department of Physics, Nanchang University, Nanchang 330031, China
2Key Laboratory of Materials Physics, Institute of Solid State Physics,
Chinese Academy of Sciences, Hefei 230031, China
3Department of Physics, Capital Normal University, Beijing 100037, China
(Dated: February 1, 2011)
Theinfluencesofintensecoherentlaserfieldsonthetransportpropertiesofasinglelayergraphene
are investigated by solving the time-dependent Dirac equation numerically. Underan intense laser
1
field, the valence band and conduction band states mix via the optical Stark effect. The chiral
1
symmetryofDiracelectronsisbrokenandtheperfectchiraltunnelingisstronglysuppressed. These
0
properties might beuseful in thefabrication of an optically controlled field-effect transistor.
2

n PACSnumbers: 42.65.-k,68.65.-k,73.40.Gk
a
J

1 Graphene has attracted much attention due to its re- of Dirac electrons will be completely changed, or even
3 markable electronic properties [1–3]. The low-energy disappear. Unlike the resonant case [13], in OSE the

quasiparticles, which have linear dispersion and nontriv- coherent excitons are virtual excitons, which exist only
]
r ial topological structure in their wave function, can be when the optical field is present. Thus the light-induced
e
described by using a Dirac-like equation. This unique shiftlastsonlyforthedurationofthepumppulse,which
h
t bandstructure of grapheneleads to many importantpo- allowsforopticalgatesthatmightonlyexistforfemtosec-
o
tential applications in nanoelectronics [4–9]. onds. Furthermore, since there is no real absorption in
.
at One of the peculiar transport phenomena in graphene the nonresonant case, the absorption of photons is quite
m is the chiraltunneling [4,5, 10]. In single layergraphene small and low power consumption is expected.
- a perfect transmission through a potential barrier in the In this Letter, we study the tunneling rate of Dirac
d normal direction is expected. This unique tunneling ef- electrons in graphene through a barrier with an intense
n
fect can be explained by the chirality of the Dirac elec- electromagnetic field. We consider a rectangular poten-
o
c trons within each valley, which prevents backscattering tial barrier with height V0, width D in the X direction,
[ in general. This kind of reflectionless transmission is in- and infinite length in the Y direction [see Fig. 1 (a) and

dependent of the strength of the potential, which lim- Fig. 1 (b)]. The Fermi level (dashed lines) lies in the
3
v its the development of graphene-based field-effect tran- valence band in the barrierregionand in the conduction
0 sistors (FET) [4]. The perfect transmission can be sup- band outside the barrier. The gray filled areas indicate
3 pressedeffectivelywhenthechiralsymmetryoftheDirac the occupied states. The optical field is propagated per-
4 electrons is broken. For instance, in a magnetic field, a pendicular to the layer surface and is linearly polarized
3
quantized transmission can be observed in graphene p- along the Y direction with a detuning ∆ = 2E ~ω.
. 0 b
7 n Junctions [11]. Recently, Elias et. al. proposed that −
0
thehydrogenationcouldconvertthesemimetalgraphene
0
into an insulator material [12].
1 (a) (b)
: Theintenseopticalfieldcanalsobreakthechiralsym- X
v B B B B
metryofDiracelectronsingraphene,e.g.,FistulandEfe- Y a in out a
i Z
X tovhaveshownthatwhenthen-pJunctionsingraphene
ω
r is irradiated by an electromagnetic field in the resonant Eb D
a condition, the quasiparticle transmission is suppressed Eb I II III
V
[13]. The optical field control on carrier transport of- E 0 (c)
k
C (i) C (i+1/2)
fers several advantages. Optical fields can control not A B
only the charge carriers but also the spin carriers, es-
X
pecially which can be performed over femtosecond time
scale. Another fundamental method of optical control is
FIG.1: (coloronline). (a)SchematicofthespectrumofDirac
the optical Stark effect (OSE) [14–18]. The OSE in tra-
electrons in single-layer graphene. The optical field is propa-
ditional semiconductors is due to a dynamical coupling
gatedperpendiculartothelayersurfaceandandislinearlypo-
of excitonic states by an intense laser field. The OSE larizedalongtheY direction. (b)Schematicofthescattering
have shown many useful applications in optoelectronics ofDiracelectronsbyasquarepotential. Ba,Bin,andBoutde-
and spintronics [19–23]. notethe absorbing boundary,incident boundary,and output
In graphene, the valence band and conduction band boundary,respectively. (c) Schematicof theone-dimensional
states can also mix strongly via OSE. Thus the chirality Yeelattice in graphene.
2
We choose ∆ > 0 to ensure that there is no interband Thus the Eq. (3) andEq. (4) canbe replacedby a finite
0
absorption inside the barrier. Meanwhile, ~ω 2E is set of finite differential equations
k
≪
used to guarantee that the influence of the optical field
outside the barrier can be neglected.
Since the Coulomb interaction between electrons and Ck+1/2(i) 1 V0(i) = 1 + V0(i) Ck−1/2(i)
A ∆t − 2i ∆t 2i A
holes in OSE is negligible when the detuning is large (cid:20) (cid:21) (cid:20) (cid:21)
[17, 19], we did not take into account the electron-hole vF V1k2(i+1/2) Ck(i+1/2)
Coulomb interaction or many body effect in our calcu- − ∆x − 2i B
(cid:20) (cid:21)
lation. Thus, neglecting the scattering between different v Vk(i 1/2)
valleys, the scattering process of Dirac electrons in K + ∆Fx + 12 2−i CBk(i−1/2), (7a)
point is described by the time-dependent Dirac equation (cid:20) (cid:21)
i~∂ Ψ(r,t)=[H0+V0(r)I+Hint]Ψ(r,t), (1) Ck+1(i+1/2) 1 V0(i+1/2) = 1 + V0(i+1/2)
∂t B ∆t − 2i ∆t 2i ×
where Ψ(r,t)=[CA(r,t),CB(r,t)] is the wave function, (cid:20) v Vk+1/(cid:21)2(i+(cid:20) 1) (cid:21)
Hσ 0==(−σi~,σvF)σ•ar∇e itshtehePuanupliermtuartbriecdesD, ivracHa1m0i6ltmon/sianis, CBk(i+1/2)−"∆Fx − 21 2i #CAk+1/2(i+1)
x y F
the Fermi velocity, V0(r) is the height of ≈the potential v Vk+1/2(i)
barrier, I is the unit matrix, and H is the interaction + F + 21 Ck+1/2(i), (7b)
int ∆x 2i A
Hamiltonian. H can write as [24] " #
int
0 V (t) Forcomputationalstability,thespaceincrement∆xand
H = ~ev [A(x,t)σ +A(y,t)σ ]=~ 12 ,
int − F x y V21(t) 0 the time increment ∆t need to satisfy the relation ∆x>
(cid:18) (2) (cid:19) vF∆t [25]. Furthermore, the space increment ∆x must
where e is the electron charge and [A(x,t),A(y,t)] = far smaller than the wavelength of electrons ∆x<λe/8,
[A eiωt,A eiωt] are the vector potentials of the electro- and the time increment ∆t must be far smaller than the
x y
magnetic field. When the Dirac electrons is incident on period of the electromagnetic field Tl.
the barrier perdenicularly, we can rewrite Eq. (1) as a At the boundary Ba, one-dimensional Mur absorbing
set of partial differential equations boundary conditions are used [26]. At the input bound-
ary B , a Gaussian electronic wave packet is injected
in
i∂CA(x,t)/∂t= −ivF∂C+BV(1x2,(tt))/C∂Bx(+x,Vt)0,CA(x,t) (3) CA =CB = √12exp(cid:20)−4π(tτ−2t0)2(cid:21), (8)
i∂C (x,t)/∂t= iv ∂C (x,t)/∂x+V C (x,t) where t and τ denote the peak position and the pulse
B F A 0 B 0
−
+V (t)C (x,t). (4) width, respectively.
21 A
Thus, by solving Eq. (7a) and Eq. (7b) directly in
Sincethetunnelingtimeissub-picosecondandthepo-
the time domain we can demonstrate the propagationof
tential V (t) and V (t) vary as fast as the frequency of
12 21 a wave packet through a barrier in real time. Numeri-
incident light beams, this scattering process is strongly
cal simulations are shown in Fig. 2. The following pa-
time-dependent. In order to study such a strongly
rameters are used in our calculation: the peak position
time-dependent scattering process, we employ the finite-
t = 1.5 ps, the pulse width τ = 1.0 ps, the space incre-
0
differencetime-domain(FDTD) methodto solveEq. (3) ment ∆x = 0.1 nm, the time increment ∆t = 5 10−5
and Eq. (4) numerically in the time-domain [25]. In the ×
ps,andtheheightofthepotentialbarrierV =400meV.
0
traditional FDTD method, the Maxwell’s equations are
When there is no pump beams, a perfect chiral tunnel-
discretized by using central-difference approximations of
ingcanbe found[see Fig. 2(a)]. Thisresultis consistent
thespaceandtimepartialderivatives. Asatime-domain
with that of Geim et. al. [4]. But when the sample is
technique,theFDTDmethodcandemonstratetheprop-
irradiated by an intense nonresonant laser beam, a re-
agationof electromagneticfields througha modelin real
flected wave packet appears [see Fig. 2(d)]. The perfect
time. Similar to the discretization of Maxwell’s equa-
transmission is suppressed. By analyzing the the trans-
tions in FDTD, we denote a grid point of the space and
mittedwavepacketandthereflectedwavepacket,wecan
time as (i,k) = (i∆x,k∆t) [see Fig. 1(c)], and for the
obtain the tunneling rate.
any function of space and time F(i∆x,k∆t) = Fk(i).
Toexplainthesuppressionofchiraltunneling,Wefirst
the firstorder in time or space partialdifferential canbe
investigate the OSE in the barrier within a rotating-
expressed as
wave approximation [15, 22, 23]. Figure 2(a) shows the
∂F(x,t) Fk(i+1/2)−Fk(i−1/2), (5) renormalized band as a function of momentum k with
∂x |x=i∆x ≈ ∆x intensity I = 30 MW/cm2. In the case of nonreso-
ω
∂F(x,t) Fk+1/2(i)−Fk−1/2(i). (6) nant excitation, ~ω < 2E and the dressed states are
∂t |t=k∆t ≈ ∆t b
3
2|CB1.0 (a) t=2.0ps t=2.5ps t=3.5ps graphene can be broken. For instance, at very small de-
+| 0.5 tuning, the wave functions of these dressed states can
2|CA0.0 be approximately written as the superposition of un-
|
2|+|CB01..50 (b) t=2.0ps p(Ψer+tu+rbΨed−)c/o√nd2u=ctio(1n,0a)n.dTvhaleesnecedrwesasveedfsutnactteisona,reΨno=t
2|CA0.0 the eigenstates of the helicity operator. The chiral sym-
| metry is broken and perfect chiral tunneling is strongly
2|CB12..50 (c) t=2.5ps suppressed. Numerical results are shown in Fig. 2(c)
2||+|CA001...050 nwmith. pFurmomp iFnitge.nsi2ty(c)Iωw=e c3a0nMfinWd/cthma2t atnhde Dtran=sm3i0s0-
2|CB1.0 (d) sion is strongly suppressed, even with lager detuning
+| 0.5 t=3.5ps (e.g., ∆0 = 10 meV, the transmittance is about 0.025).
2|CA0.0 When detuning increases, the light-induced mixing be-
| -1200 -900 -600 -300 0 300 600 900 1200 1500 comes weak [see Fig. 2(b)], the reflectance decreases,
X (nm)
and the transmittance increases. Fig. 2(d) shows the
transmittance as a function of pump intensity with dif-
FIG. 2: (color online). (a) numerical simulations of a wave
ferent barrier widths. The strong laser field can enhance
packettunnelingthroughabarrierwithoutpumpbeams. (b)-
band mixing and reduce the transmittance. From Fig.
(d)Timesequenceofawavepackettunnelingthroughabar-
rier with pump intensity Iω =3 MW/cm2, ∆0 =5meV, and 2(d) we also see that the wide barrier can prolong the
D=300 nm. The light grey shows thebarrier area. interaction time between electrons and photons, reduce
the tunneling rate, and lower the threshold of the pump
laser power.
(a) 0.5 (b) In conclusion, we have calculated the influence of the
0.2
V) 0.4 OSE on the chiral tunneling in graphene by using the
(eEk00..01 nk00..23 FstDroTnDglymseutphporde.ssWedebfiyntdhethoapttipcearllfyecitntduuncnedelibnagncdanmibxe-
-0.1
ing, even at large detuning. These properties might be
0.1
-0.2
useful in device applications, such as the fabrication of
0.0
0.30 0.-315 0.40 0.30 0.3-15 0.40 an optically controlled field-effect transistor that has ul-
K (n m ) K (n m ) trafast switching times and low power consumption.
nce01..80 (c) 01..80 ance ance01..80 (d) D=200nm This work was supported by the NSFC Grant Nos.
Reflecta000...246 000...246 Transmitt Transmitt000...246 DD==350000nnmm 1PS0tra9ot0ve4in0Kc5ee9y,21L00a09b90oG4r0Qa9tWo7r,0ya0no1df7M1,1it0lhl0iem41eO9t9epr,etnWheaRvNeeSsseFNarforc.ohmKF2Ju0ina0dn9g0ox1fi.
0.0 0.0 0.0
0 5 10 15 20 25 30 0 10 20 230 40
0 (meV) I (MW/cm)
FIG. 3: (color online). (a) Sketch of the renormalized
∗
bandenergies(solidlines)andtheunperturbedbandenergies Electronic address:[email protected]
(dashed lines) as a function of momentum k. (b) Sketch of [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,
thefermion distributionfunctionnk asafunctionofmomen- Y.Zhang,S.V.Dubonos,I.V.Grigorieva, A.A.Firsov,
tumk. (c)Thereflectance(redcircles)andthetransmittance Science 306, 666 (2004).
(blacksquares)ofthebarrierasafunctionofthedetuningfor [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,
Iω = 30 MW/cm2 and D = 300 nm. (d) The transmittance M.I.Katsnelson, I.V.Grigorieva, S.V.Dubonos,A.A.
asafunctionofpumpintensityfor∆0 =5meVwithdifferent Firsov, Nature438, 197 (2005).
barrier width. [3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.
Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109
(2009).
[4] M. I.Katsnelson, K. S.Novoselov, and A. K.Geim, Na-
blueshifted. Withincreasingdetuning,thelight-induced ture Physics 2, 620 (2006).
shiftdecreases,andthedressedstatesasymptoticallyap-
[5] V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, Science
proach the unperturbed states. The intense electromag- 315, 1252 (2007).
neticfieldcanalsoinduceastrongbandmixing. Nearthe [6] Z. Z. Zhang, K. Chang, and K. S. Chan, Appl. Phys.
absorption edge, a maximum fermion distribution func- Lett. 93, 062106 (2008).
[7] V.H.Nguyen,A.Bournel,V.L.Nguyen,andP.Dollfus,
tion n 0.44 can be observed [see Fig. 1(b)].
k ≈ Appl. Phys.Lett. 95, 232115 (2009).
Under intense light beams, the dressed states are
[8] P. Michetti, M. Cheli, and G. Iannacconea, Appl. Phys.
stronglymixedwithvalencestatesandconductionstates. Lett. 96, 133508 (2010).
Therefore, the chiral symmetry of Dirac electrons in [9] E. Prada, P. San-Jose, and H. Schomerus Phys. Rev. B
4
80, 245414 (2009). [18] D. Fr¨ohlich, B. Uebbing, T. Willms, and R. Zimmer-
[10] W.-R. Hannes and M. Titov, Europhysics Letters 89, mann, Europhys.Lett. 23 489 (1993)
47007 (2010). [19] M. Combescot, Physics Reports 221, 168 (1992).
[11] D.A.AbaninandL.S.Levitov,Science317,641(2007). [20] C. E. Pryor and M. E. Flatt´e, Appl. Phys. Lett. 88,
[12] D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. 233108 (2006).
Morozov, P. Blake, M. P. Halsall, A. C. Ferrari, D. [21] S. Sanchez, C. De Matos, and M. Pugnet, Appl. Phys.
W. Boukhvalov, M. I. Katsnelson, A. K. Geim, K. S. Lett. 89, 263510 (2006).
Novoselov, Science 323, 610 (2009). [22] W.Yao,A.H.MacDonald,andQ.Niu,Phys.Rev.Lett.
[13] M. V. Fistul and K. B. Efetov, Phys. Rev. Lett. 98, 99, 047401 (2007).
256803 (2007). [23] J. T. Liu, F. H. Su, and H. Wang, Phys. Rev. B 80,
[14] A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W. 113302 (2009).
T.Masselink, andH.Morkoc, Phys.Rev.Lett.56, 2748 [24] E. J. Mele, P. Kr´al, and D. Toma´nek, Phys. Rev. B 61,
(1986). 7669 (2000).
[15] S.Schmitt-Rink,D.S.Chemla,andH.Haug,Phys.Rev. [25] K. S. Yee, IEEE Transactions on Antennas and Propa-
B 37, 941 (1988). gation, 14 302 (1966).
[16] C. Ell, J. F. Mu¨ller, K. El Sayed, and H. Haug, Phys. [26] G.Mur,IEEETransactionsonElectromagneticCompat-
Rev.Lett. 62, 304 (1989). ibility EMC-23, 377 (1981).
[17] M.CombescotandR.Combescot,Phys.Rev.B40,3788
(1989).