Energy shift and conduction-to-valence band transition mediated by a time dependent potential barrier in graphene

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by  Andrey Chaves

Energy shift and conduction-to-valence band transition mediated by a time dependent
potential barrier in graphene
Andrey Chaves,∗ D. R. da Costa,† G. O. de Sousa,‡ J. M. Pereira Jr., and G. A. Farias
Departamento de F´ısica, Universidade Federal do Ceara´,
Caixa Postal 6030, Campus do Pici, 60455-900 Fortaleza, Ceara´, Brazil
(Dated: January 5, 2016)
We investigate the scattering of a wave packet describing low-energy electrons in graphene by a
6 time-dependent finite step potential barrier. Our results demonstrate that, after Klein tunneling
1 throughthebarrier, theelectron acquires an extraenergy which dependson therateof change the
0 barrierheight in time. Ifsucha rateisnegative, theelectron loses energy andendsupasavalence
2 bandstateafterleavingthebarrier,whicheffectivelybehavesasapositivelychargedquasi-particle.

n
PACSnumbers: 81.05.U-,73.63.-b,72.80.Vp,03.65.Pm
a
J

4 The phenomenon of tunneling has long been a text- group velocities.

book example of the contrasts between quantum and Inthisworkweinvestigatetheinteractionofwavepack-
]
l classical behaviors. The fact that a particle can, with etscorrespondingtomasslessFermionstatespropagating
l
a a certain probability, be transmitted through a classi- in graphene with time-dependent potential barriers. We
h cally forbidden region is not only a manifestation of the show that the gapless and linear aspects of the disper-
- dual wave-particlecharacter of microscopic systems, but sioningrapheneleadtoanoutcomethatisquitedistinct
s
e isalsoatthebasisoftheexplanationofphenomenasuch from the non-relativistic case: the electron-hole symme-
m as the alpha emission. In most discussions of tunneling try in graphene implies that, within a single-particle de-
. throughone-dimensionalpotentialprofiles,the barrieris scription,depending onthe initialenergyandtherateof
t
a assumedtobestaticandtheusualsolutionforthetrans- changeofpotential,theresultingenergyshiftcanconvert
m mission coefficients can be found by comparing the am- an incoming conduction band electron into an outgoing
- plitudes ofincoming andoutgoingplane waves. Further- particlewithinthevalenceband,whichbehavesasapos-
d more,textbookcalculationsareusually presentedwithin itively charged quasi-particle. In addition, the fact that
n the framework of non-relativistic quantum mechanics1. at low energies the group velocity of the charge carriers
o
Recentlytherehasbeenarenewedinterestinthestudy is independent of the energy, together with the perfect
c
[ of tunneling in the relativistic regime.2–6 This was moti- transmissionatnormalincidence,resultsthatthe outgo-
vated by the production of graphene,7 a bi-dimensional ingwavepacketspreservetheshapeoftheincomingwave.
1
layer of carbon atoms organized in a honeycomb lat- We consider a low-energy electron propagating in
v
tice. Oneunusualaspectoftheelectronicbandstructure an infinite graphene sample, so that the system mim-
6
3 of that material is the fact that at the vicinity of the ics a propagating massless Dirac particle, obeying
5 Fermi level the spectrum is gapless with a linear disper- i~∂Ψ(~x,t)/∂t = HΨ(~x,t), for the Dirac Hamiltonian of
0 sion, such that the charge carriers can be described as graphene H = v p~·~σ. The electron is described by a
F
0 ultra-relativistic massless fermions, albeit with an effec- Gaussian wave packet, multiplied by a spinor that ac-
.
1 tive light speed given by the Fermi velocity of the ma- countsfortheprobabilitydistributionsoverthetwosub-
0 terial (v ≈ c/300).8,9 This feature has important con- lattices of graphene (labeled A and B), and by a plane
F
6 sequences, such as the perfect transmission of normally wave, which gives the particle a non-zero average mo-
1 incident electrons through p-n and p-n-p junctions, an mentum k :
0
v: effect known as Klein tunneling.5 Thus graphene, apart
i frombeing ofinterestfordevice applications,alsoallows A (x−x0)2 (y−y0)2
X theinvestigationofquantumrelativisticeffectsbymeans Ψ(~x,0)=N(cid:18)B (cid:19)exp(cid:20)− d2 − d2 +ik0x(cid:21),(1)
x y
r
of tabletop experiments.
a
Quantum mechanical tunneling through one- where N is a normalization factor, (x0,y0) are the coor-
dimensional oscillating or moving potential barriers dinatesofthecenteroftheGaussianwavepacket,dx(dy)
has been previously investigated in the context of isitswidthinthex(y)-direction,k0 =E0/~vF,andE0 is
non-relativistic electrons described by Schr¨odinger’s theinitialwavepacketenergy. Thetimeevolutionofsuch
equation.10–14 These studies have shown that a awavepacketiscalculatedbymeansofthesplit-operator
wavepacket incident on an oscillating barrier develops technique, which is explained in details in Refs. [5,15–
multiple peaks which propagatewith different velocities. 18]. However, in order to deal with a time-dependent
This has been explained as resulting from an energy potential V(~x,t), one must adapt the technique so that
exchange between the incident particle and the oscil- the time evolutionoperatorin Ψ(~x,t)=Uˆ(0,t)Ψ(~x,0)is
lating potential. Due to the parabolic dispersion of the written as
tunneling particles in these calculations, an energy shift
of the wavepacket components will result on different Uˆ(0,t)=e−~i R0tHdt. (2)
2
The integral in the argument of the exponential in Eq. of the potential in time. In this case, the integral term
(2) can be separated into a sum of integrals for each in Eq. (2) becomes t+∆tVdt = α(∆t2 +2t∆t)/2 for
t
time step,sothat, inprinciple,onecouldmakeU(0,t)= 0<x<W. R
e−~i Rtt−∆tHdte−~i Rtt−−2∆∆ttHdt...e−~i R∆2∆ttHdte−~i R0∆tHdt, After undergoing Klein tunnelling through a barrier,
which is then just an sequence of exponential operators the propagatingwavepacketacquiresa phase21 givenby
applied recursively to an initial wave packet. However,
if the Hamiltonian at two different time steps t and 1 ∞
1 φ= V(x,t)dx (5)
t2 does not commute, [H(t1),H(t2)] 6= 0, one has, in ~vFZ−∞
general,
which, for the step barrier potential considered here
e−~i Rtt01Hdt−~i Rtt12Hdt 6=e−~i Rtt01Hdte−~i Rtt12Hdt (3) yields φ=Wαt ~vF. However, such a phase appears in
the wave functio(cid:14)n as exp(−iφ) = exp −i αWt , which
Nevertheless, the multiplication between exponentials in (cid:16) ~ vF (cid:17)
is equivalent to an extra energy factor exp −iE t ,
the right side of the equation can be seen as a first or- ~ x
der approximation, according to the Suzuki-Trotter ex- where Ex =αW/vF. This suggests that after t(cid:0)unnellin(cid:1)g
throughsuchatimevaryingpotential,theelectronwould
pansion of the exponential of a sum of non-commuting
acquire anextra energyE . If α<0,the tunnelled elec-
operators.19,20 Theerrorinthisapproximationispropor- x
tronlosesenergyandmayevenendupinavalenceband
tional to ∆t2. Therefore, assuming a small ∆t, applying
state, provided the magnitude of α is high enough. This
theexponentialsrecursivelytoaninitialwavepacketpro-
isindeedthecase,aswewilldemonstrate. Inallcasesin-
vides, to a good approximation, an accurate description
vestigatedhere,time-dependence oftheHamiltoniandid
oftheactualtime-evolutionofthesystem. Moreover,one
notaffecttheKleintunnellingeffect,andalltransmission
can re-write
probabilitiesarefoundtobe1(withinthenumericalpre-
Uˆ(t,t+∆t)=e−2i~Rtt+∆tVdte−~iTˆ∆te−2i~Rtt+∆tVdt+O(∆t3),c(4is)ion of the calculation method).
Let us first investigate a one-dimensional problem by
where Tˆ is the kinetic energy operator. We further ne- considering dx = 100 ˚A and dy → ∞, which represents
glect the O(∆t3) terms in this expansion by using a suf- a wave front propagatingin the x−direction. The initial
energy of such a wave front is considered as E = 100
ficiently small time step: since the argument of the ex- 0
ponential basically involves ∆t(T +V)/~, T and V are meV. The average kinetic energy E =hΨ|T|Ψi is shown
of the order of hundreds of meV and ~ is of the order of in Fig. 2 as function of time for (a) a fixed width
hundreds of meVfs, a fractionof fs (namely, ∆t=0.1 fs)
is enough to give an accurate result.
The wave packet starts at x = −300 ˚A and propa-
0
gates through a potential barrier at x = 0, with width
W, as illustrated by the shaded area in Fig. 1, where
mmeeVV//fsfs
the barrier height is considered to vary linearly in time mmeeVV//fsfs
as V (t)=αt if 0<x<W,and V =0 otherwise. As we mmeeVV//fsfs
expl0ain in greater detail below, such a time-dependent V)V)V) mmeeVV//fsfs
potential can be generated e.g. by a microwave field so mememe mmeeVV//fsfs
(((
that,forthetypicalspatialdimensionsofthesystem,the
periodofoscillationofthepotentialismuchsmallerthan
the traversal time, thus retaining the linear dependence
AAoo
AAoo
AAoo
V0(t) E+ V)V)V) AAoo
E0 E- mememe
W (((
FIG. 1: Sketch of the potential barrier (shaded area) con-
sidered in our calculations, which has width W and a time-
dependent height V0(t). The red circle represents the incom-
ing electron, with energy E0, which is able to go across the (((((ffffssfsss)))))
barrier, represented as a dashed line, even if E0 < V0(t) for
a given time t, due to Klein tunnelling. The outgoing elec-
tronleavesthebarrierregioneitherwithhigher(E+)orlower FIG.2: Kineticenergyofthewavepacketasfunctionoftime
(E−) energy, depending on how the barrier height varies on for(a)afixedwidthW =200˚A anddifferentvaluesofα,and
time (see text). for(b)afixedα=7meV/fsanddifferentvaluesofwidthW.
3
Figure3helpsustovisualizetheeffectshowninFig. 2
55ee--0055 byshowingcontourplotsofthewavepacketinreciprocal
(a) (b) 44..55ee--0055
44ee--0055 space (upper panels) as function of time. After leaving
33..55ee--0055
t(fs)t(fs) (fs) 222233..eeee55----ee0000--55550055 ptheeakbaserreinerinretghioenw, anvaemfeulnyctfoiornta't k60=fs0,.0t0h9e2c˚Aon−s1tafnort
t 11..55ee--0055
11ee--0055 t=0 moves either (a)to the left, representinganenergy
55ee--0066
00 decrease for α = −7 meV/fs, even reaching the region
k kk(((AAAooo---111))) k kk(((AAAooo---111))) of negative k, or (b) to the right, representing a energy
enhancement for α = 7 meV/fs. Despite the negative
110000 00..0044
(c) (d) 00..0033 hki in (a), we observe that in both cases, the group ve-
8800
00..0022 locity remainspositive,as the wavepacketkeepsmoving
)6600 00..0011
t (fst(fs)t(fs)22440000 ----000000....00001122 fhoarpwpaerndsitnortehael spphaacsee. vIetloiscitthyeinnitnhteerhekstiin<g0tocasseee.wFhiagt-
--00..0033 ures 3(c) and (d) show the contour plots of the real part
00 --00..0044
--660000--440000--220000 00 220000440000 --440000--220000 00 220000440000660000 of the wave function as function of time, for the same
x xx(((AAAooo))) xxx (((AAAooo))) parameters as in Figs. 3(a) and (b), respectively. By
the number of peaks, one can estimate the wave length,
which is inversely proportional to |k|. After the barrier,
the number of peaks remains essentially the same in (c)
FIG. 3: Contour plots of the time evolution of (a, b) the and increases in (d). The latter is consistent with the
squared modulus of the wave packet in reciprocal space, as
larger hki (smaller wave length) observed for t ' 60 fs
wellasthe(c,d)realpartofthewavepacketintherealspace.
in Fig.3(b), whereas the former shows an outgoing wave
The potential barrier lies within 0 ≤ x ≤ 200 ˚A and two
packetwhose wavevectork has changedin sign,but not
values are considered for the linear dependence of its height
in time: α = -7meV/fs (a, c) and +7 meV/fs (b,d). significantly in magnitude. We observe that phase ve-
locity is alwayspositive after the barrier(the position of
each peak increases with time), which confirms that the
outgoingwavepacketinFig. 3(a)behavesasaholewith
W =200˚A and severalvalues of variationrate of the po-
E < 0, since the phase velocity is v = E/~k > 0 and
tential height α, and for (b) a fixed α and several values p
hki is shownto be negative after the barrierin this case.
of W. It is seen that the final energy E, after the wave
packet leaves the barrier region, differs from the initial Our previous analyticalcalculationsshow that the ex-
wave packet E0 if α 6= 0. In fact, the final energy is tra energy Ex provided by the time-dependent potential
higher(lower)thanE whenα>0(α<0). Thatagrees barrier depends linearly both on α and W. This is con-
0
with the argument that the energy shift is proportional firmed by the results in Fig. 4, which show the final
to the rateofvariationofthe barrierheight. Notice that energy Efinal as function of these quantities. In Fig.
in the α = −5 and -7 meV/fs cases, the final energy 4(a), the numerically obtained final energy E (symbols)
is even negative, suggesting that the electron ended up iscomparedto the one predictedby the Kleintunnelling
in the valence band after traversing the time-dependent phase for a time dependent barrier (lines), as explained
barrier. Notice that in all cases investigatedhere, we as- above, where very good agreement is observed. Figure
sumeemptybands,sothatallstatesareaccessiblebythe 4(b) confirms that this extra energy depends linearly on
electron and no Pauli blocking is involved. That would W and demonstrates that electronic wave packets can
be the case, for example, of hot electrons in a system become hole-like for stronglynegativeα - consideringan
with a low Fermi level. initial wave packet energy E = 100 meV and α = −5
meV/fs, e.g., the final energy is E <0 for W &150 nm.
In the α ≤ 0 cases, we have considered an initial po-
tentialbarrierof200meV att=0fs. This explains why Further indication of the conduction-to-valence band
even for α = 0 (i.e., for a constant 200 meV potential states conversion in the case of high negative α can be
barrier), the wave packet loses kinetic energy while in- gained by analysing the behavior of the outgoing wave
side the barrier region: such energy reduction is related packetin the presence of an externalmagnetic field. For
only to the additionof the averagepotentialenergy hVi, instance,iftheoutgoingpacketrepresentsavalenceband
which is non-zero only when the wave packet lies within state, which behaves as positively charged quasi-particle
the barrier. Nevertheless, the final energy does not de- (similartoaholeinsemiconductorsorapositroninhigh-
pend on this initial potential barrier height, but rather energy systems), its behavior under such field should be
only on α, since it is a consequence of a different mecha- opposite to that of an electron. In order to check that,
nism, namely,the equivalence betweenaKleintunneling we carried out the following simulation: a circular wave
phase that depends linearly in time and an extra energy packet dx = dy = 200 ˚A with energy E0 = 100 meV
for the particle. Indeed, after leaving the barrier region starts at x = −600 ˚A, reaches a W = 200 ˚A barrier,
0
in the α=0 case,the wavepacketrecoversits initial en- placed between x = −200 ˚A and x = 0, and enters a
ergy [see Fig. 2(a), green-dotted curve], no matter how B = 5 T magnetic barrier for x > 0. The trajectories
high or low the barrier height is, as expected. of the center of mass of such wave packet, calculated
4
by (hxi,hyi), resulting from such simulation, are shown
in Fig. 5, considering α = 0 (green dotted), 4.1 (red
dashed) and -8.2 meV/fs (black solid). By the fact that
the trajectory for α = 4.1 meV/fs exhibits larger radius
ascomparedtotheoneforα=0,onealreadyinfersthat
the time-dependent barrier has boosted the electron en-
ergy, since the radii of the circular trajectories coming
from Lorentz force are directly proportional to the par-
ticle energy E as R = E evB.16 In fact, the trajectory
for the α=0 case resemb(cid:14)les a semi-circle with a diame-
ter 2R≈450 ˚A inside the magnetic barrier, whereas for
α=4.1meV/fs,whichleadstoanextraenergyE ≈100
x
meV and, consequently, doubles the electron energy, the
resulting circular trajectory has twice larger diameter.
Moreover, after passing through the α = −8.2 meV/fs x (Å)
barrier,thewavepackettrajectoryiscurvedtotheoppo-
site direction, which is expected for a positively charged FIG. 5: Trajectories of a circular wave packet dx =dy =200
particle,butwithasimilarradiusastheoneobservedfor ˚A tunnelling through a barrier at −200 ˚A ≤ x< 0 (purple,
α = 0 (green dotted). In fact, α = −8.2 meV/fs leads dark-shaded area) and reaching a magnetic step barrier for
to an energy loss of ≈ 200 meV, which makes the wave x ≥ 0 (yellow, light-shaded area). The wave packet starts
packet, which initially has E = 100 meV, end up with with initial energy E0 = 100 meV, and the barrier height
0 varieswithratesα=−8.2meV/fs(blacksolid)andα=+4.1
E ≈ −100 meV, i.e. as a hole. As the final energies for
meV/fs (red dashed). The result for α=0 (green dotted) is
α = 0 and α = −8.2 meV/fs differ practically only by a
shown for comparison.
negative sign, it is then expected that their circular tra-
jectories inside the magnetic barrier exhibit almost the
same radii, but in opposite directions.
definitionofholesinsemiconductorphysics. Thelatteris
It is worthy to point out the difference between the
rathera collectiveeffect ofvalencebandelectrons: if one
valence bandquasi-particlesobservedhereandthe usual
ofthevalencebandstatesisunoccupied,thedynamicsof
the remaining electronsin realand reciprocalspaces can
be effectively described by that of a positively charged
particle,whicheveninteractsviaCoulombpotentialwith
conduction band electrons, forming excitons, trions, etc.
Holes in semiconductorscanbe also seenas ananalogof
)))
meVmeVmeV a positron in the context of high-energy physics.2 In our
((( 110000 A oAo study, the valence band is empty, so, we cannot have a
220000 A oAo hole strictu sensu. What is observed is rather regarded
330000 A oAo asavalencebandelectron,whosechargeisstillnegative,
440000A oAo so that charge conservation is respected. However, in-
terestingly enough, such state turns out to behave as a
positivelychargedparticle(justlikeholesandpositrons),
((mmeeVV//ffss))
say, in the presence of electromagnetic fields, due to its
negative effective mass.
It is important to address the issue of the feasibility
V)V)V) of experimental detection of such an effect. The time
mememe dependent barrier could be provided by a time depen-
((( dent electric field perpendicular to the layer and applied
mmeeVV//ffss
just in a finite region, so that electrons in this region
mmeeVV//ffss
mmeeVV//ffss (barrier)wouldhavehigherenergy. Asmentionedabove,
mmeeVV//ffss suchatimedependentelectricfieldcanbeobtainedfrom
an electromagnetic wave. Linear dependence of the po-
WW ((AA(ooA))o) tential, however, would only be obtained by approxima-
tion of the sinusoidal form of the electromagnetic wave
FIG.4: Finalenergy,after thewavepacketleavesthepoten- as sin(ωt)≈ωt, which is only valid for t≪1/ω. There-
tialbarrier,(a)asfunctionofthevariationrateofthepoten- fore, the time for the electron to cross the whole barrier
tialheightα,consideringdifferentvaluesofbarrierwidthW, t = W/v must be much smaller than the period of the
F
and(b)asfunctionofW,fordifferentvaluesofα. Symbolsin electromagnetic wave in order to produce a barrier with
(a) are the numerically obtained results, whereas curves are approximately linear time dependence, so ω ≪ v /W.
F
an analytical estimate. As an example, a barrier with width W = 1000 ˚A, for
5
instance, would require ω ≪ 8.2 THz, i.e. in the higher has an important meaning, as it can be seen as an ex-
frequency limit of the GHz (microwaves) range. On the tra energy for this charge carrier. If this extra energy
other hand, such electromagnetic wave would produce a is negative, an incident electron, initially in the conduc-
potential V (t) = −Φ sin(ωt) ≈ Φ ωt, where Φ = Fd, tion band, can be converted into a valence band state
0 0 0 0
d is the distance from the source to the graphene sam- after tunnelling, which behaves similarly to a positively
ple and F is the electric field intensity, therefore Φ ω charged quasi-particle. Such transition is theoretically
0
plays the role of α in our theory. If one considers specif- verifiedboth by analysingthe signof the tunnelled wave
ically the conduction-to-valence band states transition, packetsenergyandthetrajectoryofthecenterofmassof
the amountofenergyE reducedby the time dependent such wave in the presence of an external magnetic field.
x
barrierinthiscasemustbelargerthantheinitialelectron Results predicted here are possible to be experimentally
energy, E < E . After some algebraic manipulations, observed e.g. by investigating the scattering of hot elec-
x
onefindsω =E v Φ W ≪t,whichyieldsΦ ≫E. In trons by an oscillating potential barrier, whose height is
x F 0 0
otherwords,the con(cid:14)duction-to-valencebandstatestran- modulatedbyamicrowave22focusedonafiniteregionof
sitioncanbeobservedprovidedtheelectromagneticwave the graphene sample.
frequency is in the GHz range (to guarantee linearity of
the potential in time) and its intensity is high enough.
Insummary,weinvestigatedtheKleintunnellingofan
Acknowledgments
electronicwavepacketthroughapotentialbarrierwhose
height varies linearly in time, in an infinite monolayer
graphene sample. Our results demonstrate that for this DiscussionswithF. M.Peetersaregratefullyacknowl-
specificformoftime dependenceofthepotentialbarrier, edged. This work was financially supported by CNPq,
the phase acquired by the electron after the tunnelling under the PRONEX/FUNCAP grants and CAPES.
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