Photon-assisted electron transport in graphene: Scattering theory analysis
B. Trauzettel,1,2Ya. M. Blanter,3and A. F. Morpurgo31Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 3Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
共Received 3 October 2006; revised manuscript received 9 November 2006; published 4 January 2007兲 Photon-assisted electron transport in ballistic graphene is analyzed using scattering theory. We show that the presence of an ac signal共applied to a gate electrode in a region of the system兲 has interesting consequences on electron transport in graphene, where the low energy dynamics is described by the Dirac equation. In particular, such a setup describes a feasible way to probe energy dependent transmission in graphene. This is of substan-tial interest because the energy dependence of transmission in mesoscopic graphene is the basis of many peculiar transport phenomena proposed in the recent literature. Furthermore, we discuss the relevance of our analysis of ac transport in graphene to the observability of zitterbewegung of electrons that behave as relativ-istic particles共but with a lower effective speed of light兲.
DOI:10.1103/PhysRevB.75.035305 PACS number共s兲: 73.23.Ad, 03.65.Pm, 73.63.⫺b
I. INTRODUCTION
Since the discovery of an anomalous quantum Hall effect in graphene,1,2 promising possibilities to observe quantum
dynamics of Dirac fermions in such systems have been pro-posed. The most prominent one is the quantum-limited con-ductivity 共of order e2/ h兲, where measurements1,2 and
theo-retical predictions3–8 are still inconsistent with each other.
Other examples of unusual quantum transport phenomena in mesoscopic graphene are a maximum Fano factor of 31,9
se-lective transmission of Dirac electrons through n-p
junctions,10the phenomenon of Klein tunneling,11and
trans-port phenomena at interfaces between graphene and a superconductor.12–15 Recently, the effect on the longitudinal
and Hall conductivity of an applied microwave signal has been analyzed in bulk graphene.16 In many of these works, interesting phenomena arise because transport through graphene is in general energy dependent. Therefore, it is de-sirable to directly probe the energy dependence of transport. We show that the Tien-Gordon problem17,18 of
photon-assisted electron transport for Dirac electrons is a powerful tool to quantitatively probe energy dependent transmission in graphene. The idea is to apply an ac signal to a region of graphene and a bias with respect to a neighboring region to allow electron transport from the one to the other共see Fig. 1兲.19
In the dc limit, electron transport then allows to directly determine the energy dependent transmission coefficients of the underlying scattering problem. Additionally, we find that resonance phenomena 共sharp steps in dG/dV, where G = dI / dV is the differential conductance兲 arise if the applied bias V equals multiples of the applied ac frequency. These resonance phenomena are due to the vanishing of propagat-ing modes directly at the Dirac point共the point in the spec-trum of graphene, where the valence band and the conduc-tion band touch each other兲. This implies an interesting application of graphene: It can be used as a spectrometer for high frequency noise—similar to the superconductor-insulator-superconductor 共SIS兲 junction in Ref. 20 with the advantage that there is no frequency limit.
Another motivation to look at the Tien-Gordon problem in graphene is its relevance to the observability of an interesting and unobserved phenomenon, present for free Dirac fermions but absent for free Schrödinger fermions, called
Zitter-bewegung 共ZB兲.21,22 ZB manifests itself, for instance, in a
time-dependent oscillation of the position operator of a Dirac electron in the Heisenberg picture. A pioneering attempt to describe an experimental way to observe ZB in III–V semi-conductor quantum wells has been proposed in Ref.23. Oth-ers have adapted this idea to carbon nanotubes,24spintronic,
graphene, and superconducting systems.25Another situation
where nonrelativistic electrons experience ZB is the presence of an external periodic potential.26 In fact, there is a wide
class of Hamiltonians共where the corresponding wave func-tion is a spinor兲 which all exhibit ZB.27 Nevertheless—
although almost omnipresent—the phenomenon has never been observed in nature. Therefore, it is desirable to propose
an unambiguous signature of ZB in a particular setup. To observe ZB, one needs to prepare a wave packet containing superposition of states with negative and positive energies. A graphene sheet biased close to the Dirac point is a good candidate, since energies of both signs are available, and, indeed, we show that such a superposition can be created using the Tien-Gordon effect. However, an unambiguous de-tection of ZB is impossible via measuring the oscillating current, since the oscillating current is also generated for the free Schrödinger equation, provided the wave function is a superposition of states with positive and negative energies. Thus, it is necessary not just to generate an appropriate wave packet, but also to find a specific phenomenon sensitive to ZB. We leave this difficult question beyond the scope of this article.
Additional studies are needed to single out effects of ZB and ensure their unambiguous observation. The paper is or-ganized as follows: In Sec. II, we introduce the setup under consideration and the model to describe it. The photon-assisted current is analyzed in Sec. III. Afterwards, in Sec. IV, we discuss the relevance of the previous analysis to the observability of ZB in graphene. Finally, we conclude in Sec. V.
II. SETUP AND MODEL
The setup under consideration is illustrated in Fig.1. It contains two regions of graphene called Gr共ac兲 and Gr共in兲. A gate electrode in region Gr共ac兲 shifts the Dirac points of the two regions with respect to each other by an amount eVS. Additionally, we apply a small ac signal to the gate electrode. An ideal sheet of graphene共in the absence of K−K
⬘
in-tervalley mixing兲 can be described in the effective mass ap-proximation by a two-dimensional Dirac equation共DE兲 for a two-component wave-function envelope ⌿=共⌿1,⌿2兲共sub-script 1,2 refers to pseudospins, whose origin can be traced to the presence of two carbon sublattices兲
冋
− ivប冉
0 x− iyx+ iy 0
冊
−共x兲
册
⌿e/h= iបt⌿e/h, 共1兲 wherev is the Fermi velocity. The dimensions of the sampleare assumed to be large enough such that boundary effects can be neglected. It is, however, straightforward to include boundary effects along the lines of Ref. 9. The index e / h refers to electronlike共energy ⬎0 with respect to the Dirac point兲 and holelike 共energy ⬍0 with respect to the Dirac point兲 solutions to the DE. The chemical potential of the two regions Gr共ac兲 共x⬍0兲 and Gr共in兲 共x⬎0兲 is
共x兲 =
再
eVS+ eVaccos共t兲, for x⬍ 0,0, for x⬎ 0. 共2兲
The potential共x兲 is chosen such that a solution to the DE with components of different energies is generated in Gr共ac兲 and transmitted to Gr共in兲. The solution to the wave equation 共1兲 for x⬍0 关in region Gr共ac兲兴 may then be written as
⌿e/h共ac兲共x,t兲 = ⌿0,e/h共ac兲共x,t兲e−i共eVac/ប兲sin共t兲
=
兺
m=−⬁ ⬁
Jm
冉
eVac
ប
冊
⌿0,e/h共ac兲共x,t兲e−imt, 共3兲where⌿0,e/h共ac兲 共x,t兲=⌿0,e/h共ac兲 共x兲e⫿it/បand
冋
− ivប冉
0 x− iyx+ iy 0
冊
− eVS
册
⌿0,e/h共ac兲 = ±⌿0,e/h共ac兲 .共4兲 共We have dropped the argument x of the wave function in the latter equation.兲 In Eq. 共3兲, Jmis the mth order Bessel func-tion. From now on, we focus without loss of generality on electronlike solutions only and drop the index e. This is jus-tified because, in ballistic transport from one region of graphene to another region, particles with a fixed energy 共which can be either positive or negative with respect to the Dirac point兲 are transmitted and their energy is conserved in the absence of inelastic scattering.
The electronlike plane wave solutions of Eq. 共4兲 can be written as linear combinations of the basis states12
⌿0,+共ac兲= eiqy+ikacx
冑
cos␣ac冉
e−i␣ac/2 ei␣ac/2冊
, 共5兲 ⌿0,−共ac兲= eiqy−ikacx冑
cos␣ac冉
ei␣ac/2 − e−i␣ac/2冊
, 共6兲 where ␣ac= arcsin冉
បvq + eVS冊
, 共7兲q is the transversal momentum, and kac=共
+ eVS兲cos␣ac/共បv兲 the longitudinal momentum in region
Gr共ac兲. Likewise, electronlike plane wave solutions of Eq.
共1兲 in region Gr共in兲 can be written as linear combinations of the basis states
⌿0,+共in兲= eiqy+ikinx
冑
cos␣in冉
e−i␣in/2 ei␣in/2冊
, 共8兲 ⌿0,−共in兲= eiqy−ikinx冑
cos␣in冉
ei␣in/2 − e−i␣in/2冊
, 共9兲with ␣in= arcsin共បvq/兲 and kin=共/ បv兲cos␣in. Now, we solve the transmission problem from region Gr共ac兲 to region
Gr共in兲. An incoming wave function from region Gr共ac兲 is
given by ⌿i共ac兲共x,t兲 =
兺
m=−⬁ ⬁ Jm冉
eVac ប冊
⌿0,+共ac兲e−i共+បm兲t/ប.⌿r共ac兲共x,t兲 =
兺
m=−⬁ ⬁ rmJm冉
eVac ប冊
⌿0,−共ac兲e−i共+បm兲t/ប,where rmis the energy-dependent reflection coefficient. Fur-thermore, a transmitted wave function in region Gr共in兲 can be written as ⌿tr共in兲共x,t兲 =
兺
m=−⬁ ⬁ tmJm冉
eVac ប冊
⌿+,m共in兲e−i共+បm兲t/ប, 共10兲 where ⌿+,m共in兲= eiqy+ikin,mx冑
cos␣in,m冉
e−i␣in,m/2 ei␣in,m/2冊
, 共11兲 with ␣in,m= arcsin冉
បvq + ប m冊
共12兲and kin,m=共+ បm兲cos␣in,m/共បv兲. In Eq. 共10兲, tm is the energy-dependent transmission coefficient. In order to deter-mine tm and rm, we need to match wave functions at x = 0, namely
⌿i共ac兲共x = 0,y,t兲 + ⌿r共ac兲共x = 0,y,t兲 = ⌿tr共in兲共x = 0,y,t兲.
The solutions to the resulting set of equations are
rm= ei␣ac− ei␣in,m 1 + ei共␣ac+␣in,m兲, tm= e−i共␣ac−␣in,m兲/2
冑
cos␣in,m cos␣ac 1 + e2i␣ac 1 + ei共␣ac+␣in,m兲, 共13兲where␣acis given by Eq.共7兲 and␣in,mby Eq.共12兲. It is easy to verify that unitarity holds兩rm兩2+兩tm兩2= 1. Furthermore, Eq. 共13兲 shows that if the angle of incidence is zero 共q=0兲, then all transmission coefficients are 1 and all reflection coeffi-cients vanish. This is known as Klein tunneling in relativistic quantum dynamics.11
III. PHOTON-ASSISTED CURRENT
In order to determine the transmitted current, we need to calculate the current density operator in the x direction,
Jx共x,t兲 = ev⌿*共x,t兲x⌿共x,t兲, 共14兲 integrate over a cross section in y direction, and over angles of incidence. In the following, we assume that a dc bias V is applied between regions Gr共ac兲 and Gr共in兲 and that kBT is the lowest of all energy scales. The average current is then given by I =4e h W
冕
0 qmax dq冕
0 eV d兺
m,m⬘=−⬁ ⬁ Jm冉
eVac ប冊
⫻Jm⬘冉
eVac ប冊
tm * tm⬘ei共m−m⬘兲t, 共15兲 where W is the width of the sample and qmax= min兵兩eV+បm兩 / បv,兩eV+eVS兩 / បv其 is the upper bound of the trans-versal momentum of propagating modes. The bias is applied such that the Fermi function共at zero temperature兲 in region
Gr共ac兲 reads fac共E兲=共eV−E兲 and, in region Gr共in兲, fin共E兲
=共−E兲, where共x兲 is the Heaviside step function. Since the scattering matrix is energy dependent, the current I depends on the way the bias is applied and not just on its magnitude
V. A factor 4 has been added to the right hand side of Eq.
共15兲 to take into account for spin and valley degeneracy. Apparently, there are terms in Eq.共15兲 that do not depend on time共where m=m
⬘
兲 and terms that oscillate as a function of time共where m⫽m⬘
兲.We will now show that the dc limit of Eq.共15兲 contains all the information of the energy dependent transmission of the underlying relativistic quantum dynamics. In that limit, only terms with m = m
⬘
survive in Eq.共15兲 and the differen-tial conductance readsG =4e 2 h W
冕
0 qmax dq兺
m=−⬁ ⬁ Jm 2冉
eVac ប冊
兩tm共 = eV兲兩2. 共16兲 The latter equation shows that the combination of the pres-ence of the bias V and the ac signal Vacallows to extract theenergy dependence of the transmission coefficients共in prin-ciple兲 channel by channel. Typical values for that can be used experimentally in practice are up to 10– 30 GHz. The bias can be controlled with any desired precision relative to
kBT down to mK temperature.
sensitive detector for finite frequency noise, similar to the SIS junction in Ref. 20. The advantage of graphene as a finite frequency noise detector is that it has no frequency limit, whereas the SIS detector is limited by the size of the superconducting gap.
At this point, we mention that all the above results are obtained using scattering theory for noninteracting Dirac fer-mions. There are several effects of Coulomb interaction that we neglect, most notably the linear energy dependence of the inverse quasiparticle lifetime.28 However, in our situation,
the most relevant interaction effect for ac transport must originate from screening. Indeed, as shown by Pedersen and Büttiker29 for the corresponding problem based on the
Schrödinger equation, screening by nearby gates can consid-erably alter the predictions of a noninteracting scattering theory. Close to the Dirac point in graphene, screening is expected to be reduced,30and we thus expect that the above results also hold in the presence of interactions. Investigation of the screening effects is beyond the scope of this paper, and we expect it to be well relevant away from the Dirac point.
IV. RELATION TO ZITTERBEWEGUNG
Let us first explain the signature of ZB in the current in the absence of an oscillating potential关as introduced in Eq. 共2兲兴 and then show why our particular choice of the oscillat-ing potential has some relevance as far as ZB is concerned. In the absence of the potential共x兲, the time evolution of the electron field operator that obeys Eq.共1兲 can be written as
⌿p共t兲 = 1 2关⌿p共+兲共t兲 + ⌿p共−兲共t兲兴, 共17兲 with p =共px, py兲, p=
冑
px 2 + py 2 , and ⌿p共±兲共t兲 = e⫿ivpt/ប关1 ± 共p ·兲/p兴⌿p,where=共x,y兲 is a vector of Pauli matrices. A straight-forward calculation of the current operator in the Heisenberg picture6 shows that an oscillatory component in time exists
due to an interference of⌿p共+兲共t兲 and ⌿p共−兲共t兲 solutions of the DE. Therefore, in order to see a signature of ZB in the cur-rent, it is important that electronlike and holelike solutions interfere. More generally, an oscillatory component of the current arises if the electron field operator contains solutions to the DE at different energies. Consequently, if we calcu-lated the current operator corresponding to a wave function solution to the DE with a fixed energy, then there would be no oscillatory component of the current operator left and, thus, no sign of ZB in the current. Importantly, this is the general situation in ballistic transport in graphene if a plane wave solution 共of a particle with energy 兲 to the DE is injected from one region to another. Thus, we conclude that
ballistic dc transport in graphene shows no direct signature of ZB.
In principle, the Tien-Gordon setup of photon-assisted electron transport for Dirac electrons, illustrated in Fig. 1, can be used to generate the desired state.The reason is that the ac signal stimulates the absorption and emission of pho-tons, which in turn lead to a population of different side-bands around the Fermi energy. Then, the resulting electron field operator in region Gr共in兲, Eq. 共10兲, is precisely what is needed to observe ZB in graphene. However, the detection of the resulting oscillating current, Eq.共15兲, is not sufficient to prove the existence of ZB in graphene. The reason is that a preparation of a state that is a superposition of different en-ergy solutions to the free Schrödinger equation also yields a similar oscillating current关see, for instance, Eq. 共16兲 of Ref. 29兴. The only difference to the free Schrödinger case as com-pared to the Dirac case is the peculiar energy dependence of the transmission in the latter. Thus, the setup analyzed in this paper is potentially relevant to the detection of ZB in graphene, but our analysis indicates that the truly fundamen-tal signature of ZB needs to be identified more precisely.
V. CONCLUSIONS
We have emphasized in this article that the energy depen-dence of the transmission共typical for mesoscopic graphene systems兲 plays a crucial role for the unexpected transport phenomena predicted in these devices; for recent examples see Refs. 6 and 9–15. Therefore, it is desirable to directly probe the energy dependence of the transmission. We have demonstrated that photon-assisted transport is the natural way to do so. This is so because the differential conductance of a biased setup共in the presence of a small ac signal applied to a gate兲 is directly proportional to the energy-dependent transmission at the energy corresponding to eV, where V is the dc bias applied. Furthermore, we have pointed out that a possible application of our setup is to use graphene as a spectrometer for finite frequency noise. Finally, we have dis-cussed the relevance of photon-assisted transport in graphene to the observability of ZB. Our conclusion is that photon-assisted transport can be used to create the conditions to observe ZB. However, our analysis also shows that the truly fundamental signature of ZB共which is important for its de-tection兲 needs to be pinpointed in future studies.
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