Ballistic transmission through a graphene bilayer

Ballistic transmission through a graphene bilayer

Snyman, I.; Beenakker, C.W.J.

Citation

Snyman, I., & Beenakker, C. W. J. (2007). Ballistic transmission through a graphene bilayer.

Physical Review B, 75(4), 045322. doi:10.1103/PhysRevB.75.045322

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Ballistic transmission through a graphene bilayer

I. Snyman and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 共Received 11 September 2006; revised manuscript received 17 October 2006; published 12 January 2007兲

We calculate the Fermi energy dependence of the共time-averaged兲 current and shot noise in an impurity-free carbon bilayer共length LⰆwidth W兲, and compare with known results for a monolayer. At the Dirac point of charge neutrality, the bilayer transmits as two independent monolayers in parallel: Both current and noise are resonant at twice the monolayer value, so that their ratio 共the Fano factor兲 has the same 1/3 value as in a monolayer—and the same value as in a diffusive metal. The range of Fermi energies around the Dirac point within which this pseudodiffusive result holds is smaller, however, in a bilayer than in a monolayer共by a factor l_⬜/ L, with l_⬜the interlayer coupling length兲.

DOI:10.1103/PhysRevB.75.045322 PACS number共s兲: 73.50.Td, 73.23.⫺b, 73.23.Ad, 73.63.⫺b

I. INTRODUCTION

Undoped graphene has no free electrons, so an infinite sample cannot conduct electricity. A finite sample can con- duct, because electrons injected at one end can be transmitted a distance L to the other end via so-called evanescent modes.

These are modes that decay⬀e^−L/␭with a penetration depth

␭ bounded from above by the width W of the sample. For a wide and narrow sample共WⰇL兲, there are many evanescent modes that contribute appreciably to the conductance. Be- cause the transmission of an electron via an evanescent mode is a stochastic event, the current fluctuates in time—even in the absence of any scattering by impurities or lattice defects.

Tworzydło et al.¹found that the shot noise produced by the evanescent modes in an undoped carbon monolayer 共of length LⰆwidth W兲 is pseudodiffusive: The Fano factor F

= P / 2eI¯ 共ratio of noise power P and time-averaged current I¯兲 has the same value F = 1 / 3 as in a diffusive metal共while F

= 1 for independent current pulses兲.²

A carbon bilayer has an additional length scale, not present in the monolayer of Ref. 1, namely the interlayer coupling length l_⬜. It is an order of magnitude larger than the interatomic distance d within the layer共Refs.3–5兲:

l_⬜= បv t_⬜= 3t_储

2t_⬜d⬇ 11d 共1兲

共with v⬇10⁶m / s, d⬇1.4 Å, and t储⬇3 eV, respectively the carrier velocity, interatomic distance, and nearest-neigbor hopping energy within a single layer, and t_⬜⬇0.4 eV the nearest-neighbor hopping energy between two layers⁶兲. Since L is typically large compared to l_⬜, the two layers are strongly coupled. In this paper we investigate what is the effect of interlayer coupling on the average current and shot noise.

The model and calculation are outlined in Secs. II and III.

Our main conclusion, presented in Sec. IV, is that an un- doped graphene bilayer has the same current and noise as two monolayers in parallel. The Fano factor, therefore, still equals 1 / 3 when the Fermi level coincides with the Dirac point共at which conduction and valence bands touch兲. How- ever, the interval⌬EF⯝បvl_⬜/ L²in Fermi energy around the

Dirac point where this pseudodiffusive result holds is much narrower, by a factor l_⬜/ L, in a bilayer than it is in a mono- layer.

Our results for the mean current I¯, and hence for the con- ductance in a ballistic system, agree with those of Cserti,⁷ but differ from two other recent calculations in a 共weakly兲 disordered system.^8,9 共The shot noise was not considered in Refs.7–9.兲 A ballistic system like ours was studied recently by Katsnelson,¹² with different results for both conductance and shot noise. We discuss the origin of the difference in Sec.

V. We conclude by connecting with experiments¹¹in Sec. VI.

II. MODEL

We use the same setup as in Refs.1and10, shown sche- matically in Fig. 1. A sheet of ballistic graphene in the x-y plane contains a weakly doped strip of width W and length L, and heavily doped contact regions for x⬍0 and x⬎L. The doping is controlled by gate voltages, which induce a poten- tial profile of the form

U共x兲 =

再

^{− U⬁ if x}⬍ 0 or x ⬎ L,

0 if 0⬍ x ⬍ L.

冎

^共2兲

We use an abrupt potential step for simplicity, justified by the fact that any smoothing of the step over a distance small

FIG. 1. Schematic of the graphene bilayer. Top panel: Two stacked honeycomb lattices of carbon atoms in a strip between metal contacts. Bottom panel: Variation of the electrostatic potential across the strip.

1098-0121/2007/75共4兲/045322共6兲 045322-1 ©2007 The American Physical Society

compared to L becomes irrelevant near the Dirac point, when the Fermi wave lengthⲏL.

While Refs. 1 and10considered a graphene monolayer, governed by the 2⫻2 Dirac Hamiltonian, here we take a bilayer with 4⫻4 Hamiltonian 共Refs.3–5兲,

H =

冢

^{v共pxtU0⬜− ipy兲 v共pxU00+ ipy兲 v共pxtU0+ ip⬜ y兲 v共pxU00− ipy兲}

冣

^,

共3兲 with p = −iប⳵^/⳵r the momentum operator. The Hamiltonian acts on a four-component spinor共⌿A₁,⌿B₁,⌿B₂,⌿A₂兲 with amplitudes on the A and B sublattices of the first and second layer. Only nearest-neighbor hopping is taken into account, either from A to B sites within a layer or between different layers.共Sites from the same sublattice but on different layers are not directly adjacent.兲 The Hamiltonian 共3兲 describes low-energy excitations near one of the two Dirac points in the Brillouin zone, where conduction and valence bands touch. The other Dirac point and the spin degree of freedom contribute a fourfold degeneracy factor to current and noise power.

We have taken the same electrostatic potential U in both layers. In general, the potentials will differ,^13,14but to study the special physics of undoped graphene it is necessary that they are both tuned to the Dirac point of each layer. This can be achieved by separate top and bottom gates共not shown in Fig.1兲.

For free electrons in bilayer graphene, the relation be- tween energy ␧ and total momentum k=共kx2+ k_y²兲^1/2 as de- scribed by this Hamiltonian consists of four hyperbolas, de- fined by

␧ = ±1

2t_⬜±

冑

¹4t_⬜² + k², 共4a兲

␧ = ⫿1

2t_⬜±

冑

¹4t_⬜² + k², 共4b兲 plotted in Fig.2. 共For notational convenience, we use units such thatបv=1 in most equations.兲

We calculate the transmission matrix t through the graphene strip at the Fermi energy, and then obtain the con- ductance and noise power from the Landauer-Büttiker formulas²

G = G₀Tr tt^†, P = P₀Tr tt^†共1 − tt^†兲, 共5兲

→F =Tr tt^†共1 − tt^†兲

Tr tt^† , 共6兲

with G₀= 4e²/ h, P₀= 2e兩V兩G0, and V the voltage applied be- tween the contact regions. The results depend on the degree of doping in the graphene strip 共varied by varying EF兲, but they become independent of the degree of doping of the contact regions if U_⬁Ⰷt_⬜.

III. TRANSMISSION PROBABILITIES

We calculate the transmission matrix by matching eigen- states of the Hamiltonian共3兲 at the two interfaces x=0 and x = L. This procedure is similar to a calculation of nonrelativ- istic scattering by a rectangular barrier in a two-dimensional waveguide. There are two differences. Firstly, the Hamil- tonian共3兲 is a first-order differential operator, and hence only the wave function and not its derivative is continuous at the interface. Secondly, the spectrum contains both positive and negative energy eigenstates.

The eigenstates of H for U = 0 have been given in Ref.13.

They may be characterized as follows. For given energy ␧ and transverse momentum ky, we define two longitudinal momenta

k_x±=

冑 冉

^{␧ ±1}2t_⬜

冊

²⁻¹4t_⬜² − k_y². 共7兲 The square root is taken with argument in the interval关0,␲兲.

Associated with each real k_x+ there are two propagating modes, one left-going ␾_␧,+^L and one right-going ␾_␧,+^{R . Two} more propagating modes ␾_␧,−^L and ␾_␧,−^R are associated with each real k_x−. These eigenstates of H are given by

␾^R_␧,±共x,y兲 = N±

冢

^{⫿kkx±x±⫿␧␧+ ik± ikyy}

冣

^{ei共kx±x+kyy兲, 共8a兲}

FIG. 2. Energy spectrum共4兲 of the graphene bilayer, according to the Hamiltonian共3兲.

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␾^L_␧,±共x,y兲 = N±

冢

^{− k±kx±x±⫿␧␧± ik+ ikyy}

冣

^{ei共−kx±x+kyy兲, 共8b兲}

with N_±=共4W␧kx±兲^−1/2 a normalization constant such that each state carries unit current

I = ev

冕

0 W

dy␾^†

冉

^␴0x ␴⁰x

冊

^{␾, 共9兲}

in the positive or negative x direction.

For each k_y we have two left-incident scattering states

␺_␧,± at energy ␧. In the region x⬍0 to the left of the strip they have the form

␺_␧,±⁼␾_␧+U⬁,±

R + r₊^±共␧,ky兲␾_␧+U⬁,+

L + r₋^±共␧,ky兲␾_␧+U⬁,−

L ,

共10兲 while to the right of the strip共x⬎L兲 one has

␺_␧,±^{= t}+

±共␧,ky兲␾_␧+U⬁,+

R + t₋^±共␧,ky兲␾_␧+U⬁,−

R . 共11兲

For␧⫽0 the form of the solution in the region x苸关0,L兴 is self-evidently a linear combination of the four solutions␾_␧±^L,

␾_␧±^R. Care must however, be taken in analytical work to use proper linear combinations of these modes that remain lin- early independent exactly at␧=0 共the Dirac point兲. 共See Ap- pendix A for explicit formulas.兲

The four transmission amplitudes t_±^± for given ␧ and ky

can be combined in the transmission matrix t共␧,ky兲 =

冉

^tt+^{++共␧,ky兲 t−+共␧,ky兲}

−共␧,ky兲 t₋⁻共␧,ky兲

冊

^{. 共12兲}

We consider a short and wide geometry LⰆW, in which the boundary conditions in the y direction become irrelevant. For simplicity, we take periodic boundary conditions, such that kyis quantized as k_y,n= 2␲n / W, n = 0 , ± 1 , ± 2 , . . .. In the re- gime LⰆW, 兩␧兩ⰆU_⬁ considered here, both the discreteness and the finiteness of the modes in the contact region can be ignored. As a consequence, the traces in Eqs.共5兲 and 共6兲 may be replaced by integrals through the prescription

Tr共tt^†兲^p→W

␲

冕

0

⬁

dk_y␴=±

兺

^{关T␴共EF,ky兲兴p, 共13兲}

where T_±are the two eigenvalues of tt^†.

IV. RESULTS

Figure3 contains a gray-scale plot of the total transmis- sion probability Tr共tt^†兲 as a function of ky and ␧. Darkly shaded regions indicate resonances of high transmission, similar to those found in Ref.15.

The location␧resof resonances can be estimated by equat- ing k_xL /␲ to an integer n. This yields the curves

␧res共n兲共ky兲 = ⫿1

2t_⬜±

冑

¹4t_⬜² +

冉

^␲Ln

冊

^{2+ ky2, 共14兲}

indicated in the figure by dashed lines. It is seen that good agreement is reached for兩ky兩Ⰶ1/L and again for 兩ky兩Ⰷ1/L.

For 兩kyL兩⯝1 there is a cross over. In regions 共c兲 and 共d兲, demarkated by the curves ␧_res^共0兲, the transmission generally drops to zero, since in these regions the longitudinal momen- tum kxis imaginary.

There is however a curious feature close to␧,ky= 0. The resonance closest to the Dirac point behaves differently from all the other resonances. When 兩ky兩 is increased, it moves closer to the Dirac point rather than away from it, eventually crossing into regions 共c兲 and 共d兲 of evanescent modes. It is this resonance of evanescent modes that is responsible for the pseudodiffusive transport at the Dirac point.

At␧=0, the exact formula for the eigenvalues of tt^†in the U_⬁→⬁ limit is

T_±共␧ = 0,ky兲 = 1

cosh²共ky⫿ kc兲L, 共15兲

kc= 1

Lln

冋

^2lL⬜+

冑

^{1 +}4l^L2_⬜²

册

^{. 共16兲}

In Fig.4the two transmission coefficients T_±共0,ky兲 are com- pared to the single transmission coefficient T_monolayer共0,ky兲

= 1 / cosh²共kyL兲 of the monolayer.^1,10 Details of the calcula- tion may be found in Appendix A.

FIG. 3. Total transmission probability Tr共tt^†兲 as a function of ␧ and k_yfor U_⬁= 50t_⬜and L = 50l_⬜. Darkly shaded regions indicate high transmission. Gray dashed lines indicate the estimate共14兲 for the occurrence of resonances in regions共a兲 and 共b兲, while solid lines indicate the boundary between propagating and evanescent modes.

Arrows point to the resonances of evanescent modes close to the Dirac point, responsible for the pseudodiffusive transport.

Since the two bilayer coefficients are displaced copies of the monolayer coefficient, any observable of the form A

= Tr f共tt^†兲, with f an arbitrary function is twice as large in a bilayer as it is in a monolayer. From Eqs. 共5兲 and 共13兲 we obtain

G_bilayer= 2G_monolayer=2G₀

␲ W

L, 共17兲

P_bilayer= 2P_monolayer=4e兩V兩G0

3␲ W

L, 共18兲

F_bilayer= F_monolayer=¹₃. 共19兲 Figure 5 contains plots of both the conductivity ␴

= GL / W and Fano factor of the bilayer around the Dirac point. At energies associated with resonances at normal inci- dence,

␧_res^共n兲共0兲 = ±␲²បv

L

冋

^{lL⬜n2+O共l⬜/L兲3}

册

^{, 共20兲}

the conductivity and Fano factor show abrupt features. The width ⌬EF= 2␧_res^共1兲= 2␲²បvl_⬜/ L² of the energy window be- tween the resonances that straddle the Dirac point in the bilayer is smaller by a factor l_⬜/ L than in the monolayer.

V. DEPENDENCE ON THE POTENTIAL IN THE CONTACT REGION

So far we have assumed that the potential U_⬁in the con- tact region is large compared to the band splitting t_⬜near the Dirac point of the graphene bilayer. We believe that this is the appropriate regime to model a normal metal contact to the graphene sheet, which couples equally well to the two sublattices on each layer.

It is of interest to determine how large the ratio U_⬁/ t_⬜ should be to reach the contact-independent limit of the pre- vious section. Note that for U_⬁⬎t_⬜ there are two left- incident propagating modes in the leads for each ␧ and ky. When U_⬁ becomes smaller than t_⬜ one of the two modes

becomes evanescent, leading to an abrupt change in the con- ductivity and the Fano factor. This is evident in Fig. 6. For U_⬁− t_⬜ⲏបv/L, the conductivity and Fano factor have al- most reached their U_⬁→⬁ limits. For U_⬁ⱗt_⬜ the conduc- tivity is smaller and the Fano factor larger than when U_⬁

⬎t_⬜. Both quantities vanish when the Fermi momentum

冑

U_⬁t_⬜/v in the contact region drops belowប/L and the con- tact region is effectively depleted of carriers.

These finite-U_⬁ results can be used to make contact with the previous calculation of Katsnelson,¹² who found a con- ductivity␴^{= G}0/ 2 and a Fano factor F = 1 − 2 /␲at the Dirac point, in the regime បv/LⰆ

冑

U_⬁t_⬜Ⰶt_⬜. These values are indicated in Fig.6by horizontal lines. The intersection point with our curves occurs at nearly the same value of U_⬁/ t_⬜for both quantities. The intersection point moves closer and closer to U_⬁= 0 as the sample length L is increased, but there is no clear plateau around the intersection point. Moreover, as shown in Appendix B, the intersection point does not cor- respond to a minimum or maximum as a function of the Fermi energy, so that these values would be difficult to ex- tract from a measurement.

We do believe that the results of Ref. 12 describe the asymptotic limit L / l_⬜→⬁ at EF⬅0. However, because in this limit the width ⌬EF⯝បvl_⬜/ L² of the resonance at the Dirac point vanishes, it seems unobservable.

FIG. 4. Solid curves: Transmission coefficients T_±of the bilayer according to Eq.共15兲 at L=50l⬜. These coefficients are displaced copies of the monolayer result共dashed兲.

FIG. 5. Conductivity␴ 共top兲 and Fano factor F 共bottom兲 of the bilayer, as a function of the Fermi energy E_F measured from the Dirac point for U_⬁= 50t_⬜ and L = 50l_⬜. Abrupt features occur at E_F⯝␧res共n兲共ky= 0兲 关vertical lines, given by Eq. 共20兲兴.

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VI. CONCLUSION

In conclusion, we have demonstrated that the pseudodif- fusive transport at the Dirac point, discovered in Ref.1for a carbon monolayer, holds in a bilayer as well. All moments of the current fluctuations have the same relation to the mean current as in a diffusive metal. In particular, the Fano factor has the 1 / 3 value characteristic of diffusive transport, even though the bilayer is assumed to be free of impurities or lattice defects.

Although we found that an undoped bilayer transmits as two undoped monolayers in parallel, the two systems behave very different away from charge neutrality. The resonance of evanescent modes around the Dirac point of zero Fermi en- ergy has width⌬EF⯝បvl_⬜/ L²in a bilayer, which is smaller than the width in a monolayer by the ratio of the interlayer coupling length l_⬜and the separation L of the metal contacts.

Since l_⬜⬇1.5 nm, one would not be able to resolve this resonance in the␮m-size samples of Ref.11. These experi- ments found no qualitative difference in the conductance ver- sus gate-voltage dependence of monolayer and bilayer graphene, both showing a minimum conductivity at the Dirac point of G₀. Smaller junctions in the 10– 100 nm range as are now being fabricated should make it possible to resolve the transmission resonance of evanescent modes predicted here, and to observe the unusual pseudodiffusive dynamics asso- ciated with it.

ACKNOWLEDGMENTS

This research was supported by the Dutch Science Foun- dation NWO/FOM. We have benefitted from discussions with E. McCann.

APPENDIX A: TRANSMISSION EIGENVALUES AT THE DIRAC POINT

In this Appendix we give some detail of the calculation that leads to the transmission coefficients T_±共␧=0,ky兲 of Eq.

共15兲. At the Dirac point and in the limit of large U_⬁, the left-incident eigenstates of the Hamiltonian 共3兲 are of the form

␺±共x兲

=

冦

^{eeikikyyyy关共c关␰±R±1e␹iU1⬁+ cx+±2共r␹2±+兲e␰L+k+ ryx+±−␰共cL−±3兲e␹−iU3+ c⬁x兴4±␹4兲e−kyx}兴 0 ⬍ x ⬍ L,^{x⬍ 0,} e^ikyye^{iU⬁共x−L兲}关t+±␰+

R+ t₋^±␰−

R兴 x⬎ L,

冧

共A1兲

with the definitions

FIG. 7. Conductivity共top兲 and Fano factor 共bottom兲 around the Dirac point, for L = 100l_⬜and U_⬁= 0.2t_⬜.共These parameter values correspond to the intersection point of our curves with the predic- tion of Ref.12 in Fig.6.兲 The solid lines were obtained using the four-band Hamiltonian 共3兲, while the dashed lines were obtained from the two-band Hamiltonian共B1兲.

FIG. 6. Dependence of the conductivity and Fano factor at the Dirac point on the potential U_⬁ in the contact region, for L

= 100l_⬜. Thin horizontal lines indicate the values of Ref.12. The values obtained in this paper correspond to a plateau reached for U_⬁/ t_⬜ⲏ1.

␰±

R=

冢

^⫿1⫿111

冣

^{, ␰±L=}

冢

^{⫿1− 1±11}

冣

^{, 共A2兲}

␹1=

冢

⁰¹⁰⁰

冣

^{, ␹2=}

冢

^{− it001⬜x}

冣

^{, ␹3=}

冢

^{− it100⬜x}

冣

^,

␹4=

冢

⁰⁰⁰¹

冣

^{. 共A3兲}

These eigenstates must be continuous at x = 0 and x = L, lead- ing to an 8⫻8 system of linear equations Mb±= c_±with

M =

冢

^{− 1 − 1 0− 1100001 − 1 0000101 10000z − iLt010000z⬜z − iLt0110000 ⬜ 0 − 1 − 1101 − z0000 10000z − 1− z− z0000}

冣

^,

b_±=

冢

^{crrccctt±−+±3421±−±+}

冣

^{, c±=}

^冢

^{⫿1⫿1110000}

^冣

^{. 共A4兲}

We abbreviated z = e^kyL. By solving these equations, one finds the transmission matrix

t = 2i

2 +共L/l_⬜兲²+ 2 cosh共2kyL兲

⫻

冉

^共L/l⬜− 2i兲cosh共kyL兲 共L/l_⬜兲sinh共kyL兲

−共L/l_⬜兲sinh共kyL兲 −共L/l_⬜+ 2i兲cosh共kyL兲

冊

^.

共A5兲

The eigenvalues of tt^†are then given by Eq.共15兲.

APPENDIX B: FOUR-BAND VERSUS TWO-BAND HAMILTONIAN

In this appendix we verify that the difference in the results obtained here and in Ref. 12 is not due to the different Hamiltonians used in these two calculations.

In Ref.12the limit t_⬜→⬁ was taken at the beginning of the calculation, reducing the 4⫻4 Hamiltonian 共3兲 to the effective 2⫻2 Hamiltonian⁴

H_eff= −v²

t_⬜

冉

共px+ ip⁰ y兲^{2 共px− ip}0 ^y兲2

冊

^{+ U共x兲}

冉

^{1 0}0 1

冊

^.

共B1兲

Only the two lowest bands near the Dirac point are retained in H_eff, as is appropriate for the regime U_⬁Ⰶt_⬜.

We have repeated the calculation of conductance and Fano factor using both Hamiltonians 共3兲 and 共B1兲, for pa- rameter values corresponding to the intersection point of Fig.

6, and find good agreement共see Fig.7兲.

1J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J.

Beenakker, Phys. Rev. Lett. 96, 246802共2006兲.

2For a review and a tutorial on shot noise we refer, respectively, to:

Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1共2000兲; C. W.

J. Beenakker and C. Schönenberger, Phys. Today 56共5兲, 37 共2003兲.

3P. R. Wallace, Phys. Rev. 71, 622共1947兲.

4E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805共2006兲.

5J. Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. Guinea, Phys. Rev. B 73, 214418共2006兲.

6This value of the interlayer coupling strength␥1refers to graph- ite; the value for a bilayer is not yet known.

7J. Cserti, cond-mat/0608219共unpublished兲.

8M. Koshino and T. Ando, Phys. Rev. B 73, 245403共2006兲.

9J. Nilsson, A. H. Castro Neto, F. Guinea, and N. M. R. Peres, cond-mat/0604106共unpublished兲.

10M. I. Katsnelson, Eur. Phys. J. B 51, 157共2006兲.

11K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I.

Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Nat. Phys. 2, 177共2006兲.

12M. I. Katsnelson, Eur. Phys. J. B 52, 151共2006兲.

13J. Nilsson, A. H. Castro Neto, F. Guinea, and N. M. R. Peres, cond-mat/0607343共unpublished兲.

14E. McCann, Phys. Rev. B 74, 161403共2006兲.

15M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620共2006兲.

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