Signatures of evanescent mode transport in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Signatures of evanescent mode transport in graphene
A.D. Wiener and M. Kindermann School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
We calculate the shot noise generated by evanescent modes in graphene for several experimentalsetups. For two impurity-free graphene strips kept at the Dirac point by gate potentials, separatedby a long highly doped region, we find that the Fano factor takes the universal value F = 1 /
4. Fora large superlattice consisting of many strips gated to the Dirac point interspersed among dopedregions, we find F = 1 / (8 ln 2). These results differ from the value F = 1 / PACS numbers: 72.80.Vp, 73.23.Ad, 73.50.Td, 73.63.-b
I. INTRODUCTION
The nonequilibrium current fluctuations, or shot noise,in ballistic graphene have received much attentionsince the seminal paper by Tworzydlo et al. In thatwork, it was shown that shot noise can be generated evenin a completely impurity-free sheet of graphene. Thisresult is surprising at first sight, as the shot noise van-ishes in conductors without electron scattering.
Theunanticipated noise is caused by evanescent (exponen-tially damped) waves that backscatter electrons, even inclean graphene.It has been shown by Tworzydlo et al. that in a cleansheet of graphene at its Dirac point, that is, at zero chem-ical potential, the shot noise is not only nonzero, but,moreover, it has universal characteristics. The shot noisenormalized by the mean current and expressed in units ofthe electron charge e – this quantity is commonly referredto as the Fano factor F – takes the value F = 1 /
3. Thisprediction has generated much theoretical interest in thetopic, and considerable experimental activity.
So far, the prediction of Tworzydlo et al. has beentested by two experiments, as reported in Refs. 18 and 19,and a Fano factor close to F = 1 / is identical tothe one expected for conventional disordered conductorsin the diffusive regime. It has been confirmed in nu-merous regimes, both with and without electron-electron interactions, that a Fano factor close to F = 1 / F = 1 / et al. , an additionalsignature of evanescent mode transport was observed;namely a strong dependence of the measured Fano factoron the chemical potential. Such dependence on the chem-ical potential is expected for clean graphene. As thechemical potential departs from the Dirac point, moreand more evanescent waves become propagating andcease to backscatter electrons, decreasing the Fano factor.The observed dependence of F on the chemical potential does not occur in generic diffusive conductors, mak-ing it a more distinctive signature of evanescent modetransport. Nevertheless, other scenarios are also con-sistent with the doping dependence of F reported byDanneau et al. , such as energy-dependent scattering.Additional experimental signatures of transport throughevanescent modes in graphene are therefore desirable.In this article, we propose experimental geometries forwhich the transport through evanescent modes at theDirac point of graphene has unambiguous signatures. Wefirst study a sheet of graphene subject to gate potentialsthat induce two strips of graphene with chemical poten-tial at the Dirac point, separated by a highly doped re-gion. We show that the Fano factor takes a universalvalue in this geometry, as in that of Ref. 5: F = 1 / Similar results are obtained for longer cascades of p > p → ∞ of a long graphenesuperlattice with a piecewise constant potential. Suchsuperlattices have recently received much attention as away to engineer the bandstructure of graphene - tothe point of creating new Dirac cones in the electronicspectrum. Under certain conditions, we find for suchsuperlattices yet another universal value of the Fano fac-tor: F = 1 / (8 ln 2).This paper is organized as follows: In section II, weanalytically derive the universal Fano factor F = 1 / F from this value as asymmetries areintroduced into the setup, an additional experimental sig-nature of evanescent mode transport. In section III, wefirst generalize our approach to cascades of an arbitrarynumber p of evanescent regions in series before takingthe long superlattice limit p → ∞ . We conclude with asummary in section IV. WL R x y x V L V R V d λ FIG. 1. Schematic of a graphene strip of width W consistingof a highly doped region (gray rectangle) of length λ at voltage V d separated by weakly doped regions of length L and R .Separate gate leads (not shown) control the voltages V L and V R in the weakly doped regions. The system is contacted ateither end by ideal leads (not shown) at the lead potential V ℓ .The electrostatic potential in the strip is plotted above theschematic as a function of x . II. EVANESCENT MODE TRANSPORTTHROUGH TWO EVANESCENT STRIPS INSERIES
We analyze the transport through a graphene sheet ofwidth W subject to gates that allow one to tune twostrips of lengths L and R to a chemical potential close tothe Dirac point. We denote the electrostatic potentials inthose strips by V L and V R , respectively. The two stripsare separated by a highly doped region of length λ atelectric potential V d (see Fig. 1).We model the leads that contact the sample by highlydoped regions of graphene to the left and to the rightof the sample at voltage V ℓ , as in the calculations ofTworzydlo et al. It has been confirmed in density func-tional calculations that this model correctly describestransport into certain types of contacts. In units where ~ = 1, our model Hamiltonian takes the form H γ = v σ γ · p + V ( x ) , (1)where σ γ = ( γσ x , σ y ) is a vector of Pauli matrices withthe valley index γ = ± p is the electron momentum and v is the electron velocity. The potential V ( x ) takes thevalue V d in the highly doped regions, V ℓ in the leads andthe gate voltages V L and V R in the regions that can betuned to the Dirac point (see Fig. 1). We assume the mi-croscopic potential, which is represented by the potential V ( x ) in our long-wavelength theory, to be smooth on thelattice scale, so that it does not scatter between valleys.The plane wave solutions of the Dirac equation haveenergy ǫ = V d ± v p k d + q n in the highly doped regions and ǫ = V L ( R ) ± v q k L ( R ) + q n in the left (right) gatedregions, where the ± sign refers to the conduction and va-lence bands, respectively. We choose (cid:12)(cid:12) V ℓ ( d ) (cid:12)(cid:12) ≫ v/L, v/R such that, at the Fermi level, all relevant modes are prop-agating (real k ℓ ( d ) ) in the leads and the highly dopedregions. In the weakly doped regions, both propagating(real k L ( R ) ) and evanescent (imaginary k L ( R ) ) modes canoccur.The transverse wavenumbers q n depend on the bound-ary conditions at y = 0 and y = W . We consider a classof boundary conditions that do not couple longitudinaland transverse wavenumbers. In this case, the transversewavenumber q n is a good quantum number, and modesare not mixed at the weakly-to-highly doped region in-terfaces. For the infinite mass, metallic armchair edge, orsemiconducting armchair edge boundary conditions, thetransverse wavenumber is given by q n = ( n + α ) π/W ,where α = / / . In the continuum limit W ≫ L, R , which we take forthe remainder of this article, one expects the transportproperties to be independent of the boundary conditions.Accordingly, α drops out of the calculation. A. The universal limit
We first consider the situation in which both weaklydoped regions of the sample are tuned to the Dirac point, V L = V R = 0. Matching modes at the interfaces betweensegments of differing potential results in the transmissionamplitude t n = (cid:2) e ik d λ sinh( n L ) sinh( n R ) ++ e − ik d λ cosh( n L ) cosh( n R ) (cid:3) − (2)for mode n at the Fermi level, which we choose to be atenergy ǫ F = 0. The dimensionless “lengths” L and R aregiven by L = πL/W and R = πR/W . The Fano factoris found from the transmission probabilities T n = | t n | as F = P n T n (1 − T n ) P n T n . (3)In the continuum limit, W ≫ L, R , the sums becomeintegrals over the mode index, and the Fano factor isgiven by F = 1 − I /I with I = R ∞ T n dn,I = R ∞ T n dn. (4)In Fig. 2, the transmission probability determined fromEq. (2) is integrated numerically in order to obtain the λ/ | κ d | L F FIG. 2. Fano factor F for two graphene strips as in Fig. 1 for L = R and chemical potential at the Dirac point, as a functionof the length of the highly doped region in units of | κ d | L .The curve is calculated from Eq. (3) in the limit N → ∞ . TheFano factor approaches 0 . . . . as λ →
0, in agreement withthe calculations of Tworzydlo et al. For large values of thelength of the highly doped region, the Fano factor approaches0 . . . . , in agreement with the analytic calculations presentedin the text. Fano factor for a symmetric system ( L = R ) as a func-tion of the thickness λ of the central, highly doped re-gion. As λ →
0, the Fano factor approaches 1 /
3. Thisresult agrees with the calculations of Tworzydlo et al. , as this limit corresponds to transmission through a singlegraphene strip at the Dirac point. In contrast, the Fanofactor approaches 0 . . . . in the limit λ ≫ | κ d | L , where κ d = V d /v .This numerical result suggests that the Fano factormay be accessible analytically in the limit λ ≫ | κ d | L .Indeed, one finds that the integrals in Eq. (4) can be doneanalytically for λ ≫ | κ d | L . Consider first I . We writethe transmission probability determined from Eq. (2) inthe form T n = 1 α ( n ) 11 + β ( n ) cos[2 k d ( n ) λ ] , (5)where α ( n ) = cosh ( n L ) cosh ( n R ) + sinh ( n L ) sinh ( n R ) ,β ( n ) = 2 cosh( n L ) cosh( n R ) sinh( n L ) sinh( n R ) /α ( n ) ,k d ( n ) = q κ d − n π /W . (6)The transmission probability decays exponentially with n ( L + R ), and the integrals in Eq. (4) are thus cut-off at n ≃ max { / L , / R} .The key observation is that for λ ≫ | κ d | L , the cosinefunction in Eq. (5) oscillates rapidly on the scale of theexponential decay of T n (recall that we assume | κ d | L ≫ λ/ | κ d | L → ∞ , therefore, α ( n ) and β ( n ) are constant within one period of oscillation∆ n ( n ) = κ d W λπn (7)of the cos [2 k d ( n ) λ ] function around a given index n .Consequently, I , restricted to an interval of length∆ n ( n ) around n , becomes the integral of the func-tion [1 + β ( n ) cos ( δ − ωn )] − , which can be done analyt-ically. Here, δ and ω are found by linearizing 2 k d ( n ) λ = δ ( n ) − ω ( n ) n with δ ( n ) = 2 κ d λ (cid:0) n π / κ d W (cid:1) and ω ( n ) = 2 π λn /κ d W . In the limit λ/ | κ d | L →∞ , the periods ∆ n ( n ) are short compared to the de-cay scale of the transmission probability set by L + R ,and the requisite sum over all periods becomes a secondintegral. Noting that | β ( n ) | ≤
1, this results in I ≈ Z ∞ γ − ( n ) dn , (8)where γ ± ( n ) = cosh ( n L ) cosh ( n R ) ±± sinh ( n L ) sinh ( n R ) . (9)The integral I of Eq. (4) can be done in the same man-ner, with the result I ≈ Z ∞ γ + ( n ) γ − ( n ) dn . (10)In the symmetric case, L = R , the integrals in Eq.(8) and (10) can also be evaluated analytically, result-ing in the same Fano factor as for a ballistic quantumdot, F = 1 /
4, in accordance with the numerical resultsshown in Fig. 2. Evanescent transport in this geometrythus has an unambiguous signature, with a Fano factorthat differs from the one in a disordered sample. Forthe asymmetric case, we calculate the Fano factor nu-merically as a function of
R/L , with the results plottedin Fig. 3. The Fano factor approaches 1 / R/L → ∞ , which again corresponds to a single grapheneregion at the Dirac point, as considered by Tworzydlo etal. The dependence of the Fano factor on the ratio
R/L shown in Fig. 3 is a more distinctive signature of evanes-cent wave transport than the value F = 1 / L = R alone. In a typical experiment, however, the lengths ofthe gated regions cannot be changed easily. Alterna-tively, the gate voltages V L and V R can be controlled.We discuss the dependence of F on those gate voltagesin the following subsection. B. Voltage induced signatures of evanescent modetransport
Here, we obtain the dependence of the Fano factor onthe gate voltage V R in the symmetric configuration, L = R/L F FIG. 3. Length ratio dependence of the Fano factor fortwo graphene strips as in Fig. 1 with chemical potential atthe Dirac point. The curve is calculated from the integralsgiven in Eq. (8) and Eq. (10) in the continuum limit with λ ≫ | κ d | L . The Fano factor shows a minimum of 0 . . . . for the symmetric case ( R/L = 1) and tends toward 0 . . . . as R/L → R/L → ∞ , in agreement with the results ofTworzydlo et al. R , (cf. Fig. 1) with the left gated region at the Diracpoint, V L =0. The transmission amplitude as a functionof k R takes the form t n = k R (cid:2) e − ik d λ cosh n L ( k R cos k R L −− iκ R sin k R L ) + e ik d λ nπW sinh n L sin k R L i − , (11)where κ R ( d ) = k R ( d ) + q n with κ R ( d ) = V R ( d ) /v . We con-sider the limit λ/ | κ d | L → ∞ of the resulting expressionfor the Fano factor, following the methods of the previ-ous section. In this case, we perform the final integration,corresponding to the sum over periods ∆ n ( n ), numeri-cally, with the result plotted in Fig. 4. The Fano factoris 1 / V R is at the Dirac point, aspreviously calculated, and it approaches 1 / V R ≫ v/L . The crossover between F = 1 / F = 1 / V L ≃ v/L serves as an-other distinctive signature of evanescent mode transport. III. EVANESCENT MODE TRANSPORT IN AGRAPHENE SUPERLATTICE
In this section, we investigate the shot noise for evanes-cent transport through a graphene superlattice consistingof many graphene strips tuned to the Dirac point, alter-nating with doped regions, as depicted in Fig. 5. Weeventually take the long superlattice limit of an infinitenumber of such regions. We take the graphene regions V R L/v F FIG. 4. The Fano factor for two graphene strips as in Fig. 1for L = R and λ ≫ | κ d | L as a function of gate voltage V R in the right graphene region, measured in units of v/L . Gateleads fix the chemical potential in the left evanescent regionto the Dirac point ( V L = 0). The Fano factor is 0 . . . . when V R is at the Dirac point, as previously calculated, andit approaches 0 . . . . as V R ≫ v/L . Wλ L xV xy FIG. 5. Schematic of a segment of a graphene superlattice ofwidth W , consisting of evanescent graphene regions of length L separated by doped regions (gray rectangles) of length λ at the potential V d . Separate gate leads (not shown) fix thevoltages in the graphene regions to the Dirac point, V =0. The superlattice is contacted at either end by leads atpotential V ℓ (not shown). The superlattice potential is plottedabove the schematic as a function of x . gated to the Dirac point to be of length L , and they areseparated by doped regions of length λ at the voltage V d .The contacts at both ends are again modeled by highlydoped graphene at the lead potential V ℓ . The potential V ℓ drops out of the calculation in the limit V ℓ , V d ≫ v/L . A. Transmission through a cascade of evanescentstrips in series
In order to calculate the transmission probabilitythrough the graphene superlattice depicted in Fig. 5, weemploy the transfer matrix method. The transfer ma-trix M ( x, x ′ ) for the requisite Dirac spinors of a modewith transverse momentum q satisfies the equation ∂ x M ( x, x ′ ) = (cid:20) − i V ( x ) v σ x + qσ z (cid:21) M ( x, x ′ ) . (12)In addition, one has M ( x, x ) = I , M ( x, x ′ ) = M ( x, x ′′ ) M ( x ′′ , x ′ ), det M ( x, x ′ ) = 1 and M † ( x, x ′ ) σ x M ( x, x ′ ) = σ x . The latter conditionensures current conservation.We write the transfer matrix in the doped regions as M d ( x, x ′ ) = A d ( x ) A − d ( x ′ ). Here, A d ( x ) satisfies Eq. (12)in the doped regions, where V ( x ) = V d . Similarly, thetransfer matrix in the lead regions contacting either endof the superlattice is M ℓ ( x, x ′ ) = A ℓ ( x ) A − ℓ ( x ′ ), where A ℓ ( x ) satisfies Eq. (12) in the lead regions with V ( x ) = V ℓ . Following Ref. 35, we choose matrices A ℓ ( d ) ( x ) thatallow one to project onto right and left moving states: A j ( x ) = r κ j k j k j + iqκ j e − ik j x − k j + iqκ j e ik j x e − ik j x e ik j x ! , (13)where j = { ℓ, d } and k j = κ j − q with κ j = V j /v . The transfer matrix in the evanescent regions,where the electric potential vanishes, is M e ( x, x ′ ) =exp [ qσ z ( x − x ′ )]. The transfer matrix through asingle sequence consisting of a region at the Diracpoint followed by a doped segment is then given by M ( L + λ,
0) = M e ( L + λ, λ ) M d ( λ, p doped-evanescent segments in series, contactedat either end by ideal leads, is found using M and thematrices A ℓ of Eq. (13) as M p ( x, x ′ ) = A ℓ ( x ) A − ℓ [ p ( L + λ )] [ M ( L + λ, p ×× A ℓ (0) A − ℓ ( x ′ ) (14)at x > p ( L + λ ) and x ′ < M p ( x, x ′ ) in the leads,one extracts the transmission amplitudes through the en-tire array as explained in Ref. 35: T ( p ) = 1 / | α | , where (cid:18) α β ∗ β α ∗ (cid:19) = lim x →∞ A − ℓ ( x ) M p ( x, − x ) A ℓ ( − x ) , (15)which, with Eq. (14), becomes (cid:18) α β ∗ β α ∗ (cid:19) = A − ℓ ( p ( L + λ )) [ M ( L + λ, p A ℓ (0) . (16)The transmission probability T ( p ) n of mode n is ob-tained by setting q = q n in the above equations, and wefind T ( p ) n = 2∆ ( n ) (cid:2) cosh ( n L ) − (cid:3) cosh [2 p · arcsinh(∆( n ))] + (cid:2) cos (2 k d ( n ) λ ) cosh ( n L ) − (cid:3) . (17)Here, ∆( n ) = q cos [ k d ( n ) λ ] cosh ( n L ) −
1. As previ-ously mentioned, the transmission probability becomesindependent of the lead potential in the limit κ ℓ , κ d ≫ /L that we have taken. In the special case of p = 2, thisequation reduces to Eq. (5).The Fano factor can be calculated numerically for acascade of arbitrary length by substitution of the trans-mission probability (17) into the integrals I and I ofEq. (4). The results are plotted as a function of the cas-cade size p in Fig. 6 with error bars caused by errorsin the numerical integration. The Fano factor decreasesrapidly from its maximum value of 1 / p = 1 and ap-proaches 0 . ... as p → ∞ . We next compute the exactvalue of F in this limit p → ∞ of a long superlattice,again assuming λ ≫ | κ d | L . B. The long superlattice limit
The transmission probability T ( p ) n of Eq. (17) is plottedas a function of mode index n for various values of λ and p in Fig. 7. One observes a series of peaks in T ( p ) n , whosenumber increases with λ/ | κ d | L , as illustrated in thefirst row of plots in Fig. 7. The peaks occur at values of n where ∆ is imaginary, and T ( p ) n is exponentially dampedwhenever ∆ is a real number.When ∆ is imaginary, the second hyperbolic cosinein the denominator of T ( p ) n oscillates with a frequencyproportional to p , the number of graphene segments inthe cascade. Therefore, T ( p ) n has a series of “subpeaks”in the regions of imaginary ∆, whose number increaseswith p , as illustrated in the second row of plots in Fig. 7.We further observe in the second row of Fig. 7 that thedamping to the sides of the regions with imaginary ∆ isenhanced as p increases. This is also due to the factor of p in the second hyperbolic cosine function of Eq. (17).Motivated by the above observations, we partition the p F FIG. 6. Fano factor for a graphene superlattice, as picturedin Fig. 5, as a function of p , the length of the cascade. Thecurve is calculated from the transmission probability given inEq. (17) with λ = 10 | κ d | L . The Fano factor starts at itsmaximum value of 1 / et al. In the long superlattice limit, p → ∞ , the Fano factor approaches 0 . ... , in agreementwith our analytic calculations. The error bars are caused byerrors in the numerical integration. nπL/W T ( p ) n nπL/W T ( p ) n ad cb fe FIG. 7. Plots of the transmission probability Eq. (17) asa function of mode index n . The cascade size is p = 2 inthe first row of plots, while the doped region thickness takesthe increasing values λ = 0 . | κ d | L (a), λ = 2 | κ d | L (b)and λ = 20 | κ d | L (c). Larger thicknesses lead to a higherfrequency of peaks in the transmission probability. In thesecond row, the doped region thickness is λ = 2 | κ d | L , andthe increasing cascade sizes p = 2 (a), p = 6 (b), and p = 10(c) cause an increasing frequency of the sub-peak oscillationsand strong damping outside of the peak regions. nπL/W T ( p ) n n λ n p ∆ n λ ( n λ ) ∆ n p ( n p ) FIG. 8. Transmission probability Eq. (17) as a function ofmode index n for p = 4 and λ = 4 | κ d | L , showing peak os-cillations with period set by λ and sub-peak oscillations withperiod set by pλ . The first peak region to the right of the cen-tral region, indexed by n λ , is shown magnified in the inset.The peak period is given by ∆ n λ ( n λ ) in the neighborhood ofpeak n λ , while the period of sub-peak oscillation is ∆ n p ( n p ),around peaks indexed by n p . Both periods become infinitesi-mal in the limit p ≫ λ ≫ | κ d | L . wave numbers into a series of peak regions where ∆ isimaginary. The length of these regions is on the or-der of ∆ n ( n λ ) in Eq. (7). Each peak region is thensub-partitioned into sub-peak regions, with a length thatcorresponds to the oscillation period of the second hyper-bolic cosine of Eq. (17) at imaginary ∆. That length is ofthe order ∆ n p ( n p ) ∼ ∆ n ( n p ) /p . Fig. 8 illustrates bothperiods of oscillation, showing the transmission proba-bility Eq. (17) for p = 4 with doped region thickness λ = 4 | κ d | L .In order to calculate the Fano factor for a longgraphene superlattice ( p ≫ I and I of Eq. (4) for each peak region ∆ n λ ( n λ ) over all sub-peak regions ∆ n p ( n p ). The result is then summed overthe peak region ∆ n λ ( n λ ) by a second integration, result-ing in the desired integral over one peak region. Finally,in the limit λ ≫ | κ d | L , the peak spacing becomes smallcompared to the decay scale of the transmission proba-bility, set by L , and the full integral can be found by fur-ther summing the single peak result over all such peaks,amounting to a third integration.All of the above integration steps can be performedanalytically, giving the exact Fano factor in the limit p → ∞ with λ/ | κ d | L → ∞ : F = 1 / (8 ln 2). This re-sult is in agreement with the numerical results plotted inFig. 6, and it differs from the Fano factor calculated forthe geometry of section II, as well as the calculations ofTworzydlo et al. The shot noise for transport througha graphene superlattice as depicted in Fig. 5 thus pro-vides another unambiguous signature of evanescent modetransport.
IV. DISCUSSION AND CONCLUSIONS
Throughout this article, we have assumed translationalinvariance in the y-direction. We now briefly discuss thesensitivity of our results to a breaking of that symmetrybefore summarizing. As a typical mechanism for suchsymmetry breaking, we consider a gate edge which makesa non-zero angle ∆ φ with the vertical. At such an inter-face, transverse modes are mixed. Electrons with mo-mentum k x ∼ κ d = V d /v in the x -direction increase theirmomentum in the y -direction by an amount ∆ q ≈ κ d ∆ φ as they traverse such an edge.To avoid qualitatively changing the above calculations,this momentum shift must be small on the scale κ d /pλq of the oscillations of the transmission probability, requir-ing ∆ φ ≪ /pλq . Noting that the typical momenta con-tributing to the Fano factor in our calculation are of order q ≃ /L , this results in the condition∆ φ ≪ L/pλ. (18)In addition, we have assumed λ ≫ | κ d | L in all of our analytical calculations. These results therefore require1 ≪ | κ d | L ≪ λL ≪ p ∆ φ (19)in setups with angular imperfections.In conclusion, we have proposed several experimentalgeometries for which the shot noise provides an unam-biguous signature of transport through evanescent modesat the Dirac point of graphene. For the case of twogated graphene strips in series, separated by a long highlydoped region, the Fano factor can be controlled througheither spatial asymmetry or the gate potentials. In par-ticular, the Fano factor has a universal minimum of F = 1 / F = 1 / F = 1 / (8 ln 2). ACKNOWLEDGMENTS
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