Magnetoconductance of the Corbino disk in graphene

Magnetoconductance of the Corbino disk in graphene

Adam Rycerz

Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, PL-30059 Kraków, Poland and Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany 共Received 17 September 2009; revised manuscript received 9 February 2010; published 2 March 2010兲 Electron transport through the Corbino disk in graphene is studied in the presence of uniform magnetic fields. At the Dirac point, we observe conductance oscillations with the flux piercing the disk area ⌽d, characterized by the period⌽0= 2共h/e兲ln共R_o/R_i兲, whereR_o共R_i兲is the outer共inner兲disk radius. The oscillations magnitude increase with the radii ratio and exceed 10% of the average conductance forR_o/R_iⱖ5 in the case of the normal Corbino setup or forR_o/R_iⱖ2.2 in the case of the Andreev-Corbino setup. At a finite but weak doping, the oscillations still appear in a limited range of兩⌽d兩ⱕ⌽d

max, away from which the conductance is strongly suppressed. At large dopings and weak fields we identify the crossover to a normal ballistic transport regime.

DOI:10.1103/PhysRevB.81.121404 PACS number共s兲: 73.43.Qt, 73.63.⫺b, 75.47.⫺m

An atomically thin carbon monolayer 共graphene兲 is widely considered as a successor of silicon in future elec- tronic devices.¹ Investigations of the low-energy properties of graphene, governed by the massless Dirac equation, con- stitute new and thriving subarea of condensed matter research.² Particularly striking feature of clean, undoped graphene samples is that zero density of states is accompa- nied by a nonzero universal value of the conductivity 4e²/共␲h兲.^3–6 This is a basic signature of the so-called pseudodiffusive regime, in which transport properties of graphene are indistinguishable from those of a classical dif- fusive conductor.⁷In this regime, the applied magnetic field does not affect the conductivity^8,9 and higher current cumulants.¹⁰Prada et al.also showed that for high dopings and magnetic fields, the pseudodiffusive behavior is recov- ered at resonance with the Landau levels 共LLs兲 in the ab- sence of disorder.

Numerous studies of graphene magnetoconductance focus on nanoribbons,¹¹ Aharonov-Bohm rings,^12,13 antidot lattices,¹⁴ and weak-localization effects in chaotic nanosystems.^15,16Cheianov and Fal’ko¹⁷showed the conduc- tance of a circular p-n interface is insensitive to the weak applied field. The author, Recher, and Wimmer recently iden- tified the crossover from the pseudodiffusive to thequantum- tunneling regime,¹⁸ which is characterized by a power-law decay of the conductanceG⬀L^−␣共whereLis the length of a sample area and ␣ is a geometry-dependent exponent兲 and appears for quantum billiard in undoped graphene at zero field. In the case of the Corbino disk with the outer radiusR_o and the inner radius R_i共see Fig. 1兲, we have L=R_o−R_i and

␣= 1, leading to the reciprocal decay ofGforR_oⰇR_i. As the tunneling regime shows up generically for billiards having 共at least兲 one narrow opening,¹⁸ the discussion of magnetic field effects—at least on a basic example—is desirable.

In this Rapid Communication, we analyze theoretically magnetoconductance of the Corbino disk in graphene at ar- bitrary dopings and magnetic fields. The paper is organized as follows: we start from the mode-matching analysis for the disk attached to heavily doped graphene leads, which em- ploys the total angular momentum conservation in a similar way as early works employed transverse momentum conser- vation for the strip geometry.^3,4Then, we discuss separately

the zero-doping and finite-doping situations and present the system phase diagram in the field-doping parameter plane.

The findings of Ref. 10 for the pseudodiffusive regime are reproduced forR_o/R_iⱗ2. The feature is a periodic共approxi- mately sinusoidal兲magnetoconductance oscillation visible in undoped or weakly doped disks with larger radii ratios and recovered at LLs for high dopings. Finally, we extend the analysis to the normal-graphene-superconductor 共Andreev- Corbino兲setup.

The analysis starts from the Dirac Hamiltonian in a single valley,¹⁹which is given by

H=vF共p+eA兲·␴⁺U共r兲, 共1兲

where v_F= 10⁶ m/s is the Fermi velocity, ␴⁼共␴x,␴y兲, p=

−iប共⳵x,⳵y兲 is the in-plane momentum operator, the electron charge is −e, and we choose the symmetric gauge A=^B₂共

−y,x兲. The electrostatic potential energyU共r兲=U₀in the disk area 共Ri⬍r⬍R_o兲, otherwise U共r兲=U_⬁. Since Hamiltonian 共1兲commutes with the total angular momentum operatorJz

= −iប⳵␸+ប␴z/2, the energy eigenfunctions can be chosen as eigenstates ofJz

x y

z

Ri

Ro

B= (0,0, B)

FIG. 1. 共Color online兲The Corbino magnetometer in graphene.

The current is passed through the disk-shaped area with the inner radiusR_iand the outer radiusR_oin a perpendicular magnetic field B=共0 , 0 ,B兲. The leads共yellow/light gray兲are modeled as infinitely doped graphene regions. The gate electrode共not shown兲is placed underneath to tune the doping in the disk area.

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␺j共r,␸兲=e^{i共j−1/2兲␸}

冉

␹^␹j↓^j共^↑r^共r兲兲e^i␸

冊

^{, 共2兲}

where j is an half-odd integer, s=↑,↓ denotes the lattice pseudospin, and we have introduced the polar coordinates 共r,␸兲. The Dirac equation now reduces to H_j␹j共r兲=E␹j共r兲, where␹j共r兲=关␹j↑共r兲,␹j↓共r兲兴^Tand

Hj= −iបvF␴x⳵r+U共r兲

+បvF␴y

冢

^{j− 1/2r 0+eBr2ប j+ 1r/20+eBr2ប}

冣

^{. 共3兲}

Subsequently, the scattering problem can be solved sepa- rately for each jth angular momentum eigenstate incoming from the origin共r= 0兲. As the angular dependence of the full wave function共2兲 does not play a role for the mode match- ing, the analysis limits effectively to the one-dimensional scattering problem for the spinor ␹j共r兲. We model heavily doped graphene leads by taking the limit of U共r兲=U_⬁

→⫿⬁ 共hereinafter, the upper sign refers to the conduction band, and the lower sign refers to the valence band兲 and define the reflection共transmission兲amplitudesrj共tj兲. For the inner lead共r⬍R_i兲, the wave function can be written as

␹j共i兲=e^⫾ik⬁r

冑

r

冉

¹¹

冊

^+rje⫿

冑

^ikr^⬁r

冉

^{− 11}

冊

^{, 共4兲}

where the first term represents the incoming wave and the second term represents the reflected wave. We further intro- duced k_⬁⬅兩E−U_⬁兩/共បvF兲→⬁. For the outer lead 共r⬎R_o兲 the wave function is

␹j共o兲=tj

e^⫾ik⬁r

冑

r

冉

¹¹

冊

^共5兲

and represents the transmitted wave. Defining k₀⬅兩E

−U₀兩/共បvF兲for the disk area共Ri⬍r⬍R_o兲, we write the wave function in a similar form as considered by Recher et al.²⁰ for the eigenvalue problem, namely,

␹j共d兲=Aj

冉

⫾iz^␰_j,1^j共↑1兲␰j共1兲↓

冊

^+Bj

冉

⫾iz^␰_j,2^j共↑2兲␰j共2兲↓

冊

^{, 共6兲}

wherez_j,1=关2共j+sj兲兴^−2sj,z_j,2= 2共␤/k₀²兲^sj+1/2关withsj⬅¹₂sgnj,

␤=eB/共2ប兲兴, and

␰js共␯兲=e^−␤r2/2共k0r兲^兩ls兩

再

^{MU共共␣␣jsjs,,␥␥jsjs,,␤␤rr22兲兲,, ␯␯= 2,= 1}

冎

^共7兲

with ls=j⫿¹₂ for s=↑,↓, ␣js=₄¹关2共l−s+兩ls兩+ 1兲−k₀²/␤兴 and

␥js=兩ls兩+ 1.M共a,b,z兲andU共a,b,z兲are the confluent hyper- geometric functions.²¹ Solving the matching conditions

␹j共i兲共Ri兲=␹j共d兲共Ri兲and␹共jo兲共Ro兲=␹共jd兲共Ro兲, we find the transmis- sion probability for jth mode

Tj=兩tj兩²= 16共k₀²/␤兲^兩2j−1兩

k₀²R_iR_o共X2j+Y_j²兲

冋

^{⌫共⌫共␥␣jj↑}↑^兲兲

册

^{2, 共8兲}

where⌫共z兲is the Euler Gamma function and

X_j=w_j⁻_↑↑+z_j,1z_j,2w⁻_j↓↓, Y_j=z_j,2w_j⁺_↑↓−z_j,1w_j⁺_↓↑,

w_jss^⫾_⬘=␰共1兲js共R_i兲␰^共2兲js⬘共R_o兲⫾␰共1兲js共R_o兲␰js^共2兲⬘共R_i兲. 共9兲 Without loss of generality, we choose B⬎0. For B⬍0 one getsTj共B兲=T_−j共−B兲.

First, we consider the zero doping limit, for which Eq.共8兲 simplifies to

Tj共k0→0兲= 1

cosh²关L共j+⌽d/⌽0兲兴, 共10兲 whereL= ln共Ro/R_i兲,⌽d=␲共R_o²−R_i²兲Bis the flux piercing the disk area, and ⌽0= 2共h/e兲L. We observe varying the ratio

⌽d/⌽0 affects Tj共k0→0兲 similarly as changing boundary conditions affects the corresponding formula for the strip geometry.^3,4 关Notice that Eq. 共10兲 is insensitive to the flux piercing the inner lead.兴 The disk conductance follows by summing over the modes

G=g₀

兺

j

Tj共k0→0兲=

兺

m=0

⬁

Gmcos

冉

^2␲⌽^m⌽0 ^d

冊

^{, 共11兲}

where g₀= 4e²/h is the conductance quantum 共the factor 4 includes spin and valley degeneracy兲and the Fourier ampli- tudes are

G₀=2g₀

L , Gm= 4␲²共−兲^mmg₀

L²sinh共␲²m/L兲 共m⬎0兲. 共12兲 The conductance given by Eq. 共11兲shows periodic oscilla- tions with the average valueG₀equal to the pseudodiffusive disk conductance.¹⁸ 共Thus, the averaging over ⌽d for the disk corresponds to the fictitious averaging over boundary conditions for the strip.兲 The approximate formula G⬇G₀ +G₁cos共2␲⌽d/⌽0兲 reproduces the full expression with the 1% accuracy for R_o/R_iⱕ10. The oscillations magnitude

⌬G⬅G_max−G_min⬇2兩G₁兩 converges rapidly to 0 withR_o/R_i

→1 共the pseudodiffusive transport regime兲, in agreement with earlier works^8–10 reporting no field dependence of the conductance. For instance, we obtain ⌬G⬍4⫻10⁻⁵G₀ for R_o/R_iⱖ2. In the tunneling regime, the oscillations magni- tude of ⌬Gⲏ0.1G₀ is reached for moderate radii ratios R_o/Riⱖ5. In this regime, when ⌽d/⌽0 is half-odd integer, the major contribution to the conductance originates from a single mode 共T−⌽d/⌽0= 1兲, and we haveG=G_max共withG_max

→g₀forR_oⰇR_i兲. For other values of⌽d, the conductance is generally dominated by the two modes, with j_⫾=

−int共⌽d/⌽0⫿¹₂兲−¹₂, which became equivalent for⌽d/⌽0in- teger, when G=G_min 共and G_minR_o/R_i→8g₀ for R_oⰇR_i兲 re- producing the zero-field situation.¹⁸

We now complement the discussion by analyzing a finite doping case to find out how stable are the conductance os- cillations when the gate voltage is controlled with a finite precision. For k₀⬎0 Eq. 共8兲 is well defined for arbitrary j provided that ¹₄k₀²/␤⁼共k₀l_B兲²/2⫽n= 1 , 2 , . . . 共LLs兲 with l_B

=

冑

ប/eBas the magnetic length. In such case, the asymptotic form for large fields isT^共_j^n兲⬇cosh⁻²关L共j− 2n+⌽d/⌽0兲兴, lead- ing to conductance oscillations as obtained above, see Eq.

共11兲. In fact,Tj共k0→0兲given by Eq.共10兲are reproduced for

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n= 0, showing the conductance oscillations for an undoped disk can be rationalized in terms of resonant transport through the zeroth LL pinned at the Dirac point.

The results for G, obtained by numerical summation of Tjs given by Eq.共8兲forR_o/Ri= 10, are shown in Fig.2. We first compare, in Fig. 2共a兲, the zero-doping magnetoconduc- tance 共red line兲 given by Eq. 共11兲 with those obtained for dopings varying fromk₀R_i= 10⁻⁴– 0.1, with the steps of one order of magnitude. Weak-doping curves follow the zero- doping one for first few periods when

⌽dⱗ ⌽d max=2h

e ln共k0R_i兲. 共13兲

For higher fields,Gdecays ase^{−共Ro−Ri兲2/共2lb2兲}. The high-Garea limited by Eq.共13兲shrinks rapidly with increasingk₀. How- ever, for high dopings 共k0共Ro−R_i兲ⲏ␲兲 the results start to follow the semiclassical picture, similarly as for the two- dimensional electron gas 共2DEG兲.²² The high-G area ex- pands with k₀ 关see Fig. 2共b兲兴 as it is now limited by the condition 2rc⬎R_o−R_i共withrc=k₀l_B² the cyclotronic radius兲, characterizing the ballistic transport regime. In particular, for 2r_c⬎R_o+R_i we have G/g₀⬇2k₀R_i, approaching the re- sult for zero field.^18,23 At high magnetic fields 共for which

2rc⬍R_o−R_i兲we enter thefield-suppressed transportregime, in which G⬃e^{−共Ro−Ri兲2/共2lb2兲} again, except from the isolated peaks 关see Fig. 2共c兲 for the plot in a logarithmic scale兴, which correspond to the resonances with LLs, and shrink with the field in the absence of disorder. At each resonance, the zero-doping field dependence ofGis approached for the high field.

The behaviors described above are presented in a con- densed form in the phase diagram shown in Fig.3. Colored areas represents the regions in the field-doping parameter plane where G⬎G₀ 共with the borders G=G₀ marked by solid lines兲. We also show 共with dashed lines兲 the limiting values of the magnetic field, at which the crossovers from the tunneling 共left兲 and from the ballistic 共right兲 to the field- suppressed transport regime occur. For the first LL, we dem- onstrate in a quantitative manner共see the inset兲how, starting from the ballistic regime and enlarging the field关but keeping 共k₀l_B兲²/2⬇1兴, one restores the tunneling behavior, character- ized by a chain of isolated islands ofG⬎G₀on the diagram.

So far, we have considered the disk attached to normal- metallic leads. For the Andreev-Corbino setup, with one nor- mal and one superconducting lead, the conductance is ex- pressed in terms of T_js given by Eq.共8兲as²⁴

G^NS= 2g₀

兺

_{j 共2 −}^Tj2_Tj兲^{2. 共14兲}

In particular, G^NS is still a periodic function of ⌽d at the Dirac point, and its Fourier decomposition G^NS共⌽d兲=G₀^NS +兺_m=1^⬁ G_m^NScos共2␲m⌽d/⌽₀兲leads to

G₀^NS共L兲=G₀共L兲, G_m^NS共L兲= 2Gm共2L兲. 共15兲 Although we haveG_m^NS/G_m→1 forR_o/R_i→⬁共and any m兲, magnetoconductance oscillations are noticeably amplified for moderate radii ratios. For instance, the magnitudes ⌬G/G₀

0 0.5 1.0 1.5

0 1 2

0 1 2 3 4 5 6

0 1

0 1 2 3 4 5 6

Φd/Φ0

Φd/Φ0

G[4e2 /h]G[4e2 /h] G[4e2/h]

10-⁶ 10-⁴ 10-² 1

k0Ri= 0

k₀ Ri=0

.1

10− 2

10− 3

10− 4

k0Ri=0.6

(a)

(b)

0.2 0.3 0.4 0.5

Φd=0 Φ0

2Φ0

3Φ0

4Φ0 (c)

k0Ri

FIG. 2. 共Color online兲 Conductance as a function of the mag- netic field and the doping forR_o/R_i= 10.共a兲Magnetoconductance at weak doping 共specified for each curve by k₀R_i= 10⁻⁴– 10⁻¹ with k₀=兩E−U₀兩/បv_F兲. The zero-doping magnetoconductance is also shown 共red/gray line兲. 共b兲 Magnetoconductance at large doping 共k₀R_i= 0.2– 0.6兲.共c兲Conductance as a function of doping at fixed magnetic field 关specified by the flux piercing the disk area ⌽d

= 0 – 4⌽0, with⌽0= 2共h/e兲lnR_o/R_i兴.

5 10 15 20

0 2 4 6 8

k0(Ro−Ri) 1

2r^c> R^o−Rⁱ

10⁻² 10⁻¹

n=1

n=2 n=3 n=4

Φ

d

/ Φ

0

5 6 7 Φd/Φ0

10⁻⁴ 0 δ1

Φ_d>Φmax d

FIG. 3. 共Color online兲Phase diagram representing the tunneling field-suppressed and ballistic transport regimes in the field-doping parameter plane. Solid lines corresponds toG=G₀for the radii ratio R_o/R_i= 10. Dashed lines depict borders of the tunneling 共⌽d

⬍⌽d

max, red/gray line兲 and ballistic 共2r_c⬎R_o−R_i, blue/dark gray line兲 transport regimes. 关Notice the logarithmic scale for k₀共R_o

−R_i兲⬍1.兴Inset shows the crossover into the tunneling behavior for the first Landau level 共n= 1兲 in the magnified horizontal scale共␦n

⬅¹2k₀²l_B²−n兲.

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⬎0.1 are now reached forR_o/Riⱖ2.2. At finite dopings, Eq.

共13兲for⌽d

maxholds true, and the phase diagram in the field- doping parameter plane 共Fig. 3兲 is almost unaffected. We further notice that for available ballistic graphene samples 2R_o⬍1 ␮m, and the critical field Bc typically corresponds to⌽d⬍⌽0. In effect, the zero-field conductance minimum is expected to be significantly deeper than the other minima, for which both electrodes are driven into the normal state.

In conclusion, we have identified the new transport phe- nomenon in undoped graphene, which manifests itself by periodic magnetoconductance oscillations for the Corbino geometry. The relative field-induced conductance change reaches experimentally accessible magnitudes ⌬G/G₀⬎0.1 for moderate radii ratios. At weak doping, the oscillations remain observable for a finite range of applied fields.²⁵Ad- ditionally, we have presented the complete phase diagram in a field-doping parameter plane, illustrating the crossover from the field suppressed to the ballistic transport regime, as well as the resonances through Landau levels, at which the oscillatory behavior is restored.

We hope our analysis shall raise some interest in Corbino

measurements within the graphene community. Although the discussion is limited to the system with a perfect circular symmetry and the uniform field, particular features of the results, including共i兲the conductance dependence on thetotal flux piercing the sample area and 共ii兲 the formal analogy between dimensionless flux⌽d/⌽0and the boundary condi- tions at zero field suggest that magnetoconductance oscilla- tions should appear in more general situation as well. The work primarily focuses on graphene, but the recent study on effective Dirac fermion model for HgTe/CdTe quantum wells²⁶ suggests that our findings may also be relevant to such systems.

Note added in proof. Recently, we became aware of a work on zero-doping Corbino magnetoconductance in graphene employing conformal mapping technique.²⁷

We thank K. Richter, P. Recher, and J. Wurm for discus- sions. The support from the Alexander von Humboldt Stiftung-Foundation and the Polish Ministry of Science 共Grant No. N–N202–128736兲is acknowledged.

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25For example, the inner radius R_i= 20 nm, the outer radius R_o

= 100 nm, and the doping fixed at兩E−U₀兩= 10⁻⁵ eV allow one to observe ten full oscillation periods in the field range

−2.2 TⱕBⱕ2.2 T.

26M. J. Schmidt, E. G. Novik, M. Kindermann, and B. Trauzettel, Phys. Rev. B 79, 241306共R兲 共2009兲.

27M. I. Katsnelson, Europhys. Lett. 89, 17001共2010兲.

ADAM RYCERZ PHYSICAL REVIEW B81, 121404共R兲 共2010兲

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