Adiabatic quantum pumping in buckled graphenenanoribbon driven by a kink ∗

Dominik Suszalski and Adam Rycerz

Institute of Theoretical Physics, Jagiellonian University, ul. S. Łojasiewicza 11,PL–30348 Kraków, Poland

(Received February 18, 2020)

We propose a new type of quantum pump in buckled graphene nanorib-bon, adiabatically driven by a kink moving along the ribbon. From a prac-tical point of view, pumps with moving scatterers present advantages ascompared to gate-driven pumps, like enhanced charge transfer per cycleper channel. The kink geometry is simplified by truncating the spatial ar-rangement of carbon atoms with the classical φ4 model solution, includinga width renormalization following from the Su-Schrieffer-Heeger model forcarbon nanostructures. We demonstrate numerically, as a proof of concept,that for moderate deformations a stationary kink at the ribbon center mayblock the current driven by the external voltage bias. In the absence ofa bias, a moving kink leads to highly-effective pump effect, with a chargeper unit cycle dependent on the gate voltage.

1. Introduction

The idea of quantum pumping, i.e., transferring the charge between elec-tronic reservoirs by periodic modulation of the device connecting these reser-voirs [1], has been widely discussed in the context of graphene nanostructures[2, 3, 4, 5, 6, 7, 8]. Since early works, elaborating the gate-driven pumpingmechanism in graphene [2] and bilayer graphene [3], it becomes clear thatthe transport via evanescent modes may significantly enhance the effective-ness of graphene-based pumps compared to other quantum pumps. Otherpumping mechanisms considered involves laser irradiation [4], strain fields[5], tunable magnetic barriers [6], Landau quantization [7], or even slidingthe Moiré pattern in twisted bilayer graphene [8].

It is known that quantum pump with a shifting scatterer may showenhanced charge transfer per cycle in comparison to a standard gate-driven

∗ Presented at 45. Zjazd Fizyków Polskich, Kraków 13–18 września 2019.

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Fig. 1. Buckled graphene nanoribbon as a quantum pump. (a) Typical pumpingcycles (schematic) for a single channel characterized by the complex transmissionamplitude, t = Re t + i Im t. Blue solid line enclosing the area Ac shows one-parameter cycle allowed for a standard gate-driven pump, red dashed line corre-sponds to the cycle involving a shift of a scatterer. (b) Flat ribbon with armchairedges attached to heavily-doped graphene leads (shaded areas). The ribbon widthW = 5 a (with a = 0.246 nm the lattice spacing) and length L = 11.5

√3 a are

chosen for illustrative purposes only. A cross section of buckled ribbon, charac-terized by the reduced width W ′ < W and the buckle height H > 0, and theelectrostatic potential energy profile U(x), are also shown. (c) Band structure ofthe infinite metallic-armchair nanoribbon of W = 10 a width, same as used in thecomputations.

pump [1], see also Fig. 1(a)1. Motivated by this conjecture we consider

1 In the simplest case of a one-parameter driven, one-channel pump, the charge percycle is given by Q = (e/π)Ac, with Ac being the ±area enclosed by the contour inthe (Re t, Im t) plane (where t is a parameter-dependent transmission amplitude).

werekink printed on February 21, 2020 3

a buckled nanoribbon, see Fig. 1(b), similar to the one studied numericallyby Yamaletdinov et. al. [9, 10] as a physical realization of the classical φ4

model and its topological solutions (kinks) connecting two distinct groundstates. Our setup is supplemented with two heavily-doped graphene leads,attached to the clamped edges of a ribbon, allowing to pass electric currentalong the system.

2. Model and methods

The analysis starts from the Su-Schrieffer-Heeger (SSH) model for theribbon, including the hopping-matrix elements corresponding to the nearest-neighbor bonds on a honeycomb lattice [11, 12, 13, 14]

Hribbon = −t0∑〈ij〉,s

exp

(−β δdij

d0

)(c†i,scj,s + c†j,sci,s

)+

1

2K∑〈ij〉

(δdij)2 , (1)

with a constrain∑〈ij〉 δdij = 0, where δdij is the change in bond length,

and d0 = a/√

3 is the equilibrium bond length defined via the lattice spacinga = 0.246 nm. The equilibrium hopping integral t0 = 2.7 eV, β = 3 is thedimensionless electron-phonon coupling, and K ≈ 5000 eV/nm2 is the springconstant for a C-C bond. The operator c†i,s (or ci,s) creates (or annihilates)a π electron at the i-th lattice site with spin s.

In order to determine the spatial arrangement of carbon atoms in a buck-led nanoribbon, {Rj = (xj , yj , zj)}, we first took the {xj} and {yj} coordi-nates same as for a flat honeycomb lattice in the equilibrium and set {zj}according to

z = H tanh

(y − y0

λ

)sin2

(πxW

), (2)

representing a topologically non-trivial solution of the φ4 model, with Hthe buckle height, W the ribbon width, y0 and λ the kink position and size(respectively). Next, x-coordinates are rescaled according to xj → xjW

′/W ,withW ′ adjusted to satisfy

∑〈ij〉 δdij = 0 in the y0 → ±∞ (“no kink”) limit.

In particular, H = 2a and W = 10a corresponds to W ′/W = 0.893. Wefurther fixed the kink size at λ = 3a (closely resembling the kink profileobtained from the molecular dynamics in Ref. [9]); this results in relativebond distorsions not exceeding |δdij |/d0 6 0.08. Full optimization of thebond lengths in the SSH Hamiltonian (1), which may lead to the alternatingbond pattern [14], is to be discussed elsewhere.

Heavily-doped graphene leads, x < 0 or x > W ′ in Fig. 1(b), are modeledas flat regions (dij = d0) with the electrostatic potential energy U∞ =−0.5t0 (compared to U0 = 0 in the ribbon), corresponding to 13 propagating

4 werekink printed on February 21, 2020

modes for W∞ = 20√

3a and the chemical potential µ0 = EF − U0 = 0.The scattering problem if solved numerically, for each value of the chemicalpotential µ0 and the kink position y0, using the Kwant package [15] allowingto determine the scattering matrix

S(µ0, y0) =

(r t′

t r′

), (3)

which contains the transmission t (t′) and reflection r (r′) amplitudes forcharge carriers incident from the left (right) lead, respectively.

The linear-response conductance can be determined from the S-matrixvia the Landuer-Büttiker formula [1], namely

G = G0Tr tt† =2e2

h

∑n

Tn, (4)

where G0 = 2e2/h is the conductance quantum and Tn is the transmissionprobability for the n-th normal mode. Similarly, in the absence of a volt-age bias, the charge transferred between the leads upon varying solely theparameter y0 is given by

∆Q = − ie2π

∑j

∫dy0

(∂S

∂y0S†)

jj

, (5)

where the summation runs over the modes in a selected lead. Additionaly,the integration in Eq. (5) is performed for a truncated range of −Λ 6 y0 6L+ Λ, with Λ = 250 a� λ for L = 75.5

√3 a.

3. Results and discussion

In Fig. 1(c) we depict the band structure for an infinite (and flat) metallic-armchair ribbon of W = 10 a width. It is remarkable that the second andthird lowest-lying subband above the charge-neutrality point (as well as thecorresponding highest subbands below this point) show an almost perfect de-generacy near their minima corresponding to E(2,3)

min ≈ 0.25 t0, which can beattributed to the presence of two valleys in graphene. For higher subbands,the degeneracy splitting due to the trigonal warping is better pronounced.

The approximate subband degeneracy has a consequence for the conduc-tance spectra of a finite ribbon, presented in Fig. 2: For both the flat ribbon(H = 0) and a buckled ribbon with no kink (H = 2a, y0 → − ∞) theconductance G ≈ G0 for |µ0| < E

(2,3)min and quickly rises to G/G0 ≈ 2.5−3

for µ > E(2,3)min . For µ < 0 the quantization is not so apparent, partly

werekink printed on February 21, 2020 5

0.0 0.2 0.4

0.0

1.0

2.0

3.0C

ondu

ctan

ce [

2e2 /

h]

0.00 0.04 0.08 0.12 0.160.0

0.4

0.8

1.2

flat ribbon, H= 0kink at the center, H= 2ano kink, H= 2a

Chemical potential, μ0/t0

Con

duct

ance

[2e

2 /h]

0.4 0.2

Fig. 2. Conductance of the ribbon with W = 10 a as a fuction of the chemicalpotential. The remaining parameters are L = 75.5

√3 a and Wlead = 20

√3 a.

Different lines (same in two panels) correspond to the flat ribbon geometry, H = 0,W ′ = W (black dashed), the buckled ribbon with H = 2a and y0 = L/2 (bluesolid) or y0 → −∞ (orange solid); see Eq. (2). Bottom panel is a zoom-in of thedata shown in top panel.

due the presence of two p-n interfaces (at x = 0 and x = W ′) leading tostronger-pronounced oscillations (of the Fabry-Perrot type), and partly dueto a limited number of propagating modes in the leads. For a kink posi-tioned at the center of the ribbon (H = 2a, y0 = L/2) the conductance

6 werekink printed on February 21, 2020

0.00 0.05 0.10 0.15

0

1

2

3

4

5

6C

harg

e pu

mpe

d pe

r kin

k, Q

/e H = 2.0 a

H = 1.5 a

H = 1.0 a

Chemical potential, μ0/t0

Fig. 3. Charge transferred upon adiabatic kink transition calculated from Eq.(5), for varying buckling amplitude H/a = 1, 2 (specified for each line in theplot), displayed as a function of the chemical potential. Dashed line shows theapproximation given by Eq. (6). Remaining parameters are same as in Fig. 2.

is strongly suppressed, G � G0, in the full |µ0| < E(2,3)min range (with the

exception from two narrow resonances at µ0 = ±0.1327 t0, which vanishesfor y0 6= L/2) defining a feasible energy window for the pump operation.

In Fig. 3 we display the charge transferred between the leads at zerobias, see Eq. (5), when slowly moving a kink along the ribbon (adiabatic kinktransition). In the energy window considered, the ribbon dispersion relationconsists of two subbands with opposite group velocities, E(1)

± = ±~vFky, seeFig. 1(c). Therefore, the total charge available for transfer at the ribbonsection of Leff = L−Winfty length can be estimated (up to a sign) as

Qtot

e≈ |µ0|

~vFLeff

π= 35.3

|µ0|t0

, (6)

where vF =√

3 t0a/(2~) ≈ 106 m/s is the Fermi velocity in graphene andwe put L−Winfty = 55.5

√3 a. For the strongest deformation (H = 2a) the

kink almost perfectly blocks the current flow, and the charge transferred isclose to the approximation given by Eq. (6); see Fig. 3.

4. Conclusion

We have investigated, by means of computer simulations of electrontransport, the operation of adiabatic quantum pump build within a topo-

werekink printed on February 21, 2020 7

logical defect moving in buckled graphene nanoribbon. We find that thepump effectiveness is close to the maximal (corresponding to a kink per-fectly blocking the current flow) for moderate bucklings, with relative bonddistortions not exceeding 8%. As topological defects generally move withnegligible energy dissipation, we hope our discussion will be a starting pointto design new graphene-based energy conversion devices.

Acknowledgments

We thank Tomasz Romańczukiewicz for discussions. The work was sup-ported by the National Science Centre of Poland (NCN) via Grant No.2014/14/E/ST3/00256.

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