Quantized transport of interface and edge states in bent graphene

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Quantized transport of interface and edge states in bent graphene Cong-Cong Li, Lei Xu, Jun Zhang

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Received date: 24 December 2014 Revised date: 2 February 2015 Accepted date: 3 February 2015 Cite this article as: Cong-Cong Li, Lei Xu, Jun Zhang, Quantized transport of interface and edge states in bent graphene, Solid State Communications, http://dx. doi.org/10.1016/j.ssc.2015.02.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Quantized transport of interface and edge states in bent graphene Cong-Cong Lia , Lei Xua , Jun Zhanga,b,∗ a School b State

of Physical Science and Technology, Xinjiang University, Urumqi 830046, China Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

Abstract We explore the transport behavior of interface and edge states in bent graphene under a magnetic field. The bending angle can change the distribution of interface and edge states, resulting in an interesting evolution of quantized conductance. The interface state vanishes when the bending angle is not less than π/2, whereas the edge state remains. In the presence of Zeeman splitting, the transport properties are also considered and a quantum spin Hall effect is found. These results may provide a way to control the interface current. Keywords: A. bent graphene, D. interface state, D. quantum Hall effect 2010 MSC: 00-01, 99-00

∗ Corresponding

author. Fax:86-991-8582405 Email address: [email protected] (Jun Zhang)

Preprint submitted to Journal of LATEX Templates

February 7, 2015

1. Introduction Graphene has attracted great attention since it was successfully fabricated in experiment. It is the first truly two-dimensional material and has many unconventional properties,[1, 2] such as the linear dispersion[3, 4], quantum Hall 5

effect.[5, 6] Due to the linear dispersion near zero energy, the quantum Hall effect in graphene is characterized by chiral edge states and the quantized conductance is given by σxy = 4(n+1/2)e2 /h with n an integer.[5] However, the quantum spin Hall (QSH) effect which is characterized by helical edge states, is also predicted in graphene[7, 8] and was later experimentally realized in HgTe/CdTe quantum

10

well.[9, 10] These helical edge states are fully spin-polarized formed by counter propagating edge states with opposite spins and are preserved by time-reversal symmetry.[7, 8] Moreover, electron transport properties of graphene have been extensively studied in electric and magnetic fields,[1, 2] and some interesting results have been obtained, such as particular transport properties in graphene

15

with multiple magnetic barriers[11, 12] and tunable controllable electronic states in graphene by both electric and magnetic fields.[13] Two-dimensional graphene can be bent into the third dimension without degradation to its structural properties and electron transport.[14, 15, 16] Therefore, the bent graphene has also been a focus of intense interest. The bend effec-

20

tively produces a new type of Hall edge state along the bent region,[17, 18, 19] leading to some novel transport properties in bent graphene.[20, 21, 22] The effect of bending curvature has been studied in bent graphene.[18, 23] The tight-binding approximation in planar graphene only considers the π and π ∗ bands due to the hopping between pz orbitals perpendicular to graphene plane.

25

However, there is a curvature-induced misalignment of the pz orbitals in bent graphene. Thus, the hopping integral should be modified as t0 = t cos α (α is the misalignment angle between pz orbitals).[18] This provides a train of thought for us to study the transport properties of bent graphene. In this work, we consider the bent graphene ribbon depicted

30

schematically in Fig. 1(a). It consists of a graphene ribbon with a zigzag edge

2

(b)

(a) z y

B x (c)

(d)

x

B

Figure 1: (Color online) (a) A zigzag-edge graphene ribbon is bent along the x direction. The ⃗ is perpendicular to the bent graphene ribbon. (b) A topological uniform magnetic field B equivalent geometry obtained by unbent graphene ribbon. TR, BR and MR correspond to top planar region, bottom planar region and middle bent region respectively in Fig. 1(a). Lx represents the width of the MR. (c) The plane projection of Fig. 1(a) and θ is the bending angle. (d) The profile of the effective magnetic field in an equivalent unbent graphene ribbon.

bent along the x direction. The graphene ribbon is bent into a wedge shape, as shown in Fig. 1(c). The bending angle θ between two planar regions is an arbitrary value. The bent graphene ribbon is divided into three regions, top planar region (TR), bottom planar region (BR) and middle bent region (MR). 35

⃗ is in the z direction and perpendicular to the BR of bent The magnetic field B graphene, as shown in Fig. 1(c). We study the effect of bending angle to interface states, edge states and quantized conductances in a magnetic field. By changing of the bending angle, different interface and edge states are found in the bent graphene ribbon. Besides, the effect of curvature-modified hopping integral and

40

width of the MR is also considered. The results show that they hardly affect the spatial distributions of interface and edge states. Finally, we discuss the electron transport property in the presence of Zeeman splitting and find that the system can host a QSH phase.

2. Model and method 45

⃗ = (0, 0, Bz (x)), the Hamiltonian of In a perpendicular magnetic field B tight-binding model takes the form H = −t

∑

(eiϕij c†i cj + h.c.),



3

(1)

where t is the nearest-neighbor hopping integral on the honeycomb lattice and ϕ is the gauge potential for the nonuniform magnetic field. The operator c†i (ci ) creates (annihilates) an electron at site i, and < ij > denotes nearest-neighbor 50

sites. In terms of the Landauer-B¨ uttiker formula, the conductance with spin σ can be calculated by the equation Gσ = Tσ e2 /h.[24, 25, 26] The transmission coefficient Tσ from lead r to lead l with spin σ is described by A Tσ = Tr[Γlσ GR σ Γrσ Gσ ],

(2)

A where Γℓσ = i[ΣR ℓσ − Σℓσ ] is the coupling between conductor and lead ℓ (ℓ = l, r) R/A

55

with Σℓσ

the self-energy. The retarded Green’s function of the sample GR σ has

A † c R R −1 a form GR , where Hσc is the Hamiltonian of σ = [Gσ ] = [E − Hσ − Σlσ − Σrσ ]

the conductor region.

3. Results and discussion In Fig. 1(a), we show the bent graphene ribbon under a uniform magnetic 60

field. The system can be described by a topological equivalent geometry suffering an effective magnetic field shown in Fig. 1(b) and 1(d). The magnetic field ⃗ is perpendicular to BR, whereas the field is −B cos θ [θ is the bending angle B shown in Fig. 1(c)] in TR. In MR, the effective magnetic field is the normal ⃗ ·n component of magnetic field, B ˆ ,[17], where n ˆ is normal to the ribbon. The

65

angle between the tangent of each point in MR undergoes a uniform change by assuming a smooth arc in the MR. To analyze all possible interface and edge states, we choose a graphene ribbon with zigzag edge, where a open (periodic) boundary condition is taken in the x(y) direction, as shown in Fig. 1(a). By diagonalizing the Hamiltonian Eq.1

70

on a rectangular sample under a nonuniform magnetic field, we obtain energy spectrums for different bending angles, shown in the top panel of Fig. 2. Due to the particle-hole symmetry, the electron energy spectrum is symmetrical about zero energy which is similar to that of pristine graphene.[1] However, the band

4

E(t)

0.1

(a)

(c)

(b)

a

c d

b

a

c

b

d

(d)

a

a

b

b

0.0

-0.1 0.3

0.12

0.6

0.3

Ky/

0.6

0.0

0.3

1.2

0.6 Ky/

Ky/

(e)

1.8

(h)

(g)

(f)

1.5 Ky/

0.08 BR

0.04

TR

MR

c

d

ab

0.00

BR

200

400

600

200

c

400

MR

BR

TR

b

600

200

N

400

MR

b

600

200

(j)

400

600

N

x

x

TR

a

a

N

x

(i)

G(e2/h)

BR

TR

ab

d

N

16

MR

x

(k)

(l)

8

0

-0.1

0.0 E(t)

0.1

-0.1

0.0

0.1

-0.1

0.0 E(t)

E(t)

0.1

-0.1

0.0

0.1

E(t)

Figure 2: (Color online) Top panel: The electron energy spectrum of bent graphene ribbon with ϕ = 0.002 and Lx ≈ 28.4nm. Middle panel: The spatial distributions of interface and edge states indicated in the top panel labelled by the letters a,b,c and d. Nx represents lattice indexes. Bottom panel: The corresponding calculated conductance for the bent graphene ribbon.

structure of bent graphene is different from the case of pristine graphene in a 75

uniform magnetic field, because the bend changes the distribution of effective magnetic field leading to additional interface states which are located at the magnetic interface. This suggests that these interface and edge states may redistribute in bent graphene ribbon. We now focus on these edge states. The spatial distributions of edge states

80

|Ψ|2 can be numerically obtained, as shown in the middle panel of Fig. 2. When θ < π/2, original edge states are located at the sample boundaries, whereas the generated interface states emerge and locate at the magnetic interface. The appearance of new interface states is because of the magnetic field changing direction in MR. When θ > π/2, the generated interface states disappear, but

85

the edge states remain. In particular, when θ = π/2, the effective magnetic field is zero in TR, and the edge state in this region becomes an extended state.

5

The evolution of interface and edge states also results in an interesting transition of quantized conductances. Using the Green’s function method, conductances of bent graphene with different bending angles can be calculated, shown 90

in the bottom panel of Fig. 2. It shows that, for θ < π/2, the quantized conductance is G = 4ne2 /h with n = 1, 2, 3, · · · (hereafter). Due to the contribution of interface channels, the quantized conductance near zero energy is 4e2 /h, which differs from the monolayer graphene.[1] For θ > π/2, the quantized conductance is G = 4(n − 1/2)e2 /h, where the interface channels disappear. However, for

95

θ = π/2, there is only edge channel left on one of two boundaries corresponding to a quantized conductance G = 2ne2 /h. In terms of this, the edge state can be controlled in principle by varying the bending angle. Therefore, this suggests an alternative way to manipulate the quantized transport property in bent graphene by varying the bending angle.

100

Another natural question arises: whether the width of MR (Lx ) can affect the transport property in bent graphene. Thus, we shorten the width of MR (Lx ≈ 2.84nm) to re-calculate the quantized conductance. However, it gives rise to the same results as shown in the left panel of Fig. 3, so that the width of MR cannot affect transport property in the present case. Moreover, the

105

bending curvature changes the hopping integral due to the misalignment angle between pz orbitals.[18, 23] Therefore, the curvature-modified hopping integral is given by t0 = t cos α (α is the misalignment angle between pz orbitals). Again we calculate the electron energy spectrum and the conductance by taking the hopping integral as t cos α instead of t in MR, and the same quantized

110

conductances are obtained, as shown in the right panel of Fig. 3. Thus, the effect of bending curvature can be neglected in this model. In the presence of Zeeman splitting M = 0.01t,1 the spin gap can be opened as depicted in the top panel of Fig. 4. When the chemical potential lies within 1 Here,

we enlarge the Zeeman splitting in order to see the band structure near zero en-

ergy more clearly. In graphene, the Zeeman splitting can be enhanced by electron-electron interactions and electron exchange,[27, 28] reaching ∼ 100K for B = 10T .

6

E(t)

0.1

(a) θ = π/3 a

0.0

b

c

d

(b) θ = π/3 b a

c

d

-0.1 0.3

0.12

Ky/π

0.6

0.3

(c) θ = π/3

|ψ|

2

BR

MR

0

c

220 Nx

G(e2/h)

d

c

ab

420

(e) θ = π/3

TR

MR

BR

TR

ab

16

0.6

(d) θ = π/3

0.08 0.04 d

Ky/π

200

Nx

400

600

(f) θ = π/3

8

0

-0.1

0.0 E(t)

-0.1

0.1

0.0 E(t)

0.1

Figure 3: (Color online) Similar to that of Fig.2. Left panel: the width of MR Lx ≈ 2.84nm. Right panel: the hopping integral of MR t cos α.

the spin gap, we calculate the spatial distributions of wave functions for the 115

corresponding edge states, shown in the middle panel of Fig. 4. Comparing with the above case without Zeeman splitting, we find that the interface states are apparently different. For θ ≥ π/2, the interface states are present, whereas they are absent for θ < π/2. Moreover, owning to Zeeman splitting, the spin degeneracy is lifted and the corresponding conductances are also changed, as

120

shown in the bottom panel of Fig. 4. Numerical results indicate that the quantized conductance becomes G = 2e2 /h in the vicinity of zero energy for θ < π/2. In contrast to the case without Zeeman splitting, this value is halved because interface states vanish in the spin gap. Conversely, the conductance doubles with a quantized value G = 4e2 /h for θ ≥ π/2. In addition, for θ < π/2, a

125

small quantized plateau with G = 6e2 /h is also found, which comes from both interface and edge states effects. At other energies (except near zero energy), the quantized conductances are also calculated in detail. For 0 < θ < π/2 or θ > π/2, the conductances become G = 2(n + 1)e2 /h or G = 2ne2 /h, respec-

7

0.1 (b) a

0.0

12

(d)

(c)

b

a

12

b

12

12

a

12

c

b

12

d

12

12

c

b

a

12

12

12

d

12

0.004 E(t)

E(t)

(a)

-0.004

-0.1

0.72

Ky/

0.3

0.18

0.73

Ky/

0.6

0.3

(e)

0.12

MR

BR

0.6

0.3

BR

TR

Ky/

0.6

1.2

MR

TR

1.5

1.8

Ky/

(g)

(h)

BR

MR

MR

BR

TR

TR

|

|2

Ky/

(f)

0.06 b b

a a 1

1

2

1

1

2

2

2

1

d d 1

c c

2

200

N

400

200

600

G(e2/h)

x

8

0

600

0.0

0.1

-0.1

a a 1

d d

2

1

N

400

600

0.0

0.1

-0.1

1

200

a a 1

2

N

400

2

600

X

(k)

E(t)

2

c c

2

2

X

(j)

E(t)

200

x

(i)

-0.1

N

400

1

2

1

0.00

b b

b b

a a

b b

(l)

0.0 E(t)

0.1

-0.1

0.0

0.1

E(t)

Figure 4: (Color online) Similar to that of Fig.2, but with a Zeeman splitting term M = 0.01t. Black or red solid lines in the top panels represent spin-up or spin-down sectors respectively.

tively. For θ = π/2, the quantized conductance is G = (n + 1)e2 /h. However, 130

for θ = 0, both the spin-up and spin-down energy spectrum outside the spin gap are nearly overlapped so that the conductance is still G = 4ne2 /h. For θ < π/2, we find a spin filtered helical edge state, depicted schematically in Fig. 5. The spin filtered edge states have important consequences for both charge and spin transport. In this two-terminal geometry, we find that the

135

charge conductance is G = 2e2 /h. Because of the edge current density is related to the spin density, the charge current is also accompanied by spin accumulation on opposite edges of the graphene ribbon in this phase. Clearly, This is a timereversal symmetry-broken QSH effect due to the magnetic field. Therefore, Zeeman splitting can induce an interesting evolution of interface states and

140

results in a QSH effect in bent graphene. Moreover, when the bending angle θ is more than π, the transport properties can be obtained by making a space inversion of the case with 2π−θ. Thus, for θ > π, the above mentioned transport properties can persist and the QSH effect can also occur like that in θ < π case.

8

(a)

0

(b) BR

MR

TR

I

V

Figure 5: (Color online) (a) The edge currents in bent graphene. (b) Schematic diagrams of two-terminal measurement geometries. Black or red solid lines represent spin-up or spin-down sectors respectively.

Generally, the transport properties of folded graphene nanoribbons strongly 145

depend on their edge geometries.[17] However, our results show that the transport properties of bent graphene nanoribbon are dominated by the number of edge and interface channels because the direction of edge currents is parallel to the interface. In the presence of a magnetic field, the bent graphene nanoribbon with zigzag or armchair edge has the same number of edge and

150

interface channels, leading to a similar conductance in both zigzag and armchair edge cases. Thus we only consider the zigzag graphene nanoribbon in this manuscript. There is also an obstacle for experimental realization of the predicted results: how to fabricate a bent graphene ribbon. At the present experimental condition, it is difficult to fabricate a bent graphene ribbon with an

155

arbitrary bending angle. However, it is possible to fabricate a graphene ribbon with a specific bending angle in experiments, such as the self-folded graphene observed in experiments,[14, 15, 16] and putting a graphene sheet on two mesas[29] or a bent substrate which is also an alternative way to fabricate bent graphene.

4. conclusion 160

In summary, we adopt the tight-binding model and Green’s function method to study quantum transport properties of interface and edge states in bent graphene under a magnetic field. It shows that interface and edge states can be controlled by changing bending angle, and the quantized conductance exhibits an interesting evolution. The effect of curvature-modified hopping integral and 9

165

width of the MR is also considered, which cannot change the quantized conductance. In the presence of Zeeman splitting, the quantum transport properties are studied. The quantized conductance is different from the case without Zeeman splitting because of the redistribution of interface and edge states. A QSH effect is also found in the presence of Zeeman splitting. These results provide a

170

way to manipulate the interface state and transport property in bent graphene.

Acknowledgments This work was supported by NSFC Project Nos. 11347207, 11404276 and 11265015. L. X. acknowledges the support by Doctoral Fund of Xinjiang university Project No. BS130112 and NSF of the Xinjiang uygur autonomous region 175

No. 2014211A003.

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