band structure transformation as driving force behind...

Band structure transformation as driving forcebehind Anderson localization of charge carriers

in impure graphene

Yuriy Skrypnyk

Bogolyubov Institute for Theoretical Physics

International School and Conferenceon Nanoscience and Quantum Transport

Y.V. Skrypnyk (ITF) LOCALIZATION IN IMPURE GRAPHENE October 2016 1 / 29

Figure 1: Minimum conductivity of graphene, K.S. Novoselov et al.,Nature 438, 197 (2005).


Figure 2: Resistivity of five representative graphene samples as a functionof applied gate voltage, Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S.Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys.Rev. Lett. 99, 246803 (2007).


Model

Hamiltonian of the substitutional binary alloy with a diagonaldisorder in the tight– binding approximation (the Lifshitz model),

H = H0 +H imp, H imp = VL∑n,α

′c†nαcnα,

where n refers to lattice cells, α enumerates sublattices, c†nα and cnα

are electron creation and annihilation operators, and the summationis restricted to the sites occupied by impurities.

Host Hamiltonian,

H0 = t ∑<nα,mβ>

c†nαcmβ, t ≈ 2.7eV, vF =

√3at2

, E(k ′) ≈ ±vFk′.


Technique of the Green’s-function clusterexpansion

Averaged Green’s function:

G (k ,E ) =(E − E (k)− Σ(k ,E )

)−1.

Modified propagator method

Σ(E ) ≈ cVL

1− VLG00(E ),

c – impurity concentration.

R.W. Davies, J.S. Langer, Phys. Rew. 131, 163 (1963)


Coherent potential approximation:

Σ(E ) ≈ cVL

1− (VL − Σ(E ))G00(E )

P. Soven, Phys. Rev. 156, 809 (1967)

Series for the self-energy:

Σ(k ,E ) = cτ(

1− cA00 − cA200+

+c ∑l 6=0

A30l exp(ikr l ) + A4

0l

1− A20l

+ · · ·)

, A0l (E ) = τG0l (E ),

τ =VL

1− VLG00(E ), G0l (E ) =

1

N ∑kG (k ,E )e−ikr l .


Applicability criterion:

R(E ) = c

∣∣∣∣τ2

(∂

∂(E − Σ(E ))G00(E ) + G 2

00(E )

)∣∣∣∣ 6= 1

F. Ducastelle, J. Phys C 7, 1975 (1976)

Small parameter of the expansion:

R(E ) ≡ c∣∣∣∑l 6=0

A20l (E )

∣∣∣ 6 1

2

In 3D systems with parabolic dispersion,

E − Σ(E ) ≡ κ2e i2ϕ, κ > 0,

k̃(E )`(E ) =cot ϕ

2, R(E ) ' |sin ϕ| , ϕth ≈ π/6.

Yu. Skrypnyk, Phys. Rev. B 70, 212201 (2004)Y.V. Skrypnyk (ITF) LOCALIZATION IN IMPURE GRAPHENE October 2016 7 / 29

Band structure transformation in graphene withpoint defects

Effective dimensionality D = d/m

1D ε(k) ∼ k2 D = 1/2

2D ε(k) ∼ k2 D = 1

3D ε(k) ∼ k2 D = 3/2

4D ε(k) ∼ k2 D = 2

1D ε(k) ∼ k D = 1

2D ε(k) ∼ k D = 2

3D ε(k) ∼ k D = 3


Finite impurity concentration, graphene

LDOS at the impurity site has the Lorentz shape, ε = E/(√√

3πt)

ρi (ε) ≈|ε|Γ2

r

[πvLεr ]2 [(ε− εr )2 + Γ2r ]

, 1 ≈ 2vLεr ln |εr | , vL =VL√√

3πt

Characteristic scale of spatial variations of the Green’s function

∼ 1/|εr |, ⇐ g(r , ε) ∼ f (2√

π|ε|r)

Mean distance between defects

∼ 1/√c

Rough estimation for the critical concentration

c0 ∼ ε2r ⇒ c0 = −1/[2v2L ln(ζ/|vL|)]Y.V. Skrypnyk (ITF) LOCALIZATION IN IMPURE GRAPHENE October 2016 9 / 29

Applicability criterion, ε− σ = κ exp(iϕ), 0 < ϕ < π:

|R(ε)| ≈∣∣∣∣ lnκ + 1 + i(ϕ− π

2 )

lnκ + (ϕ− π2 ) cot ϕ

∣∣∣∣ 6 1

2.

Width of the transport gap,

∆R ≈ exp(− 1

4cv2L− 1), c � c0, ∆R ≈

√− c

ln(√c)

, c � c0.

Width of the transport gap in conventional systems,

Low − dimensional , ∆R ∼ c1/D, c � c0,

3D, ε(k) ∼ k2, ∆R ∼ c2, c � c0.

Yu. V. Skrypnyk, V. M. Loktev, Phys. Rev. B. 73, 241402(R) (2006)


−0.02 0 0.02 0.04 0.06 0.08

25

50

75

100

125

150

175

200

F

σ(e

/h)

2co

nd

εFigure 3: Conductivity of graphene with point defects vs. Fermi energy forvL = −8 and concentrations c = c0/2n, n = 1...5, c0 ≈ 0.0005.

Yu. V. Skrypnyk, V. M. Loktev, Phys. Rev. B. 82, 085436 (2010)


Numerical simulation

0

0.04

0.08

DO

S

VL=8t

c=0.012

−0.2 −0.1 0.1 0.2 0

0.4

0.8

R(ε

)

ε

Figure 4: DOS at the impurity perturbation VL = 8t and c = 0.012.Critical concentration – c0 ≈ 0.003. Blue curve stands for the numericalcomputation, red – the CPA, green – the ATA, black – R(ε). Triangledenotes the Fermi level.


−0.25 −0.2 −0.15 −0.1 −0.05 00

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

ε,t

IPR

c = 0.012

c = 0.003

c = 0.00075

Figure 5: Inverse participation ratio P(ε) = ∑nα

|ψnα |4 for VL = 8t and

different impurity concentrations.

S. Pershoguba, Yu. Skrypnyk, V. Loktev, Phys. Rev. B 80, 214201 (2009)


0

0.05

0.1

0.15

0.2

XY

|ψ|2

VL = 8t

c = 0.012 ε = −0.1989t P = 0.0532

Figure 6: Fragment of eigenstate for VL = 8t and c = 0.012. Theseconfiguration of impurities is characteristic to the second peak in the DOS.


0

0.02

0.04

0.06

XY

|ψ|2

VL = 8t

c = 0.012 ε = −0.1441t P = 0.0114

Figure 7: Fragment of eigenstate for VL = 8t and c = 0.012. Theseconfiguration of impurities is characteristic to the first peak in the DOS.


MIT in hydrogenated graphene

Figure 8: Energy distribution curves (EDCs) at kF , the inverse momentumwidth Lmfp, and the inverse Fermi wave vector 1/kF .


Figure 9: Bandstructure cuts as a function of nH

E. Rotenberg et al., Phys. Rev. Lett. 103, 056404 (2009)


Figure 10: Evolution of the EDC at kF with increasing the hydrogencoverage.

Yu.Skrypnyk, V.Loktev, Phys. Rev. B. 83, 085421 ( 2011)


0.00 0.02 0.04 0.06 0.08

-0.4

-0.3

-0.2

-0.1

0.0

k H1�AoL

E-

EFHe

VL

Figure 11: Contour plot of the spectral function at nH = 5× 1012 cm−2,VL = −25 eV.


0.00 0.02 0.04 0.06 0.08

-0.4

-0.3

-0.2

-0.1

0.0

k H1�AoL

E-

EFHe

VL

Figure 12: Contour plot of the spectral function at nH = 10× 1012 cm−2,VL = −25 eV.


Tunable metal-insulator transition in boronnitride heterostructure

Figure 13: Schematic view of heterostructure device and measurementgeometry.

L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V.Morozov, K. S. Novoselov, I. V. Grigorieva, E. H. Hill, V. V.Cheianov, V. I. Falko, K. Watanabe, T. Taniguchi, R. V. Gorbachev,Nature Physics 7, 958961 (2011)


Figure 14: Electron transport in graphene – BN heterostructure fordifferent doping of the control layer at 70 K.


Figure 15: Electron transport in graphene – BN heterostructure fordifferent doping of the control layer at 20 K.


Figure 16: Resistivity of the studied layer at different T for high and lowdoping of the control layer.


Figure 17: T dependence of the maximum resistivity for the same devicefor low and high doping, and for the thin spacer.


Crude estimation of the transport gap width for very strong scatterers

nthe−h ≈ ni ≡ cnc , nc ≈ 3.8× 1015cm−2

Experimental magnitude of the transport gap width

nthe−h ≈ 1010cm−2 ⇒ ∆E ≈ 0.01eV

More careful estimation of corresponding impurity concentration

ni ≈ 5.4× 1010cm−2 ⇒ c ≈ 1.4× 10−5

For impurities with potentials that are less than ≈ 50 eV

cv2L ≈ 0.05⇒ c ∼ c0


Figure 18: Defect density dependence of ID at different VD values of 100mV, 1 V, and 4 V.

S. Nakahara, T. Iijima, S. Ogawa et al., Acs Nano 7, 5694-5700 (2013)


Functionalized graphene as a model system for the two-dimensionalmetal-insulator transitionM.S. Osofsky, S.C. Hernandez, A. Nath, V.D. Wheeler, S. Walton,C.M. Krowne, and D.K. Gaskill, Scientific Reports 6, 19939 (2016)

Figure 19: Resistance/square (in units of h/e2) for graphene exposed to(a) nitrogen- and (b) oxygen-containing plasmas. Inset (b): the behaviorexpected for 2D variable range hopping.


Conclusion

Certain characteristic concentration of impurities can bespecified, at which the graphene’s spectrum undergoes aqualitative change. The magnitude of this critical impurityconcentration follows from the spatial overlap of individualimpurity states.

It has been established that the cardinal modification of thespectrum is manifested by the opening of a transport gap aroundthe impurity resonance energy.


band structure transformation as driving force behind...

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