Charge neutrality point of disordered graphene

Michael FoglerUC San Diego

Talk at UCR, 11/01/2008

Reference: arXiv:0810.1755; support: NSF

Charge carriers in grapheneWe can create electrons … or holes

2μ∝n potential chemicaldensityelectron

==

μn

μ

Transport in graphene transistors

Electron-hole symmetryLinear σ (n) at high carrier density |n|“Universal” minimum conductivity σmin

2

min (4 8) eh

σ = ÷ ×

Con

duct

ivity

, 1 /

kΩ

Gate voltage, V

np

Novoselov (2005)

Substrate and doping dependence

Mohiuddin et al. (2008)

Chen at al. (2008)

Timed K+- doping Different substrates

Experimental evidence for charge inhomogeneity

p

np

n np

p

Experiment: Martin et al., Nature (2008)Theory: Nomura and MacDonald (2007); Das Sarma et al., 2007-2008; Shklovskii, PRB (2007); This work: M.M.F., arXiv:0810.1755

MINIMUM CONDUCTIVITY PROBLEM

Part II

Theoretical work on the minimum conductivity

Early work: Gor’kov, Fradkin, P.A. Lee (d-wave superconductors)Ludwig, M.P.A. Fisher, et al.Ando et al.Aleiner et al., AltlandCheianov et al.Mirlin et al.Nomura and MacDonaldDas Sarma et al.Beenakker et al.Castro Neto et al.… (> 30 papers)

min40, , , 4,σπ

= ∞ …

Suggested answers:

Why is the problem difficult?

Perturbation theory in disorder does not apply at the Dirac point, EF = 0 All other previously proposed “self-consistent approx.” are uncontrolledElectron interactions are not weak, α ~ 1Electron interactions are long-range:V(r) ~ 1 / r

Model of disorder: charged impurities in the plane

-+

-

Substrate

+

+

+ +-

- -+-

-

-+

+++ +

+

-+

• At low energy long-range scatterers dominate• Explains the offset of the neutrality point • Explains the linear behavior of conductivity vs. density • Such disorder can be added intentionally

-+

-

Transport away from the neutrality point

Substrate

+

+

+ +-

- -+-

-

-+

+++ +

+

-+

2

2 | |( )i

nnn

σπ α

=

conductivity electron concentration in-plane impurity concentration i

nn

σ ===

perturbative resultvalid for large n

AndoNomura and MacDonaldDas Sarma et al.Ostrovskii et al.Novikov…

Charge inhomogeneity

-+

-

Substrate

+

+

+ +-

- -+-

-

-+

+++ +

+

-+

r

n(r)

Dirac point Charge-neutrality point

Numerical simulations

Rossi and Das Sarma, ArXiv:0803.0963

( ) Prob. fun. of nP n n= ( ) (0) ( )S r n n≡ r

Thomas-Fermi approximation

Route to a controlled theory

constant dielectric,2

== κυκ

α e

If 1, 1κ α HfO2, water, ethanol, …

1. Treat α as a small parameter

2. Neglect weak localization (justified by the results)

κ

NONLINEAR SCREENING

Dielectric function w/o disorder

1

12

( ) energy density( ) inverse compressibility

screening length2

nn

Re

ε ε

χ εκ χπ

−

−

=

′′≡

≡

22( ) 1 ,

1 1( ) 1 ,

k kek U Uk

k kk R R

πχκ

∈ = + =

∈ = +

1 1/ 2| |n Rχ − −∝ →∞ ∴ →∞

Graphene: *NO* metallic screening at the Dirac point!

Usually:

Long-distance properties of a disordered system

11. ( ), , are self-averaging2. is finite (but hard to compute)

n RRε χ −

4 3

2 3

2 ie n Rr

πκ

rR

?R =1 1( ) 1 , at k k

k R R∈ = +

(0) ( )rK ≡ Φ Φ r ( ) screened potentialΦ =r

1(0)χ −

Martin et al. (2008)

Finite

10nmR ≈

Experiment: Theory:

Short distances, r << R

( ) bare potential( ) screened potential

VΦrr

3

3

Rr

∝

(0) ( ) (0) ( )rK V V≡ Φ Φ r r

rR

Screening is weak, in the 1st approximation non-existent (approach of Efros and Shklovskii)

rK ln (1)R Or

∝ +

?R =

Thomas-Fermi approximation

( )

2 2

( ) ( ) 0,

( ) sgn( ) | | ,| |

e n

n n n

n

μ

μ υ

υ

Φ − =

=

Φ Φ=

r r( ) electrostatic potential( ) local chemical potentialμ

Φ ==

rr

1( )( )FR

nλ =r

r(Modified) Thomas-Fermi approx. is valid:

Short-range statistics of the potential and so that of nare known. Hence, the energy density (dominated by the short scales) and R can be computed

1 weak screening, is largeRα ∴

Key results for the statistics of the density distribution

2| |2

22 2

1, | |2 | |

( )1 1~ ln , | |

| |

n

n

e nLn

P nL n

n L L

π−⎧

⎪⎪= ⎨⎪⎪⎩

2 24

0

( ) (0) ( )1 3 1 (1 2 )arcsin ,

2/r

S r n n

K K

θ θ θ θπ

θ

≡

⎡ ⎤= − + +⎣ ⎦≡

r

1lnLα

≡

12 in Lα

≡

4R

α≡

Self-similar fractal set of electron-hole “puddles”

r

n(r)2−

α -2 electrons

/ 4R α=

1ln 1Lα

=1

2 in Lα=

~1 electronMost typical: A larger puddle:

Numerical simulations

Rossi and Das Sarma, ArXiv:0803.0963

( ) Prob. fun. of nP n n= ( ) (0) ( )S r n n≡ r

• Qualitative agreement with our theory• Quantitative comparison is not

meaningful at such small α

TRANSPORT

min1(0.50 0.05) lnσα

= ±Our result:

Random internal p-n junctions (PNJ)

p

np

n np

p

Percolation theory approach: Cheianov et al, PRL (2007)

Local conductivity away from PNJ’s

2

2 | |( )i

nnn

σπ α

=

electron concentration in-plane impurity concentration i

nn

==

perturbative resultvalid for large n

Nomura -MacDonaldAndo et al.Das Sarma et al.Novikov…

r

σ (r)ln(1/ )α

Local and macroscopic conductivities at the CN point

r

σ(r)

,p nσ σ

2

2

-

1~ ~ ln

~

p n

p n

eh

eh

σ σα

σ-p nσ

?σ =

p

np

n np

p

Internal P-N boundaries at the Dirac point are diffusive and transparent: 3/ 22

-

2

-

~ , ~

1, ~ ln

p n

n p p n

e p dG ph

eG G Gh α

⎛ ⎞⎜ ⎟⎝ ⎠

*NO* percolation physics

diameter of a droplet,its perimeter length ,

3/2 fractal dimension

dp===

Effective medium theoriesD. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935)

1( ) ( 1)d d

σσ σ

=+ −r

M. Hori and F. Yonezawa, J. Math. Phys. 16, 352 (1975)

0

( ) 1ln expz z

ddz e

dσ

σ

∞− ⎡ ⎤ =⎢ ⎥⎣ ⎦∫

r

( )min10.50 0.05 lnσα

= ±

Formulas are very different but the results are very consistent:

Predicted transport behaviorLinear σ (n) at high carrier density |n|Non-universal σminBut changes appear only at astronomically large κ

min10.5lnσα

≈

Con

duct

ivity

Electron density, n

np

inn2α

σ =

Similar to: Adam et al. (2007)

Conclusions thus farApplied a nonlinear screening theory to graphene with coplanar charged impuritiesProblem is solvable in the leading-log approximation for α << 1 Key statistical properties of the density distribution are computed analyticallyMin conductivity can be accurately estimated from the effective-medium theories; percolation-type transport is not realizedMinimum conductivity is quasi-universalExperimental min conductivity is larger by ~ 2; Other sources of disorder? Correlations between impurities?

Recent experiments with Графин[Gra ‘fin]

Набор стеклянный:Графин, поднос, 6 рюмок.Серия: Вооруженные силы РФ

Графин – rus. [Gra ‘fin] Transparent container made of thick glass for temporary storage and/or serving water and alcoholic beverages in style

(Set: “Grafin,” tray, 6 shot glasses)

Recent experiments with Графин

Transport mobility of graphene immersed in ethanol

25 < κ < 55Coulomb scatterers are not important?

Mohiuddin et al. (2008)

Thank you!

Transport in “usual” 2D electron systems

1. The lower the electron concentration, the lower the electrical conductance

2. Onset of localization at the lowest concentrations

Conductance

Gate voltagehe2

Higher T

Lower T

AlGaAs

GaAs

gate

2D electron layer

donors

• Experiments • Theory

Eytan et al. (1997)

Ilani et al. (2001)

Inhomogeneities in “usual” 2D electron systems

Efros et al (1992) Shi and Xie (2001)

Shi & Xie

Electron density (10-3 / aB2)

0 2 4 650

0

50

100

-

Fogler, PRB (2004)

Also: Allison,…, Fogler,…, PRL (2006)

2 1in s

Control parameter

spacers=

A SINGLE P-N JUNCTION IN GRAPHENE

Zhang and Fogler, PRL (2008)

n

p

Effective thickness of the p-ninterface

1/3

tuntun

1~( )F

dnxk x dx

−

=

2

- ~p n FeG k WhW

( ) ~ ( ) ?Fk x n x =

Impurity scattering

n

p

+k2eα

κ υ=( )c

kα

Λ =

For Coulomb scattering

Ando et al, Nomura & MacDonald, Das Sarma et al, Novikov, …

Ballistic vs. diffusive p-n junctions

n

p

n

p

The interface is “blocked” by impurities

p-n interface is largely open for ballistic transport

2

- 2~p ni

e nG Wh nα

′Fogler, Glazman, Novikov, and Shklovskii, PRB (2008)

21/3

- ~p neG W nh

′