Charge neutrality point of disordered...
Transcript of Charge neutrality point of disordered...
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
′