charge neutrality point of disordered...
TRANSCRIPT
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Charge neutrality point of disordered graphene
Michael FoglerUC San Diego
Talk at UCR, 11/01/2008
Reference: arXiv:0810.1755; support: NSF
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Charge carriers in grapheneWe can create electrons … or holes
2μ∝n potential chemicaldensityelectron
==
μn
μ
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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)
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Substrate and doping dependence
Mohiuddin et al. (2008)
Chen at al. (2008)
Timed K+- doping Different substrates
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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
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MINIMUM CONDUCTIVITY PROBLEM
Part II
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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:
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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
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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
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-+
-
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…
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Charge inhomogeneity
-+
-
Substrate
+
+
+ +-
- -+-
-
-+
+++ +
+
-+
r
n(r)
Dirac point Charge-neutrality point
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Numerical simulations
Rossi and Das Sarma, ArXiv:0803.0963
( ) Prob. fun. of nP n n= ( ) (0) ( )S r n n≡ r
Thomas-Fermi approximation
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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
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NONLINEAR SCREENING
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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:
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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:
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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 =
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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 ∴
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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≡
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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:
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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
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TRANSPORT
min1(0.50 0.05) lnσα
= ±Our result:
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Random internal p-n junctions (PNJ)
p
np
n np
p
Percolation theory approach: Cheianov et al, PRL (2007)
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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
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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 =
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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===
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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:
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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)
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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?
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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)
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Recent experiments with Графин
Transport mobility of graphene immersed in ethanol
25 < κ < 55Coulomb scatterers are not important?
Mohiuddin et al. (2008)
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Thank you!
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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
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• 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=
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A SINGLE P-N JUNCTION IN GRAPHENE
Zhang and Fogler, PRL (2008)
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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 =
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Impurity scattering
n
p
+k2eα
κ υ=( )c
kα
Λ =
For Coulomb scattering
Ando et al, Nomura & MacDonald, Das Sarma et al, Novikov, …
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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
′