Hopping transport and the “Coulomb
gap triptych” in nanocrystal arrays
Brian Skinner1,2, Tianran Chen1, and B. I. Shklovskii1
1Fine
Theoretical Physics Institute
University of Minnesota
2Argonne
National Laboratory
2 September 2013
UMN MRSEC
Electron conduction in NC arrays
Conventional hopping models:
Each site has one energy level:
. filled or empty.
energy
μ
coordinate
In nanocrystal arrays:
Each “site” is a NC, with
a spectrum of levels:
Conductivity is tuned by:
– spacing between sites
– insulating material
– disorder in
energy/coordinate
– Fermi level μ
Energy level spectrum is tailored by:
– size
– composition
– shape
– surface chemistry
– magnetism
– superconductivity
– etc.
Electron conduction in NC arrays
energy
μ
coordinate
Conductivity reflects the interplay between individual energy
level spectrum and global, correlated properties.
Experiment: semiconductor NCs
CdSe NCs, diameter 2 nm – 8 nm
[Bawendi group, MIT]
[talk by Philippe Guyot-Sionnest]
[talk by Alexei Efros]
15 nm
[JJ Shiang et al, J. Phys. Chem. 99:17417–22 (1995)]
Experiment: metallic NCs
precise control over
size/spacing:
tuneable metal/insulator
transition:
range of shapes:
cubes
stars
rods
wires
hollow
spheres
core/
shell
[Aubin group, ESPCI ParisTech]
[Kagan and Murray groups, UPenn]
[Talapin group, Chicago]
Experiment: magnetic NCs
Co / CoO NCs
Fe3O4 NCs
[H. Zeng et. al., PRB 73, 020402 (2006)]
[H. Xing et. al., J. Appl. Phys. 105, 063920 (2009)]
Experiment: superconductor NCs
superconductor/insulator transition
tuned by B-field or insulating barrier
[Zolotavin and Guyot-Sionnest, ACS Nano 6, 8094 (2012)]
[talk by Philippe Guyot-Sionnest]
Experiment: granular films
Disordered Indium Oxide
Indium evaporated onto SiO2
[Belobodorov et. al., Rev. Mod. Phys. 79, 469 (2007)]
[Y. Lee et. al., PRB 88, 024509 (2013)]
[talk by Allen Goldman]
Model of an array of metal NCs
Uniform, spherical, regularlyspaced metallic NCs with
insulating gaps
insulating
gaps
Large internal density of states:
spacing between quantum levels
δ0
metallic NCs
Model of an array of metal NCs
High tunneling barriers
a << d
Tunneling between NCs is
weak:
G/(e2/h) << 1
d
electron
wavefunction
a
Single-NC energy spectrum
A single, isolated NC:
e-
Coulomb self-energy:
Ec = e2/2C0
E
ground
state
energy
levels:
Ec
Ef
g1(E)
Single-NC energy spectrum
A single, isolated NC:
Coulomb self-energy:
e-
Ec = e2/2C0
+
E
ground
state
energy
levels:
2Ec
Ef
g1(E)
Single-NC energy spectrum
Multiple-charging:
Coulomb self-energy:
-
Ec = e2/2C0
Single-NC energy spectrum
Multiple-charging:
e-
Coulomb self-energy:
Ec = e2/2C0
-2
→ (2e)2/2C0
E
2Ec
ground
state
energy
levels:
Ef
each NC has
a periodic
spectrum of
energy levels
g1(E)
Same spectrum
that gives rise to
the Coulomb
blockade
Density of Ground States
Disorder randomly shifts
NC energies:
+e
E
-e
Ec
Ef
g1(E)
+e
-e
-e
+e
E
Ef
g1(E)
“Density of ground states”
(DOGS): distribution of
lowest empty and highest filled
energies across all NCs
Hamiltonian and computer model
• Simulate a (2D) lattice of NCs with random interstitial charges
δq  (-Qmax, Qmax)
• Search for the electron occupation numbers {ni} that minimize the
total energy
qi = (δq)i - eni
• Calculate DOGS by making a histogram of the single-electron ground
state energies at each NC:
• Calculate resistivity ρ as a function of temperature T by mapping the
ground state arrangement to a resistor network
DOGS - results
Main features:
1.
g(E) vanishes
near E = 0
2.
g(|E| > 2) = 0
3.
Perfect symmetry
Distribution is universal at sufficiently large disorder
The Coulomb gap
E+
E-
in 2D
Efros-Shklovskii conductivity:
Typical hop length:
1/ 2
rhop
 4 

~  2
 e k BT 
Absence of deep energy states
Usual situation:
Here:
(lightly-doped semiconductors)
g1(E)
small
disorder,
w1:
w1
E
No deep energy states for any value of disorder.
Absence of deep energy states
Usual situation:
Here:
(lightly-doped semiconductors)
g1(E)
small
disorder,
w1:
E
w1
large
disorder,
w2 :
Coulomb gap is
less prominent
E
w2
No deep energy states for any value of disorder.
E
2Ec
Here, deep states are
not possible:
E+
EfE
Ei+ = Ei- + 2Ec
g1(E)
“Triptych” symmetry
[orthodoxy-icons.com]
Ei+ = Ei- + 2Ec
DOGS is completely constrained by symmetry and Coulomb gap.
g(E) is invariant in the limit of large disorder.
Miller-Abrahams resistor network
...
i
Rij
j
Rjk
...
Rik
Rjl
...
Ril
k
Rkl
l
...
ρ is equated with the minimum
percolating resistance.
Variable-range hopping
rate of phononassisted tunneling:
D’
 2r E 
  exp 

  k BT 
ξ = localization length
ξ ~ a D’/d >> a
Variable-range hopping
rate of phononassisted tunneling:
i
D’
 2r E 
  exp 

  k BT 
Rij
ξ = localization length
j
 2r Eij 
Rij  exp 

  k BT 
Efros-Shklovskii conductivity
low T
ρ(T) is largely
universal at
sufficiently
large disorder
higher T
(T*)-1/2
T* 
2 DkBT
Ec
Model of an insulating array of
superconductor NCs
Model of an insulating array of
superconductor NCs
Uniform superconducting
pairing energy, 2Δ
Weak Josephson coupling
J ~ Δ ∙ G/(e2/h) << Ec
 heavily insulating, with
decoherent tunneling
Focus on the case where Δ and Ec
are similar in magnitude
# pairs in
NC i
pairing energy
[Mitchell et.
al., PRB 85,
195141
(2012))]
Single-electron energy spectrum
An isolated NC with Cooper pairing
(and an even number of electrons):
Coulomb self-energy:
e-
Ec = e2/2C0
E
Ec
single electron density of ground states:
Ef
g1(E)
Single-electron energy spectrum
An isolated NC with Cooper pairing
(and an even number of electrons):
Coulomb self-energy:
Ec = e2/2C0
e-
Binding energy of pair:
2∆
+
E
Ec
single electron density of ground states:
Ef
g1(E)
Ec+2Δ
Pair energy spectrum
Can also have hopping of pairs:
Coulomb self-energy:
2e-
(2e)2/2C0 = 4Ec
+/- 2
E
pair density of ground states:
Ef
4Ec
g2(E)
DOGS - results
singles
pairs
4(1 – Δ)
Δ = 0:
e  2e
Δ = 2Ec:
e  √2e
Δ = Ec:
2(Δ – 1)
Miller-Abrahams network for singles
and pairs
i
j
• ρ1 is the percolating resistance of the singles network.
• ρ2 is the percolating resistance of the pair network.
Effective charges in hopping transport
ES hopping:
ln  * 

2 D'
ln(  /  0 )
T* 
2 DkBT
Ec
Effective charges in hopping transport
Slope gives
ES hopping:
ln  * 

2 D'
ln(  /  0 )
T* 
2 DkBT
Ec
TES  C
e2

Effective charges in hopping transport
Slope gives
ES hopping:
TES  C
e2

e* = 2e
e* = √2e
e* = e
ln  * 

2 D'
ln(  /  0 )
T* 
2 DkBT
Ec
Magnetoresistance
Superconducting
gap is reduced by a
transverse field:
single e- hopping
is gapped
pair hopping
is gapped
[Lopatin and
Vinokur, PRB 75,
092201 (2007)]
( For example,
Zeeman effect:
   0 1  ( B / Bc ) 2 )
1.5
1
Δ/Ec
increasing magnetic field:
0.5
0
Conclusions
E
energy
•In NC arrays, single-particle
spectrum and global correlations
combine to determine transport
=
coordinate
•For metal NCs, the “Coulomb gap
triptych” is a marriage between the
Coulomb blockade and the
Coulomb gap [PRL 109, 126805 (2012)]
 Disorder-independent transport
e* = 2e
•For superconducting NCs, the gap
changes the “effective charge” for
hopping
[PRB 109, 045135 (2012)]
Thank you.
e* = √2e
e* = e
Reserve Slides
Publications
metal NCs: Tianran Chen, Brian Skinner, and B. I. Shklovskii, Coulomb gap
triptych in a periodic array of metal nanocrystals, Phys. Rev. Lett. 109,
126805 (2012).
superconducting NCs: Tianran Chen, Brian Skinner, and B. I. Shklovskii,
Coulomb gap triptychs, √2 effective charge, and hopping transport in periodic
arrays of superconductor grains, Phys. Rev. B 86, 045135 (2012).
semiconductor nanocrystal arrays: Brian Skinner, Tianran Chen, and B. I.
Shklovskii, Theory of hopping conduction in arrays of doped semiconductor
nanocrystals, Phys. Rev. B 85, 205316 (2012).
2D and 3D DOGS - metal
2D:
3D:
2D and 3D DOGS - SC
2D:
3D:
Disorder
+e
+e
+e
Disorder
Some impurities are
effectively screened
out by a single NC
+e
-e
+e
+e
-e
Disorder
+qA
-qA
+e -qB
+qB
-e
+e
+e
-e
Some impurities are
effectively screened
out by a single NC
Others get
“fractionalized”
Disorder
+qA
-qA
+e -qB
-e + qB
-e
+e
+e
Some impurities are
effectively screened
out by a single NC
Others get
“fractionalized”
-e
Result is a net
fractional charge on
each NC
Tunneling conductance
intra-NC density of states:
tunneling conductance:
Electron energy spectrum of a
semiconductor nanocrystal
Electron energy spectrum has two components:
1) quantum
confinement
energy:
1D
1P
1S
ΔEQ
Electron energy spectrum of a
semiconductor nanocrystal
Electron energy spectrum has two components:
Ec
2) electrostatic
charging
energy:
total Coulomb self-energy:
5e2/κD
U(Q) = Q2/κD
3e2/κD
energy to add one electron:
e2/κD
Ec = U(Q - e) - U(Q)
0
-e2/κD
-3e2/κD
-5e2/κD
Ec = (e2 - 2Qe)/κD
Random doping of NCs
D
d
D’
Regular lattice of
equal-sized NCs
Donor number Ni
is random:
Electron energy spectrum of a single
nanocrystal
no donors
E
N=1
E
N=2
E
N=3
E
N=5
N=9
E
E
1D
1P
...
EQ1S
1S
...
Typical case: ν = 5, ΔEQ = 5 Ec
N=0
N=1
N=4
N=5
N=6
N=9
E
E
E
E
E
E
1D
...
...
1P
1S
N = 10
E
Typical case: ν = 5, ΔEQ = 5 Ec
N=0
N=1
N=4
N=5
N=6
N=9
E
E
E
E
E
E
E
+1
+2
N = 10
1D
-2
-1
...
...
1P
1S
Density of states: ν = 5, Δ = 5 e2/κD
1P
1S
1D