Transport in Metallic and Semiconducting Dots
Historically, charge quantization was discovered and confirmed
experimentally earlier than quantum mechanics.
The first experiment was performed by R.A. Millikan:
[Physical Review I, 32, 349-397 (1911)]
The quantization of charge was observed in studying the
transport of oil drops in gas.
Transport in Dots
1
How about the manifestation of charge quantization in the
transport in materials?
An earliest clear demonstration of the role of “charging”
was performed by I. Giaever and H.R. Zeller when they
studied small Sn particles imbedded in an oxide film.
[Ref: Physical Review Letters, 20, 1504 (1968)]
[Ref: Physical Review, 181, 789 (1969)]
Transport in Dots
2
Tunneling, Zero-Bias Anomalies, and Small
Superconductors
H.R. Zeller and I. Giaever
Preparation of the tunnel junction containing Sn particles. Because
of the faster oxidation of Al with respect to Sn, the oxide in the space
between the particles is much thicker than the oxide at the surface of
the particles, as sketched schematically in the cross section of the
samples.
o
For a Sn particle with a radius r ~ 50 A the spacing of the electron
energy levels is of the order 0.1 meV or 1 0 K in units of k BT .
I-V of a junction with
average
particle
radius
o
r  110 A
For
at
1 .6 0 K
.
H  35 kOe all the
particles are normal.
It is the result of “vertical”
tunneling through a layer
of grains. This phenomenon
is now well known as
“Coulomb blockade”.
Transport in Dots
3
Dynamical resistance-versus-voltage characteristics for normal and
superconducting particles at T  1.6o K .
For particles with
o
r  110 A , H c is 13 kOe so that all particles are normal at
H  30 kOe .
Three possible mechanisms were considered in the paper:
1.
Direct tunneling through the aluminum oxide, avoiding the Sn particles.
This mechanism gives a constant, voltage- and temperature-independent,
background conductivity.
For properly prepared junctions with a particle
o
radius r  30 A , direct tunneling can be made completely negligible even at
1o K and at zero bias.
2.
Tunneling from one Al film onto a Sn particle, localizing the electron there
and then in turn tunneling out on to the other side. This process needs an
activation energy which in turn is responsible for the zero-bias resistance
peak.
3.
This was the mechanism mostly discussed in the paper.
Tunneling from one aluminum film through a particle and out onto the
other aluminum film without actually localizing the electron on the particle.
The particle is only involved as an intermediate state; thus, the process is of
second order.
This process can conceivably become important at low
temperatures and at low voltages; however, no experimental evidence has
been found for this process.
It is very similar in principle to Anderson’s
model for tunneling involving intermediate magnetic impurity states.
Transport in Dots
4
More on single-electron charging effects in single small
tunnel junctions:
[Physical Review Letters, 39, 109 (1987)]
T.A. Fulton, and G.J. Dolan
A scanning-electron micrograph of a typical sample. Junctions
labeled a, b, and c are formed where the vertical electrodes overlap
and contact the longer horizontal central electrode. The bar is 1 m
long. Small-area tunnel junctions had low-capacitances ( C  1 fF ).
T  1.7 K . S and L
are
junctions
with
small and large C.
Inset: Offset voltage
vs junction areas for
four different samples.
Typical junction area:
(0.03  0.01m) 2
;
thickness-to-dielectric
-constant
ratio:
0.15 nm .
Transport in Dots
5
Single tunnel junction ( C  10 18 F ) formed out of STM:
[Physical Review Letters, 60, 369 (1988)]
P.J.M. van Bentum, H. van Kempen, L.E.C. van de Leemput, and P.A.A. Teunissen
This work demonstrated that the stray capacitance far away from the tunneling
area does not contribute to the effective capacitance that determines the
Coulomb blockade.
Conditions for the observation of Coulomb Blockade
In a capacitor formed out of a single Metal-Insulator-Metal (MIM)
junction, the charge Q across the junction can be changed
continuously.
It is because that these charges are essentially
“polarized charges” from the two metal “plates”.
On the other hand, electrons might tunnel through the middle
barrier of the junction. But when an electron does so, it tunnels in
its entirety, that is, as a whole. The subsequent change in Q will
then be quantized. The tunneling electron might need to overcome
an energy barrier if the charging energy of the capacitor were to be
increased.
Transport in Dots
2
This energy is typically of order e 2C .
6
To be able to observe Coulomb blockade, two conditions must be
met:
(1) Thermal fluctuation must not be too large:
e2
Ec 
 k BT .
2C
For C  10
15
e2
1K ;
2C
F,
18
and for C  3  10 F ,
e2
 300 K .
2C
This amounts to r ~ 28 nm for grain radius and somewhat larger for
disc radius, when the relative dielectric constant is assumed unity.
Taking into consideration of the dielectric constant, the radius should
be sub-10-nm structures in order to observe coulomb blockade at
room temperature.
(2) Quantum fluctuation must not be too large:
The single electron tunneling time  T  RTC , where RT is the
tunneling resistance. Quantum fluctuation due to this tunneling
is small when
Ec 

Transport in Dots
RT  RQ 
h

T
h
 26 k
2
e
7
A single tunnel junction of capacitance C and tunnel
resistance RT biased by an ideal current source
The parameters C and RT characterizes the junction. The state
of the junction, however, is described by Q and n , which is,
respectively, the charge on the capacitor and the net number of
electron tunnel from electrode 2 to electrode 1.
The change in energy of the system when one electron tunnels across
the junction is given by
Q 2 (Q  e) 2
e
e
E 

  Q  
2C
2C
C
2

where + (-) refers to an electron tunneling from electrode 2 (1) to
electrode 1 (2). Here  e is the charge of an electron.
At T  0 , the tunneling process is favorable energy-wise when
E   0 . Therefore, coulomb blockade occurs when

Transport in Dots
e
e
Q .
2
2
8
Single electron tunneling (SET) scenario:
For a finite but small current I from the current source, Q

increases linearly with time ( Q  I ) . Tunneling will occur at Q 
e
2
e
such that Q changes suddenly to 
and the cycle continues.
2
Thus the voltage of the junction will exhibit saw-tooth oscillations,
and with oscillation amplitude
e
.
2C
The frequency of the tunneling
I
e
events is f SET  .
It is interesting to note that even though the tunneling events are
probabilistic, the effect of the coulomb blockade is to enforce
correlations between tunneling events so that tunneling occur at
uniform time intervals.

The tunneling rate  depends on the energy change E as well as
the temperature T . It was derived by Likharev, (to be shown later)
1
E 
  2
e RT 1  exp(  E  k BT )

Transport in Dots
9
Beyond the Coulomb blockade regime, when the bias current I is
large enough so that V (t ) 
e

at all times, we have E  0 and
2C
E   0 at all times also. Therefore,
 
1
E  ;
e RT
 
2
1
(  E  ) exp( E  k BT ) .
e RT
2
We have assumed that E  k BT . The tunneling transition rate
  can be expressed in terms of V , given by
C
 (V )  2
2e RT

2
 2 
e 
V  V    .
C  


The voltage V varies within the range V 
To obtain the relation between
I
e
e
V V 
.
2C
2C
and V , we invoke the
requirement that one tunneling event occur in each voltage cycle, that
is
V e / 2C

V e / 2C
dV 
 (V )  1 ,
I C
and we obtain
V  IRT 

e
.
2C
e 

I  GT V 

2C  .

The intercept on the voltage axis
e
is called the Coulomb gap.
2C
Clear experimental evidence of SET oscillations in single junctions has
remained elusive to date. The parasitic capacitances of the leads leading
to the junction far exceed that of the junction itself. An external current
source charges up the lead capacitance and tends to behave as a voltage
source rather than a current source with respect to the tunnel junction.
Transport in Dots
10

Derivation of the transition rate  for a single tunnel junction:
We consider tunneling for a metallic tunnel junction within the
single-particle picture using the transfer Hamiltonian method. A
constant voltage V is applied to the left electrode relative to the
right. The Fermi energies
E Fr  E Fl  eV
.
The charge on the left electrode is  Q and on the right electrode is
 Q . Transition rate
kr kl 
where Tkr ,kl
2
2
Tk ,k
 r l
2
1  f ( E  ( E
l)
l
 Er ) ,


 kl H t k r .
The total rate, from occupied states on the right to unoccupied states
on the left is given by
  (V ) 
2

 Tkr ,kl
f ( Er ) 1  f ( El ) ( El  Er ) .
2
k r ,k l
Assuming that the tunnel matrix element and the density of states are
constant, we have

2 2
 (V ) 
T Dlo Dro  dE f ( E  E Fr ) 1  f ( E  E Fl )

Ecm



and

2 2
 (V ) 
T Dlo Dro  dE f ( E  EFl ) 1  f ( E  EFr ) .

E cm



The total current

I (V )  e   (V )    (V )


2e 2

T Dlo Dro  dE f ( E  E Fr )  f ( E  E Fl )

Ecm
Transport in Dots


11

lim
Ecm 
 dE  f ( E )  f ( E  eV )  eV ,
Ecm

V  IRT ;
and
RT 

2e 2 T Dlo Dro
2
.
Note that in the above derivation, we have include any Coulomb
blockade effect, thus RT is the junction resistance in the regime
when Coulomb blockade is no longer significant.
The transition rates, however, are subject to modification
when the charging energy is no longer negligible.
The state of the junction is characterized by the net number n of
electrons tunneling through the junction from right to left.
The

change in energy E (n)  E (n)  E (n  1) . Using a “golden rule”
approximation for the transition rate, the rate of tunneling of
electrons back and forth through the junction is
 
2





.
T
f
(
E
)
1

f
(
E
)

E

E


E

if
i
f
i
f
 
2
ki ,k f
Again assuming weak energy dependence in the transfer matrix
element and the densities of states, we have
1
 (V )  2
e RT


 dE f ( E ) 1  f ( E  E ).

Ecm
Using the property of the Fermi function



f ( E ) 1  f ( E  E ) 
f ( E )  f ( E  E  )
1  e E

/ k BT
,
and integrating, we get
1
E 
 (V )  2

e RT 1  e E / kBT .

Transport in Dots
12
Coulomb Staircase in double tunnel junctions
[Physical Review Letters, 63, 801 (1989)]
R. Wilkins, E. Ben-Jacob, and R.C. Jaklevic
o
Average In droplet size is 300 A . Curve A is an experimental
I  V characteristic. Curve B is a theoretical fit the data for
CD  3.5  1019 F , CT  1.8  10 18 F , RD  7.2  106  , RT  4.4  109  . The
obvious asymmetric features in curve A require a voltage shift
Vs  22 mV . Curve C, calculated for Vs  0 .
A small quadratic term
was added to the computed tunneling rate for each junction.
Transport in Dots
13
Coulomb Staircase Characteristics in
Single electron transistor
Total charge Q in the middle island:
Q  Qi  Cl (VC  Vl )  Cg (VC  Vl )  Cr (VC  Vr )
i

Q  VC C   CiVi  ne
i
where n is an integer, and
C  Cl  C r  C g .
Rearranging, we have
VC 
Q   CiVi
i
C
.
The electrostatic energy U of the system is
Qi2
U 
i 2Ci
Transport in Dots
14
Eliminating VC , we have
U
Q2
1

2C 2C





2
  Ci   C jV j     CiVi   C jV j  .
 j
 j
 i
  i

In a more symmetric form with respect to the summation indices,
 C C V
Q2
1
U

2C 2C
i
j i
i
j
Vj  .
2
i
Note that U is the internal energy. But when Q is changed, via
the tunneling of electrons onto or out of the island, the voltage
sources will do work too.
For instance, if an electron tunnels through junction j into the
island,
VC  VC  VC  VC 
Q
e
 VC 
,
C
C
and the change in the charges Qi (for i  j ) are:
Qi  Ci ( VC  Vi )  
Hence
the
 QiVi 
work
done
by
the
e
Ci .
C
i -th
voltage
source
is
e
CiVi .
C

The work done by the j -th voltage source is  eV j 


e
C j Vj  .
C

The total work done by all the voltage souces is:
W j  e (Vi  V j )
i
Ci
.
C
Note that W j does not depend on Q .
Transport in Dots
15
The total energy (or enthalpy) E of the system is defined as
E  U  n j Wj
j
where n j is the net number of electrons tunnel into the island
through junction j . The first term U is the electrostatic energy
stored in the capacitor system, and the term   n j W j is the net
j
work done by the capacitor system to its environment – the voltage
sources.
Without loss of generality, we let V1  0 , V2  V and the number of
electrons in the island be n .
The change in U in the n  n  1 process is:
U


2
2

ne  n  1e
(n) 

2C
U  ( n ) 
2C


1
 2ne 2  e 2 .
2C
The work done by the voltage sources if an electron tunnel into the
island through junction 1:
W1 
e
CgVg  C2V   e (C2V  Qo )
C
C
where Qo  C gVg  Qoo and Qoo represents the shifting in the
polarization charge due to naturally occurred random charged
impurities near the island.
Similarly, the work done by the voltage sources if an electron tunnel
into the island through junction 2: ( C g  C1 )
W2  
Transport in Dots
e
Qo  C1V  CgV    e Qo  C1V 
C
C
16
The change in E for the n  n  1 process in which tunneling
occurs in junction 1 is:
E1  U (n)  U (n  1)   1  W1 
E1 
e
C
 e

 2  (ne  Qo )  C2V  .
The change in E for the n  n  1 process in which tunneling
occurs in junction 2 is:
E2  U (n)  U (n  1)   1 W2 
E2 
e
C
 e



(
ne

Q
)

C
V
o
1
 2

Tunneling will occur when either E1  0 or E2  0 .
To see the Coulomb staircase, let us consider the simpler case of
C1  C2  C J , and R2  R1 .
Tunneling occurs much faster in
junction 1 than in junction 2. Therefore, the maximum number of
electrons in the island will be determined by the condition
E1  0
 1 CV
 nmax  int    J  .
e 
 2
Here we consider the case Qo  0 and V  0 is above the
coulomb blockade threshold.
The current is determined by the tunneling in junction 2, given by
I  e2 (V )
Transport in Dots
17
I
I
1
2 R2
1
E 2
eR2

e
V  C
J

Whenever V increases by V 
1 

 nmax   .
2 

e
e
2
, the number of electrons
CJ
C
in the island will increase by 1.
Subsequently,
I
will increase
abruptly by e 2 R2C J  . Except for that abrupt increase in
I
, the
current increases linearly with V , with a slope of 1 2 R2 . This is
the Coulomb staircase.
The “charge” Qo , or equivalently, the gate V g also plays a very
interesting role in determining the I  V characteristics. Putting
Qo back into the above discussion, we have
I
1
2 R2

e 
Qo 1 
V

n

  .

max

CJ 
e 2 

What happen if Qo  e 2 ?
Well,
I increases linearly with V even when V ~ 0 !
There is no Coulomb blockade! But still there is Coulomb staircase.
Transport in Dots
18
The previous V  Qo analysis can be understood from looking at the
energy-band diagram of the double-junction system.
For a simple illustration, we let C1  C2  C J .
The maximum
threshold voltage for the coulomb blockade region is Vthreshold  e C ,
where C  2C J . The characteristic energy scale for the problem
2
is EC  e 2C  .
Consider the point A on the V  Qo diagram where Qo  e ;
V  2Vm a x 4 EC e , two alternate ways of electron transport
cycles are shown:
E1  EC ;
E1  3EC ; E 2  3EC ; E 2  EC ;
then the energy diagram is:
If an electron tunnel into the island through junction 1, then
E1   EC ;
E1   EC ; E 2  5EC ; E 2  3EC ;
and the energy diagram becomes:
However, if alternatively, an electron tunnel out of the island
through junction 2, then
E1  3EC ;
E1  5EC ; E 2   EC ; E 2   EC ;
and the energy diagram becomes:
Transport in Dots
19
From the V  Qo diagram, it is clear that:
1. For a fixed Qo :
increasing V gives rise to Coulomb staircase.
2. For a fixed V :
decreasing Qo , or equivalently increasing V g leads to
the increase in n , the number of electrons in the island.
3. For a fixed V , where V is in the low bias regime, that
is V  Vthreshold:
Current oscillates with Qo , or V g : we have the
coulomb oscillations.
[Physical Review B, 51,
12649 (1995)]
M.
Tinkham,
J.M.
Hergenrother, and J.G. Lu
A simple quasiequilibrium
model that accounts in some
detail
for
the
observed
temperature dependence in
the current through a SET
with Al island ( V  125V ).
Note the transition from 2e
to e periodicity occurs in
the
rather
temperature
narrow
range
240 ~ 270 mK near T * .
Transport in Dots
20
The above discussion is based on a theory: the Orthodox
Theory, which is proven very successful in the metallic
junctions.
The key is to calculate the ensemble distribution  (n ) of the
number of electrons on the center island. The time rate of
change of  (n ) is
 ( n, t )
    (n ) (n, t )    ( n ) (n, t )
t
tunn

 


( n  1) ( n  1, t )    (n  1) (n  1, t )

The first term on the right hand side is the outscattering
term whereas the second term is the inscattering term.
 ( n, t )
 0.
steady-state condition requires
t
The
Thus we have
 (n)  (n)    (n)   (n  1)  (n  1)   (n  1)  (n  1) .
Here
  (n)  1 (n)  2 (n) .
We have to solve numerically for
 (n ) .
The steady-state current is given by






I (V )  e   ( n )  ( n )   ( n )  e   ( n ) 1 ( n )  1 ( n ) .
n  
Transport in Dots

2

2
n  
21
[Physical Review Letters, 74, 3241 (1995)]
D.C. Ralph, C.T. Black, and M. Tinkham
Temperature T  4.2 K and H  0 . Dashed curve is the theoretical fit,
with an offset of 100G -1 .
Transport in Dots
22
[Physical Review Letters, 78, 4087 (1997)]
D.C. Ralph, C.T. Black, and M. Tinkham
Current-voltage curves displaying Coulomb-staircase structure for
three different samples, at equally spaced values of gate voltages.
Data for different V g are artificially offset on the current axis.
Transport in Dots
23
[Physical Review B, 61, 46 (2000)]
H. Imamura, J. Chiba, S. Mitani, K. Takanashi, S. Takahashi, S. Maekawa, and
H. Fujimori
Room temperature
measurement.
Transport in Dots
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Differences
structures:
between
metal
and
semiconductor
junctions
or
The Fermi wavelength of the electrons in semiconductors is of the
order of the limit of fabrication of artificial structures, 10-50 nm; in
contrast, it is of the order of the lattice spacing in metals.
Consequently, the coexistence of a discrete energy spectrum with the
charging energy will be easier to study in semiconductors than in
metal junctions. This coexistence expresses itself both in the I(V)
characteristics and in conductance oscillations as a function of gate
voltage.
Even though the single particle level spacing  may be significant in
STM-grain tunneling experiments, it will be much smaller than the
charging energy. Semiconductors offer the possibility to study the
full range of   EC to   EC . In semiconductors also the number of
free electrons can be suppressed strongly, usually with the help of
gate electrodes, to the point where only a few free electrons are
present. Here deviations from a simple electrostatic energy are
expected. Capacitances in semiconductors are subjected to change
because the potential profile (and thus the size) of the dot will
strongly change during the charging of the dot with the first few free
electrons.
Transport in Dots
25