Single-Electron Transistors 2003

7/27/2019 Single-Electron Transistors 2003

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Single Electron Transistor Instructor : Pei-Wen Li1

Single-Electron Transistors

Reference book:

Single Charge Tunneling Coulomb Blockade

Phenomena in Nanostructures

By Hermann Grabert and Michel H. Devoret, 1992

Referred Journal Review Papers:

Correlated discrete transfer of single electrons in ultrasmall tunnel juntions

By K. K. Likharev, IBM J. Res. Develop. Vol.32, p.144, 1989

Single-Electron Devices and Their Applications

By K. K. Likharev, Proceedings of the IEEE, vol.87, p.606, 1999

Single-Electron Memory for Gita-to Tera Bit Storage

By K. Yano et al., Proceedings of the IEEE, vol.87, p.633, 1999


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Contents

Single Electron Phenomena: A General Introduction

Single Electron Transistor

Fabrication and Analysis Applications


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Single Electron Phenomena: A General Introduction

.Scaling prospects for various

bit-addressable memories.

DRAM is expected to be bottlenecked atthe generation of 64 Gbit integration (70

nm technology) due to the problems with

the storage capacitor scaling.

Nonvolatile memory is going to be the

mainstream for 64Gbit-16Tbit memory.

SET/FET would be feasible starting from

~ 3 nm minimum feature sizes.


4/46Single Electron Transistor Instructor : Pei-Wen Li4

SET Evolution

The manipulation of single electrons was demonstrated in the seminal

experiments by Millikan at the very beginning of 20 century.

Single Electron Device: in which the addition or substraction of a small number

of electrons to/from an electrode can be controlled with one-electron precision

using the charging effect.

They are not interesting not only as new physical phenomena in nanostructures but also

because they offer new operating principles for future ICs. Application: Memory, switch, Thermal meter

Advantages:

Good stability: is the strong incentive to explore the possibility of the devices.

atomic scale physical dimension ULSI possible Ultralow power operation: simply because they use very small number of electrons to

accomplish basic operation.

Fast operation: only a few electrons (



Schematic of Single Electron Devices

A quantum dot is weakly coupled by tunnel barriers to two electron reserviors.Oxide



A General Introduction

Recall that the motion of electrons in an infinite potential well is in a standing

waveform. That means that the energy of the particle in the infinite potential

well is quantized. That is, the energy of the particle can only have particular

discrete values.

integerpositiveaisnwhere

2

2

222

ma

nEE n

h==



Charging Energy

Let a small conductor (island) be initially electroneutral, i.e., have exactly as

many (m) electrons as it has protons in its crystal lattice. In this state the island

does not generate any appreciable electric field beyond its borders, and a weak

external force may bring in an additional electron from outside. Now the net

charge of the island is (-e), and the resulting E field repulses the following

electrons which might be added.



Charging Energy

The charging energy of the island isEC, where Cis the capacitance of the island:

When the size of the island becomes comparable with the de Brogliewavelength of the electron inside the island energy quantization

The energy scale of the charging effects is given by a more general notion, the

electron addition energy (Ea). In most cases of interest,E

acould be

approximated by

HereEk is the quantum kinetic energy of the addition electron; for a degenerate

electron gasEk= 1/g(F)V, where Vis the island volume andg(F) is the density

of states on the Fermi surface.

CeEC

2

=

kCa EEE +=

( )m

k

mW

nEN

22

22

2

2hh

+=


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Electron Transfer in an quantum dot

The transport of electrons through the quantum dots is an interplay of resonant

tunneling and Coulomb blockade effects.

In the absence of charging effect, a conductance peak due to resonant tunneling

occurs when the Fermi energyEF in the source lines up with one of the energy

levels in the dot.

However, this condition is modified by the charging effect. The energy levelis renormalized by the charging effect

That is, the renormalized level spacing is enhanced above the

bare level spacing by the charging energy.

levelbaretheiswhere,21

2

*nnN E

CeNEE +

2

C

eEE +=

levelenergybaretheiswhere,2 2

222

nnF

Ema

nEE

h==


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Single-electron tunneling through a quantum dot

The energy levels, EN, are modified by the charging effect. That is the

charging energy regulatesthe level spacing.

The spin degeneracy is lifted by the charging energy.

CeEE

2* +=

Bare Energy level

Normalized Energy level


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Single-electron tunneling through a quantum dot

(a) , with N referring to the lowest unoccupied level in

the dot.

(b) An electron has tunneled into the dot,

)1(2

2

+=+ NeEC

eE FN

)(2

2

NeEC

eE FN +=


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Addition Energy, Kinetic Energy

For 100-nm-scale devices,Ea is dominatedby the charging energyEc and is of theorder of 1 meV (~10 K). Since the thermalfluctuations suppress most single-electroneffects unless

these device have to be operated in the

sub-1-K range. Unpractical! If the island size is below ~ 10 nm,Ea

approaches 100 meV, and some singleelectron effects become visible at RT.

However, digital SE devices requireeven higher values ofEa to avoid thermallyinduced random tunneling events, so thatminimum feature size of SET has to besmaller than ~ 1nm. RT operation

Ea =EC+ E

TkE Ba 10


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Single Electron Transistor

The resulting SET device is reminiscent of a usual MOSFET, but with a small

conducting island embedded between two tunnel barriers, instead of the usual inversion

channel.

The expression of the electrostatic energy W of this SET is

n1 and n2 are the number of electrons passed through the tunnel barriers one and two,

respectively, so that n = n1- n2, while the total island capacitance Ctotal is now a sum of

C0, C1, C2, and whatever stray capacitance the island may have. The external charge Qe

= C0Vg is just a convenient way to present the effect of the gate voltage Vg.

[ ] constCCnCneVCQneWtotaltotale

++= /2/)(1221

2

Vg


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Single Electron Transistor (SET)

Operation principle:


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Id-VdCharacteristics

Id-Vd is a function ofVg.

Coulomb Blockade Threshold Voltage Vth.

Coulomb gap


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Coulomb Gap

Large bias (Vds)Id-Vds measurements

generally probe the excitation

spectrum of the dot. The conductance

peaks are associated with the excitedelectron states in the QD, appearing

whenever such an excitation is aligned

with the Fermi level of one of the S/D

leads.

The Coulomb-blockade gap is

manifested by the flat region of theId-

Vds curve spanning Vds ~ 0V. At theedge of the gap, the large peak in

differential conductance on either side

marks the threshold voltage above

which electrons can tunnel into the dot.

( dds

CC

eVgapCoulomb

+==


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Coulomb Staircase

Unlike the Coulomb suppression of current in the neighborhood ofVds = 0 V

(Coulomb gap), the staircase is not a universal feature of the Coulomb blockade.

Rather, it is a special result of having very different tunneling rates through the

two tunneling barriers.Coulomb staircase


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Coulomb Staircase

Increasing Vds, the quasi-fermi

level on Source lead is raised by

the bias potential; initially no

current flows because electrons atthe quasi-fermi level do not yet

have enough energy to overcome

the charging energy of the QD.

Eventually, Vds reaches the point

at which an electron can tunnel

from the Source lead onto theQD. current flow and a peakin G is observed.

h f C l b S


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Physics of Coulomb Staircase

Increasing gate voltage Vgattracts more and more electrons to the island. The

discreteness of electron transfer through low-transparency barriers necessarily

makes this increase step like.

When one tunnel barrier is significantly more transmitting than the other tunnel

barrier, theI-Vbehavior of the dot can exhibit the namely Coulomb staircase

behavior, that is a stepwise curve.

What is surprising is that even such a simple device allows a reliableaddition/subtractionof single electrons to/from an island with an enormous

(and unknown) number of background electrons, of the order of one million in

typical low-temperature experiments with 100-nm-scale aluminum islands.

Id-V Characteristics


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Id VgCharacteristics

Coulomb-Blockade Oscillation inId-Vgand conductance-Vg, where conductance

d

d

V

IG =

C d P i i


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Conductance Positions

The gate-voltageposition of the conductance peaks, corresponding to charge-

degeneracy points, are determined at very low temperatures by the conditions

E(N) =E(N+1), which leads to eN+1

= (N+1/2)e2/C+ N+1.

(Recall that ,

where i represents the energy of the ith eigenstate relative to the Fermi level in

the QD and the summation is over the set of occupied states.) The spacing between conductance peaks ,

whereEa is the single-electron addition energy,

is the Coulomb charging energy andis the quantized level separation.

The gate-voltage position of the conductance peaks contains information aboutthe single-particle energies. (addition energy, charging energy, and quantized

level separation)

( )+=N

iNeC

NeNE

2)(

2

e

E

C

e

e

EV

dot

ag

+==

dotC

e

C d t P iti


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Conductance Positions

In principle, unless the single-particle levels Iare equally spaced, the

conductance peaks are not exactly periodic in Vg. This is true as long as kT


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Temperature Behavior of Conductance Peaks

At low temperatures, the heights of successive peaks in Vgvary non-monotonically and

adjacent peaks are separated by broad minima.

when kT


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p f ff

conductance

G< 0

Fine

structure

S1 Vg/ Vd=Cd/Cg S2 Vg/ Vd=Cs/Cg

Device Parameters Extracted from the Rhombus Shapes


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Device Parameters Extracted from the Rhombus Shapes

The electronic structure in the QD could be extracted from the contour plot of

the differential conductance as a function ofVgand Vd.

The ratio of the gate-dot (Cg), drain-dot (Cd), and source-dot (Cs) capacitances

can be calculated from the slope S1 and S2.

Cg: Cd: Cs = 1: S1 : S2

Then the gain modulation Cg /Ctotal, Ctotal= Cg+ Cd+ Cs The addition energyEa = Vg.

Electronic Structure extracted from I V


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Electronic Structure extracted from I-V

Addition Energy

Charging Energy

Energy level spacing

Dot diameter

Minimum tunneling resistance for single-electron


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charging

Implicit in the formulation of the Coulomb-blockade model is the condition that the

number of electrons localized in the dot island,N, is a well-defined integer. This is to say,

well defined in the classical sense, as opposed to a quantum definition which describesN

in terms of an average value , which is not necessarily an integer, and time-averagedfluctuations .

The Coulomb-blockade model requires that


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Co-tunneling

Even if the minimum resistance criterion is met and single-electron chargingeffects are manifested, small quantum fluctuations, or uncertainties, inNare notentirely ruled out. In the classical Coulomb-blockade model there is then a fixed

number of electronsNon the QD and at T= 0 the charge on the QD does notfluctuate.

However, the fact that very small quantum fluctuation inNmay be presentcorresponds to electrons momentarily tunneling onto the QD, with an energy

deficit on the scale of the classical Coulomb charging energy. Essentially, the tunneling electron resides on the QD in a virtual charge state for

a sufficiently brief interval such that the energy uncertainty of this state is largerthan its classical energy deficit, subsequently tunneling out. This process has

been referred to as co-tunneling or macroscopic quantum tunneling of charge. The rational behind is that the total charge of the system (a macroscopic

variable) undergoes a transition through a classically forbidden intermediatestate, in apparent violation of the Coulomb blockade.

Elastic co-tunneling


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Elastic co-tunneling

The tunneling of an electron into a certain energy state and the tunneling of an

electron from thesame state out of the dot. The end result of the two tunneling

events is that the state of the QD is unchanged, and as such, this is referred to as

elastic co-tunneling, which contributes a linear term to the I-V curve.

where is the average energy separation between eigenstates in the QD andE1(E

2) is the charging energy associated with adding (removing) a single electron

to (from) the dot. Note, in particular, that the resulting conductance scales

roughly as the ratio between the level spacing and the Coulomb gap U e2/C. The case of elastic co-tunneling depends, in principle, on the geometry of the

QD. This is because the electron involved has to couple to both leads; thus in a

sense it must traverse the dot in a virtual state.

VEEe

hIel

+

=

2122

21 11

8

Inelastic co-tunneling


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Inelastic co tunneling

It corresponds to an electron tunnels into a certain state in the dot and a second

electron, from a different state, tunnels out of the dot. The state of the dot is

modified, leaving an electron-hole excitation. The resulting current is nonlinear

in Vds and temperature-dependent. The case of inelastic co-tunneling gives thefollowing well-known form

( ) VeV

kTEEe

hIinel

+

+

=

22

2

21221

2

11

6

Cotunneling


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Cotunneling

The distinction is made between these two processes because their relative

contributions to the total net cotunneling current depend on the density of states

in the QD.

In metal QDs, in which the density of states is large, the elastic component of

co-tunneling is usually overwhelmed by the inelastic component.

In semiconductor QDs, in which the density of states is much smaller than in

metals, both elastic and inelastic terms can contribute to the co-tunnelingcurrent.

In practice, co-tunneling is expected to modify the classical picture of single-electron charging in the form ofexcess current in the region of the Coulomb-

blockade gap, in the case ofI-Vds measurements, or excess tunneling current

between conductance peaks in low-bias measurements.

Fabrication


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Fabrication

To apply SETs for low power ICs, (i) room-temp operation; (ii) uniformity and

(iii) compatible with the LSI processes are required.

Task:

For room temperature operation, the quantum dot diameter should be less than 10

nm, which corresponds to the total capacitance about 1 aF.

E-beam lithography:

High cost and the following etching process is not easy to control Scanning probe microscopy (SPM) to place Au atoms in nanostructure.

Only applicable in specific substrate

Metal/Superconductor SET

Semiconductor SET

Epitaxial growth quantum dots (self-assembled) or 2DEG with side gate (depletion)

E-beam + dry etching

Fabrication


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Fabrication

In addition to advanced e-beam lithography technology, matured and

controllable fabrication processes are needed to form small quantum dots (


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f g g g ( )

Si quantum dot formed by 2D oxidation


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q f y

Double-dot charge transport in SET/SHT


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g p

Ge implantation/Ge Segregation


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p g g

Application


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pp

Major application fields: Memory

Digital-data-storage

Precision Measurement

Memory >> Logic We can use SE devices only in a memory cell, whereas keep using conventional

CMOS technology in the peripheral circuitry.

Memory cell technology has continuously changed, including the emergence offlash memory technology and ferroelectric-film memory technology.

The way of storing information is rapidly changing from the old regime, relying onpapers and other analog electronic means, to the digital regime in the multimedia era.

New needs of storing information are different from the older specifications inbandwidth, storage capacity, power consumption.

Fundamental difficulty in a logic functional unit since SE devices generally havepoor current-drive capability.

Application-Data Storage System


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Quantum Information- Qubit


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Developing a quantum computer is a basic endeavor in science

and technology.

The advantage of Si-based quantum computer is

Low cost

Large scale integration

Contrast to classical bits, |0> or|1> , a quantum computer consistsof Qubits, which could be represented by a superposition of|0>or|1> , i.e.,

|0> + |1> (where 2 + 2 =1)This huge parallelism makes it possible to solve some of the most

difficult problems, such as integer factorization.

Quantum Computer Roadmap: development status


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After NTT Technical Review, June 2003

Roadmap of Quantum Computer


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status of solid-state QC


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Si-based Key Devices for QC


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Source DrainIsland

Si

Figure 1. Schematic diagram for a single electron transistor and a

coupled quantum dots.

(singlet) |0> |1>

|0> |1>

(quantum bit).


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Room-Temperature Characterization of Ge SETs


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Peak-to-valley current ratio (PVCR) of

1.92 is observed at room temperature

Clear offsets and plateaus are seen for

gate voltages corresponding to the drain

current valleys, while linear relations are

obtained for gate voltages correspondingto the drain current peaks.

Single-Electron Transistors 2003

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