Single Charge Tunneling, Coulomb Blockade Phenomena in NanostructuresNATO ASI Series, Vol. B edited by H Grabert and M H Devoret, Plenum Press New York, 1991

Single Electron TunnelingGerd SchönInstitut fur Theoretische FestkorperphysikUniversit at Karlsruhe D Karlsruhe GermanyChapter of the book onQuantum Transport and DissipationT Dittrich P Hanggi G Ingold B Kramer G Schon W ZwergerVCH Verlagrevised version October, 1997

Single-Electron Devices and Their Applications, KONSTANTIN K. LIKHAREV,PROCEEDINGS OF THE IEEE, VOL. 87, NO. 4, APRIL 1999

Single Electron Tunnelingchiidong@phys.sinica.edu.tw.http://www.phys.sinica.edu.tw/~quela/中央研究院 物理研究所 陳啟東

tunnel barrier

N1 I N2)()( xtHxt

ti Ψ=Ψ∂∂

)()(2

)( 222

xtxExm

xtE b Ψ⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−=Ψh

0)()()(2 2

22=Ψ−+Ψ

∂∂ xEExxm xbx

h

)()()( txxt tx ΨΨ=Ψ

)(xxΨ

ikxikx BeAe −+kxkx DeCe +−

)()( dxikeEAS −

0 d x

tunneling matrix element

122

)/1)(/(4)(sinh1)()(

−

⎥⎦

⎤⎢⎣

⎡−

+==bb EEEE

kdESET

for λ=2π/k

( ) ( ) ( ) ( )[ ]dEeVEfeVENEfENTAI +−+= ∫∞

∞−→121

2

21

( ) ( )[ ] ( ) ( )dEeVEfeVENEfENTAI ++−= ∫∞

∞−→ 21

2

12 1

( ) ( ) ( ) ( )[ ]

( ) ( )[ ]dEeVEfEfeR

dEeVEfEfeVENENTA

III

∫

∫∞

∞−

∞

∞−

→→

+−≈

+−+=

−=

121

2

1221

Current from N1 to N2

⎟⎠⎞⎜

⎝⎛Δ−

Δ−=

TkEE

eRI

Bexp1

1

tunnel barrier

N1 I

TunnelingRules: 1. occupied state to empty state

2. No associated energy change

eV

N2

eelectronreservoir

C1

C gC

3

C4C5C

6

C7

CΣ ≡ C1 + C2 + ••• + C7

Gap ΔV = e/CΣ

Vg

C1

C 2C

3

C4C5

C6

Vb

C7

Vb

Inject electrons via a tunnel junction

I I

Empty continuous states

Filled continuous states

ΔV = e/CΣ

A neutral isolated dot Electron BoxElectron Box

eelectronreservoir

ΔV = e/CΣ

Vb Vx= gg V

CC

Σ

An applied gate voltage can lift up island potential

particlerparticle rC επε02=

e

Vg

source drain

gate

e

C1R1

C2R2Cg

+Vb/2 +Vb/2

source I drain

Vg-Vb/2 -Vb/2

200 nm

Island

Gate

DrainSource

Junction

+Vb/2Vg

-Vb/2

Circuit and a real sample

C C C C gΣ ≡ + +1 2

E QC

Q V

eC

nC V

en

E

ii

i i

g g

n

= +

= +⎡

⎣⎢

⎤

⎦⎥ +

≡

∑ ∑2

2 22

2 Σterms independent of

C C C C

V V V

Q Q Q Q

i g

i b g

i g

=

= ±

=

1 2

1 2

2

, ,

,

, ,

e

+Vb/2Vg

source drain

gate

eVC

C1 C2Cg

-Vb/2

Single Electron TransistorSingle Electron Transistor

ggcbcbci CVVCVVCVVQQ )(/2)(/2)( 21 −+++−==∑

ΣΣ

+−≡

+=

CVCne

CVCQ

V ggggc

for C1=C2 and Vb=0

Coulomb Blockade:Coulomb Blockade: V Vb g= =0 0;

The characteristic energy scaleEc=e2/2CΣ

Ec

EcFermilevel

source island drain

occupied statesempty statesgap

Rule for tunneling: from occupied states to empty states

V E e Vb c g≥ =2 0; Effect of a bias voltage:Effect of a bias voltage:

Two possible tunneling sequences:

Ec

Ec

+eVb -e

2Ec

Ec

( )V n E e C C Vc C g g= +2 Σ

( ) ....1 1 1 n n n n→ + → → + →

n n→ + 1 n n+ →1

note

Ec

Ec

-eVb +e

2Ec

Ec

( ) ....2 1 1 n n n n→ − → → − →

n n→ −1 n n− →1

ISD

Vb

eEV Cth 2≡

slope

= 1/

(R 1+R

2)

Effect of a gate voltage:Effect of a gate voltage:

ECn V Ee

eC

V eCc

Cg

g= = ± ± = ±0

2 2, if = th en

Σ

( )V n E e C C Vc C g g= +2 Σ

Ec

Ec

+eVb

-e2Ec

Ec

Ec

n n→ + 1 n n+ →1

...1 = sequence tunneling;2 ;0 when →→→+=> nn+nCeVV ggb ~

Ec

Ec

+eVb

-e2Ec

Ec

Ec

w hen ; tunneling sequence = ... V V e C n n nb g g> = − → − → →0 2 1;~n n→ −1 n n− →1

T=4.2K

bias voltage (mV)-40 -20 0 20 40

5nA

Vg2EC

slope=1/2(R1+R2)for Vg= ne/2Cg;

Vb< 4EC/e

measuredIVb at various Vg

IV V n e Cb g g for = ± +( ) /2 1 2

IV V ne Cb g g for = ± /

Summary

source draineVg > 0eVb < 2Ec

~eVb

source drain

eVbeVg = 0eVb > 2Ec

T=4.2K

bias voltage (mV)-40 -20 0 20 40

2EC

Vg =0

-5nA

5nA

0

source draineVg > αEceVb > 0~

EC eVb

source drain

eVg > 2αEceVb > 0~

eVb

T=4.2K

bias v o ltag e (m V )-4 0 -2 0 0 2 0 4 0

2EC

Vg = e/2Cg=+0.6 V

-5nA

5nA

0

Vb≠0, Vg=0 Vb≈0, Vg→EC

at Vb > 4EC/e and Vg=e/2Cg

drain

eVb

Vg = αEceVb > 4Ec

T=4.2K

bias v o ltag e (m V )-4 0 -2 0 0 2 0 4 0

2EC

Vg = e/2Cg=+0.6 V

-5nA

5nA

0

T=4.2K

bias voltage (mV)-40 -20 0 20 40

2EC

Vg =0

Vg =- e/Cg =-1.20 V

Vg = e/Cg =+1.20 V

curr

ent (

5nA

/div

isio

n)

Vg = e/2Cg

Vg = -e/2Cg

200 nmGate

DrainSource

T = 50 mK

Sour

c e-d

r ai n

curr e

nt

-1.0 -0.5 0.0 0.5 1.0

-4

-2

0

2

4

-1 0 1 2-2

1nA

Vb ( m

V)

Vg (V)

n = Qg/e = CgVg/e

中央島

閘極

源極

接合

-Vb/2

Vg

300 nm

汲極

+Vb/2

ISD

I(Vb) vs. I(Vg)

的簡併態及 10n ≡ CgVg/eISD n=-2 -1 0 +1 +2

-2 -1 0 1 2

n=-2 -1 0 +1 +2

En/

EC

0

1

1.5

0.5

T = 50 mK

S our

ce- d

rain

cur re

n t

-1.0 -0.5 0.0 0.5 1.0

-4

-2

0

2

4

-1 0 1 2-2

1nA

Vb (m

V)

Vg (V)

n = Qg/e = CgVg/e

Coulomb Oscillation

Source

VS

DrainVDVg

Cgn=0

n=1

n=3

n=4 e/CΣ

Each oscillation corresponds to a shift in the chemical potential of the island

by an amount of e/CΣ

Current I eP G= ⋅$ $ $ , , , , , ,P P P P P P≡ − − + +L L2 1 0 1 2

∑ →++−−

Γ≡Γ

ΓΓΓΓΓ≡

i

fifn

nnn

G

, ,,,,,,~̂

21012 LL

ΔE E n E n n n Vn n f i f i bi f→ ≡ − − −( ) ( ) ( ) 2

Assumptions: 1. Sequential tunneling (no co-tunneling process)2. Global rule: Fast charge redistribution3. Fast energy relaxation

Find out Pn :the probability of having n excess electrons in the islandMaster equation

Calculate energy difference between two charge states

Calculation of single electron tunneling current

①

②

③

( )TkEE

Renn Bnn

nn

l

l

if

f

f→

→

Δ−

Δ−=Γ

→

ι

ι

exp11 rate Tunneling 2 l = S, sourceD, drain

states mequilibriuin 0 ,

1,

11 =Γ−Γ= ∑∑=

±→=

±→±DSl

nl

nnDSl

nl

nnn PP

dtdP

[ ] nn

lnn

lnn

l peI 11∑ −→+→ Γ−Γ=Current

T=4.2K

bias voltage (mV)-40 -20 0 20 40

2EC

Vg =0

Vg =- e/Cg =-1.20 V

Vg = e/Cg =+1.20 V

curr

ent (

5nA

/div

isio

n)

Vg = e/2Cg

Vg = -e/2Cg

-40 -20 0 20 40

5nA

Vb (mV)

Measurement vs. Simulation

Asymptotic IVb ( )21

2)(RR

CeVsignVI bb+

−→ Σ

Asymmetric SET Asymmetric SET (simulations)

I SD

(nA

)

VSD (mV)

R1= 50 kΩ, R2= 1 MΩ, C1=0.2fF, C2=0.15fF, Cg=16aF, EC=2.54K, T=20mK

Coulomb Staircase

appears when R1C1≠R2C2Vg=0

4EC/ee/Csmall

e/(RC)large

appears when R1≠R2

slope

= (C

g/CΣ)/R

1

slop

e =

(Cg/C

Σ)/R

2

I SD

(pA

)

Vg (mV)

VSD=e/CΣ=0.42mV

e/Cg

Stability diagram:

-1 -0.5 0.5 1Qg

-6

-4

-2

2

4

6

Ecêe

CB

-0.6 -0.4 -0.2 0.2 0.4 0.6Qg

-6

-4

-2

2

4

6

Ecêe

CB

Symmetrical Junctions

Cr = 200CgCl = 100Cg

ref “NewTrans.nb”

80

-80

-60

-40

-20

0

20

40

60

Vb (mV)25-25 -20 -15 -10 -5 0 5 10 15 20

I (nA

)IV characteristics for 1D array with C=100aF, R=20kΩ

12

3 4 56 7 8 9

# of islands

7

0123456

101 2 3 4 5 6 7 8 9

Vth

in m

V

# of islands

e/C=0.8mV

e e e e

Is=ef

e e e(a)

(b)A B C0 0 01 0 10 1 11 1 0

f

V I

Josephson oscillations2eV = hf

SET oscillationsI = ef

Quantum Hall effectV = (h/2e2)I

-Vb/2-Vb/2A B

A B(c)

+Vb/2-Vb/2

Vx

(d)

+Vb/2-Vb/2

Applications: Summary

other applications:other applications:ultra sensitive charge detector,

a small change in the island charge causes large change in current,sensitivity ~ 10-5 e/√Hz (R.J.Schoelkopf et al, 98)

memory cell,use number of charge stored in the island as an information bit,8×8 bit prototype room temperature memory cell array (K. Yano et al. 96)a non-volatile storage cell (C. D. Chen et al, 97)

Current Standard,pumping charges one by one with an accuracy rf source, I = e × ferror = 15 parts in 109, or 15 ppb (John Martinis et al, 96)

Primary thermometry,the full width at half maximum of the dI/dV⏐V~0 = αT, α = 10.878.. ×(kB/e)Good for KBT >> EC, experiment:T = 0.1~8 K, (J.P.Pekola et al, 94)

logic gate element,forming AND, OR, NAND gates

Computers: Making Single Electrons Compute, Science, v275, p. 303 (1997)

an universal constant

Coulomb blockade

Charging energy (e2/2C):Electrostatic energy associated with charging/discharging an isolated object

Criteria Criteria (for a well defined charge number) :

⇒ sm all o r lo w C T

R Rh

eK≥ ≡ ≈2 26 kΩ

electrical paths to charge/discharge the object

C = total capacitance seen from the object

R = total resistance seen from the object

1. to surmount thermal fluctuations

2. to surmount quantum fluctuations

(tunneling transport -- not diffusion)

(two resistance in parallel)

isolated object

( )eC

R C h

Et

2

2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ≥ ⇒

ΔΔ123

123

2eC

k TB2

>>

Material:any conducting materials,

including metal, semiconductor,

conducting polymer, carbon nanotube, ...

Structure:any structures with isolated

objects accessible through tunnel junctions

Single Electron Transistor (SET)

The simplest structure:

T/EC=0(dashed steps), 0.02, 0.05, 0.1, 0.2, 0.4 and 1(nearly linear)

Controlling and Detection of Charge number in a Box

∑

∑∞

−∞=

∞

−∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

n B

n B

TknETknEn

n)(exp

)(exp

At 20mK, this discrepancy is 40mK and is only partially understood.

charge noise 10-4e/ Hz at 1Hz

NATO, Single Charge Tunneling, p121

( )enQCC

CQS

−+

=~

( ))(2

~)(

2

CCQnenE

S +−

=

-0.5 0.5 1 1.5Q1 = C1 U1

-0.5

0.5

1

1.5

Q2 = C2 U2

v=0,2�

00 10

01

Control Gate 2

Vg1Vg2

Vgc1 Vgc1I2I1

NATO, Single Charge Tunneling, p128

Single Electron Pump

ref “Q8Final&.nb”

NN--junctionjunction Electron PumpElectron Pump

time

CgVg0

e

Tunneling from I2 to I3 via J3

e

Vg1 Vg2 Vg3 Vg4 I

J1 J2 J3 J4 J5

I1 I2 I3 I4

before tunneling: Q3 = - e/2, the rest = (+e/2)/4after tunneling: Q3 = +e/2, the rest = (- e/2)/4

Cg2Vg2 = +e - Q(t)Cg3Vg3 = Q(t)

Cg2Vg2 + Cg3Vg3 = e

optimal operation condition

QC = e/2

Example:

Ref: H. Dalsgaard Jensen and John M. Martinis PRB 46, 13407 (1992)

Accuracy of electron counting using a 7-junction electron pumpMark W. Keller and John M. MartinisNational Institute of Standards and Technology, Boulder, Colorado 80303Neil M. ZimmermanNational Institute of Standards and Technology, Gaithersburg, Maryland 20899Andrew H. SteinbachNational Institute of Standards and Technology, Boulder, Colorado 80303

Appl. Phys. Lett. 69 (12), 1804 (1996)

1) Current standard, (I=ef)error rate 15/109 (15 ppb)

2) Capacitance standard (C=e/ΔV)3) hold time = 600 sec.

pump ±e rate = 5.05MHz

static

C=e/ΔV=e/7.5μV~21fF

pump ±e slowly

Average time per error ~ 13 sec.

pump ±e fast

12

222

22

24)!1()!12(22),(

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=Γ

NN

T

KN

c CeeV

RR

NNN

CeeVN

ππh

Co-tunneling

NATO, Single Charge Tunneling, p115

[ ] [ ][ ]Ω×=Γ−

kV

RVeVHz

Tc

μ4

73105.2),4(

For N=4

NATO, Single Charge Tunneling, p218

P(E)

E/ECI/R

TEC

V/EC

AV

RT, C RS=γRKΩ≈≡ k

heRK 26

2

CeEC 2

2

≡

Effect of shunt resistor on a small tunnel junction:

P(E) ---- Probability for the tunneling electron to emit energy E to the environment.

RS= 0.01RK

0.1RK RK= 0

.01R K

RS= RK=0.1

R K

R S= 0

R S= ∞

Lower shunt resistance → lower junction resistanceH. Grabert and M. H. Devoret, ‘Single Charge Tunneling’, (Plenum Press, New York, 1992)

EC = e2/2C

EK = level spacing = 1/D(EF)V

n = 1022 cm-3m = meεr=410% tunnel barrier=2nm

Additive energy for an oxideAdditive energy for an oxide--encapsulated nanoparticleencapsulated nanoparticle

Likharev review article, Fig. 2

EK

EC

RT

2nm 19nm

Single-Electron Devices and Their Applications, KONSTANTIN K. LIKHAREV,PROCEEDINGS OF THE IEEE, VOL. 87, NO. 4, APRIL 1999

eVC

C2

Σ

eVC

C1

Σ

J1 J2

eVC

C2

Σ EC

ECeV

C

C1

Σ

J1 J2

eV eVafter tunneling

EC

Electron transfer through a quantum dot with charging energy EC

J1 J2

What makes electrons flow?

Fermi function:

biasing: μ1-μ2=qVD

[ ] )()(exp11)( 10

11 μμ

−=−+

≡ EfTkE

EfB

[ ] )()(exp11)( 20

22 μμ

−=−+

≡ EfTkE

EfB

Coupling strength for J1 and J2 = γ1 and γ2

( )pfeI −⎟⎠⎞

⎜⎝⎛= )( 1111 εγh

( )pfeI −⎟⎠⎞

⎜⎝⎛= )( 2222 εγh

At steady state, I1=I2

p = average number of electron in the QD 10 ≤≤ p

21

2211

γγγγ

++

=ffp

)]()([ 2121

2121 εεγγ

γγ ffeIII −+

=−==⇒h

Handbook of Nanoscience, Engineering, and TechnologySection: 12.2.2 Current flow as a balancing actBy Magnus Paulsson, Ferdows Zahid and Supriyo Datta, Edited by William A. Goddard,III et al. CRC Press, 2003

Quantum transport: atom to transistorSec. 1.2 What makes electrons flow

Filling electrons to an artificial atomn = 0 1 2 3 4 5

Discrete energy levels in a dot

Electronic Spins in nanoElectronic Spins in nano--particles: Zeeman Effectparticles: Zeeman Effect

levels probedby tunneling

filled levels

Even number of electrons

Ene

rgy

Magnetic field

μ

levels probedby tunneling

filled levels

Odd number of electrons

Ene

rgy

Magnetic field

μHz=σz(go/2) μΒΗσz=±1/2

Hamiltonian:

von Delft et al. PRL, 77, 3189 (96)

Kondo temperature( )

⎟⎠⎞

⎜⎝⎛

Γ+

Γ=U

UexpUT 00Kεπε

Duncan Haldane1978

εo

U

EF

εo= impurity level U = charging energy

sour

ce

drai

nΓ

TK TK

QD DOS electrode DOS

metalfilm

process time ~ h/⎪εo⎪Γ = level broadening

Provide a new conduction channel in Coulomb blockade regime

in Quantum dots with odd number of electronsSingle channel resonance

Kondo resonance in Quantum Dot systems