single electron tunnelingspin/course/98f/091217.pdfsingle charge tunneling, coulomb blockade...
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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
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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
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