Part 2: Nano-electronic devices Chapter 1: Single electronics 1.1 Charging

Hervé Courtois Institut Néel

CNRS, Université Joseph Fourier and Grenoble INP http://neel.cnrs.fr/spip.php?article804&lang=en

herve.courtois@neel.cnrs.fr

Quantum Nano-Electronics Part 2:

Nano-electronic devices

Chapter 1:

Single electronics 1.1 Charging

A note on capacitance

€

Q_i=

C

^ijVj

∑

j

A given conductor is at a given potential.

Charges and potentials are related through a capacitance matrix:

Energy of the system:

€

V_i=

C

ij−1Q_j

∑

j

€

C

^{ij =}

C₀₁+C₁₂+C₁₃ −C₁₂ −C₁₃

−C₂₁ C₀₂+C₂₁+C₂₃ −C₂₃

−C₂₁ −C₃₂ C₀₃+C₃₁+C₃₃

#

$

%

%

%

&

' ( ( (

1 2

3

€

C

^ij=

C

^{ji €}

∞

€

∞

€

∞

€

E=1

2

C

^ijViVj

∑

i,j

€

C₀₃

€

C₀₂

€

C₀₁

€

C₁₃

€

C₂₃

€

C₁₂

= zero potential

Practical capacitances

€

C=ε₀ε_rS Parallel plates: d

Concentric spheres:

Ex: 1 cm radius gives 0.1 pF, 1 µm gives 0.1 fF (10^-15), 10 nm gives 1 aF (10^-18)

€

C=ε₀ε_r4πr₁r₂

r₂−r₁≈ ε₀ε_r4πr₁ if r₁<<r₂

A large electrostatic energy change

Consider the electron box configuration:

The electro-static energy changes when one electron is added to the grain.

Total capacitance C_Σ = C + Cg

One needs E_c > k_BT in order to observe single-electron effects.

With 1 aF, one obtains E_c = 80 meV > k_BT at ambiant temperature (25 meV).

With 1 pF, one obtains E_c = 80 µeV > k_BT at 0.3 K.

€

E_c= e² 2C_Σ

Symbol for a tunnel junction

R_T, C C_g

C02

U

A significant tunnel junction resistance

Consider the electron box configuration:

The lifetime of a charge state is the discharge time for the capacitor:

Width of discrete levels due to charge quantization in the grain:

For charge discreteness to be observable, one needs:

€

τ=R_TC

€

δ= τ= 

R_TC

€

δ<E_c ≈ e²

2C

€

R_T>R_K = 

e²=25.8 kΩ

Symbol for a tunnel junction

R_T, C C_g

C02

U

1.2 The single electron box

Electrostatic energy in an electron box

Consider the electron box configuration:

Capacitance matrix:

Electrostatic energy of the box:

the box, potential V₁ potential V₂ = U

€

C_Σ=C+C_g Q=C_ΣV₁ Q_g=C_gU

€

C

⁼

C+C₀₂ −C −C₀₂

−C C+C_g −C_g

−C₀₂ −C_g C_g+C₀₂

#

$

%

% %

&

' ( ( (

Symbol for a tunnel junction

R_T, C C_g

C₀₂ U

€

E=1

2

C

^ijViVj

∑

i,j ⁼¹₂^$ _% &

(

^C+Cg

)

^{V12−2CgUV1+}

(

^Cg+C02

)

^U2' ( )

potential V₀ = 0

We define

Electrostatic energy in an electron box

Consider the electron box configuration:

Substitute for charges, complete the square:

+ terms independent of Q related to work done by the source

€

C_Σ=C+C_g Q=C_ΣV₁ Q_g=C_gU

the box, potential V₁ potential V2 = U Symbol for a

tunnel junction

R_T, C C_g

C02

U

E=

Q−Q_g

( )

²

2C_Σ

€

E=1

2^# _$ %

(

C+C_g

)

^{V12−2CgUV1+}

(

^Cg+C02

)

^U2&

' ( = 1 2

Q²

C_Σ−2QQ_g C_Σ +Q_g²

C_Σ−Q_g²

C_Σ+C_gU²+C₀₂U²

#

$

% %

&

' ( ( We define

potential V₀ = 0

Energy parabolas

Electrostatic energy as a function of gate charge for various Q = - ne Gate charge is a continuous variable, Q a quantized one.

Q_g E

Q=0 Q=1 Q=-1

Q=-2 Q=2

Charge in the fundamental state

Charge state defined by the minimum energy: the Coulomb staircase.

Island charge is a step function of the gate voltage.

Valid at T = 0.

<n>

Q_g

Effect of temperature

At thermal equilibrium, Boltzmann distribution:

depends on the parameter

€

n =

n exp −E n

( )

k_BT

#

$

% &

' (

n=−∞

∞

∑

exp −E n

( )

k_BT

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$

%

&

' (

n=−∞

∞

∑

€

k_BT

e² 2C_Σ

Charge in the fundamental state

The staircase is smeared by temperature.

<n>

Q_g

€

θ= kT e² C_Σ

1.4 Practical implementation

Low temperatures

Temperature of 4.2 K reached with a liquid He bath in equilibrium with its vapor, 1 K with pumping, 0.25 K same thing with pure He3.

Temperature down to about 0.02 K = 20 mK with mixture of He3 and He4: dilution refrigerator.

Angle evaporation technique

Bilayer of resist with the bottom layer more sensitive, bridges can be realized.

Directive evaporation under different angles:

e-

θ θ Overlap = junction

An oxidation can be realized between the two depositions: tunnel junction

Si

Bottom resist

A typical sample

Al oxide preferred: reliable, reproducible, despite amorphous nature.

Complex devices can be build.

A fact of nature

Device connected through rather long wires: 1-2 meters if low T.

At high frequency, wires must be regarded as transmission lines.

Impedance close to the vacuum characteristic impedance Z₀.

Fact of Nature: free space impedance much lower than Quantum resistance.

Fine structure constant:

The free space effectively shunts the tunnel junction.

Need to have a highly resistive element close to the device: a resistor or another tunnel junction. Very close: distance less than photon wavelength.

€

Z_L

R_K = e² 2πhcε₀ ≈α

π

€

α= 1 137.036

€

Z₀= 1

ε₀c ≈376Ω

…

VOLUME70,NUMBER7 PHYSICAL REVIEW LETTERS 15FEBRUARY1993

2 0.5^—0

0-

-0.5

I

-0.3 0 0.3

2h,/e

CO

ic

I

ic 0—

.~ I

pc

I I I I I I I I I

-0.6 -0.3 0 0.3 0.6 ^{I I}

-2

I

3

+~e

I I I

-4 -3 0 1 2 4 5

V (mV)

FIG. 3. 1(V)curvesfor theSNSelectrometer atT=25mK, andzero magnetic field, for three values ofthe gate voltageUo corresponding to maximum, intermediate, and minimum gap.

The minimum gap corresponds to thebare superconducting gap 2hoftwoNSjunctions inseries. The dot indicates the optimal bias point for maximum sensitivity. Inset: 1(V)^curvefor a sin- gleSNjunction under the same conditions.

C,Uie

FIG. 4.Variations ofthe average value nofthe number of extra electrons inthe box as a function ofthe polarization C, U/e, atT=25mK.TraceiV:normal island. Trace5:super- conducting island. For clarity, traceShas been oAset vertically by4 units.

calculations of the box parameters which gave C~

=0.2+0.05fF, C,=25+5aF,andC,=11~2aF.The experiments were done with the sample mounted in a shielded copper box thermally anchored to the mixing chamber of a dilution refrigerator. All voltage and current lineswere carefully filtered [14].When neces- sary, thesample was put in its normal statebya1T magnetic fieldproduced bya superconducting coil.

To perform the measurements ofn vsU the biasand gate voltages V and Uoof the electrometer werefirstad- justed to maximize r)I/BUo(dotinFig.3).The electrom- eter current Iwasthenrecorded as a function of U. The resulting sawtooth signal isa measurement, apartfroma gainfactor,of the second term of Eq.(I).We obtained n by adding tothis sawtooth signal a linear term whose coemcient wasadjusted tonullout theslopeof the teeth.

InFig. 4weshowthe measured equilibrium value nas a function of the polarization C,U/efor thesample inboth the normal and the superconducting state, at 20 mK.

Theeven-odd symmetry of the stepsinthenormalstateis clearly broken inthe superconducting state. Note that themiddle of thesteps inthe superconducting state coin- cideswiththemiddle of the stepsinthenormal state, as predicted bytheory [seeFigs.2(b)and2(d)]inthecase D(T)&E,. Our previous experiments ona boxwith an SSjunction never showed any even-odd asymmetry [5].

Webelieve that this was due to the presence of afew long-lived, out-of-equilibrium quasiparticles which inthe present experiment are "purged" bythe normal metal lead.

Becauseof theunavoidable electrostatic cross talk be- tween the U voltage andtheelectrometer island, which was only partially corrected forinour setup, thegainof theelectrometer depends onthe U voltage. This leadsto thenoticeable step height variations as U departs from

zero. Nevertheless, these vertical scale distortions donot aAect the conclusions wedraw fromourdata, which are basedonly onthelength of the stepsalongthe horizontal axis. The scaling factorused for this axis corresponds to C,=21~0.⁵aF,ingood agreement withour numerical estimates. Whenthe temperature wasincreased the steps became gradually rounded (datanotshown). Fromafit of the temperature dependence of the datainthenormal stateusing Eq.(I)weobtained a direct measurement of C~=0.20~0.05fF,alsoingood agreement withournu- merical estimates.

%'e_have measured theodd-even steplength ratio_{p as} a function of temperature, thereby obtaining D(T)/E,. The experimental results are shown inFig. 5together with thetheoretical predictions inthecaseof a continu- ous BCSdensity of states (dashed line). Since N~ ^is known from thesample dimensions, theonly adjustable parameters are Cq'=0.19fFand d,"'/e=195pV. The parameter C~'is intheerror range of C~whilethe uncer- tainty range forh."'_isadjacent to the error range of6de- duced from the electrometer 1(V). Apart from this minor discrepancy which may bedueto thefact that the island, contrary to theSleadsof the electrometer, isnot covered byanormal layer, there isgood agreement be- tween theory and experiment for temperatures higher than 50 mK. At lower temperatures, thedata deviate significantly fromtheory, inamanner which could be ex- plained by afailureof the box tofollowthetemperature ofthethermometer. However, we find thisexplanation unlikely. Inaprevious run onaNN boxwithparameters adapted to calibration purposes, the staircase sharpness precisely followed thetemperature down to35 mK, A more likelyexplanation isthat thedensity of states of the island may notbe astrictly smooth BCSone. Toillus- trate this point, we show inFig.5a complete fitof the data (fullline)using aminimal model: Inaddition to the 996

VOLUME70,NUMBER7 PHYSICAL REVIEW LETTERS 15FEBRUARY1993

E

n=1 2

a)

which p(e)isnonzero. Inthis limit, p(T)canbe evalu- ated analytically for mathematically simple p. Ifwe assume a continuous BCS density of states, p(T)

=N,a(T)e ^~where

&n&

5 4 3 2.. 1 -~

E

n=11 2 5 6

C,U/e

b)

6 C,U/e

E,^-

c)

&f1&

5 2--3 1--

III

I I II

II

I I

IIIIIIII

I III

I I

I I

I I

I I

I I

I

III I

I I

I I

I I

C,U/e

C,U/e

FIG. 2.Ground-state energy of the boxinthe(a)normal and(c)superconducting states as a function ofthe polarization C, U/e, for several valuesoftheexcess number nofelectrons in the island. E,istheelectrostatic energyofoneexcess electron on the island for U=0. In an ideal superconductor, the minimum energy for oddnis5above theminimum energy for evenn.The dots correspond tolevelcrossings where single elec- tron tunneling ispossible. Equilibrium value 1nl vsC, U/eis shown inthe(b)normal and (d) superconducting states, at T=0.

(n)=^{C U}_e +CPe^Cz t)U,⁸ ,+ne ^{pE,(n—c,v—/e)2'} where P=I/k~T and where Z„^isthe partition function ofthe island withnexcess electrons. WenowfollowRef.

[4]:Weassume Fermistatistics for the quasiparticle ex- citations ofthisisolated system and weset the parity of thenumber of quasiparticles equal to the parity ofn.

We get Z„=[Z++(^—1)"Z—]/2, with Z+-=+q[1

~exp(—_Pe&)],where q denotes ageneric quasiparticle statewith energy ^t.~.

At temperatures such that k~T&&E„the (n)vs U staircase isjustslightly rounded. Thelength of thesteps is nowdefined fromthevaluesof Uwhere (n)isahalfin- tegerandDointheexpression of theodd-even steplength ratio isnowreplaced byD(T)=V~—_{Po,the difference}

between thefreeenergies 7„=^—k~TlnZ„ofthe island with an odd and an evennumber of electrons [9].^Intro-

ducing thetransform p(T)=_{fo p(e)}in[coth(Pe/2)]de/2 ofp(e),thedensity of quasiparticle states, onecan ex- pressD(T)= —k8Tln[tanhp(T)]. Wenow suppose that exp(—_e;„/kaT)«I,where e;„^isthe lowest energy for

Neff(T)=No(2xka T/5) +O[(T/6) ] isthe eAective number of quasiparticle states available forexcitation [10]and where No=p~N~/).,p~ being the normal density of states at the Fermi energy peratom and N~ the number of atoms inthe island. Because lnN,~ depends weakly onthesample parameters and on temperature, D(T)isapproximately given at tempera- tures such that Noexp( 5/kgT—)«1byh(1—_T/Tp),

with Tp=k/(kii lnNo) intherange 200-300mKfor real- isticAlislands. Moregenerally, if there isinside the gap discrete quasiparticle states with energies eq.and de- generacies g~„ ^they each contribute to p(T) by g~,exp(—_Pe~,).Their effectisto reduce D(T),which is given inthelimit T=0byD(T) =eq,—_{kBTIngz,,where}

qo isthelowest discrete quasiparticle state. Finally, ^we must point out that the 2e-periodic behavior of the SN boxissimilar to the 2e periodicity which has beenob- served for thecurrent through theSSS[4,11]andNSN [12]Coulomb blockade electrometers asa function ofthe charge induced onthegate. However, note that when D(T)&E„the box experiment, in contrast with the transport experiments on Coulomb blockade electrome- ters,givesaccess totheratioD(T)/E,andnot simply to the temperature atwhich itvanishes.

The sample was fabricated using e-beam lithography anddouble-angle e-beam evaporation through asuspend- ed mask [13].Firstwedeposited a30nm thickalumi- num filmtoform the superconducting island of the box, withlateral dimensions 2.2pm&0.1pm, aswellas the leadsof the electrometer. Thisfirstlayerwas thenoxi- dized in300 Paofoxygen for 15min atroomtempera- ture. A50nm thick layer of Cualloyed with 3%by weight ofAlwasthendeposited toformthenormal lead connected totheboxandtheisland of the electrometer.

Thetwo nominally identical junctions of the electrometer had anareaof—8x10 pm, and were much larger thantheboxjunction. Thesuspended mask was designed sothat there was nooverlap ofthe Al island of the box withitsCu-Al copy,which isinherent to thedoubleevap- oration technique. The current-voltage curve (inset of Fig.3)ofa singlejunction fabricated withthesame tech- nique showed asharp current riseat6/e=180~10 pV, with thesquare-root voltage dependence characteristic of NS junctions. Figure 3shows acurrent-voltage charac- teristicof the electrometer: When the gate charge^isad- justed soastosuppress Coulomb blockade forpositive voltage, thesharp current riseat2A/e=360+10 pVin- dicates that the electrometer consists indeed oftwo NS junctions inseries. Detailedanalysis of these1(V)curves yielded thecapacitance parameters of the electrometer.

They served ascalibrations for numerical electrostatic 995

VOLUME70,NUMBER 7 PHYSICAL REVIEW LETTERS 15FEBRUARY1993

Measurement ofthe Even-Odd Free-Energy Di8'erence ofanIsolated Superconductor P. Lafarge,P.soyez, D. Esteve, C. Urbina; andM. H.Devoret

Service de Physique deI'EtatCondense,Commissariat a I'Energie Atomique Sac!ay, 9II9l,Gifsur-Yv-ette, France (Received 16 November 1992)

We have measured the diAerence between thefree energiesofanisolated superconducting electrode withoddand evennumber of electronsusinga Coulomb blockadeelectrometer. The decreaseof this en- ergy diAerence withincreasing temperature is ingood agreement withtheoretical predictions assuming a BCSdensity of quasiparticle states, except at the lowest temperatures where the results indicate the presenceofanextra energylevelinsidethe gap.

PACSnumbers: 74.50.+r, 73.40.Rw,74.25.Bt

The key concept of the Bardeen-Cooper-SchrieAer (BCS)theory ofsuperconductivity [I]isthe pairing of electrons. A surprising feature of thetheory appears whenoneconsiders amacroscopic piece of superconduct- ing metal witha fixed number of electrons N.IfNis even, allthe electrons cancondense intheground state.

IfNisodd, however, oneelectron should remain asa quasiparticle excitation. In principle, ifone wouldmea- suretheenergy required toaddoneelectronto thesuper- conductor, thereshould beadiff'erence betweenthe cases ofeven and oddN. Thisfundamental even-odd asym- metry, which might vanish duetosample imperfections [2],does not manifest itselfinconventional experiments on superconductors because these experiments areonly sensitive toa finite fraction of quasiparticles. In this Letter, we report a new experiment based on single- electrontunneling [3]withwhich wemeasured theeven- oddfreeenergy diAerence introduced by Tuominen etal.

[4].

Consider asuperconducting-normal (SN)tunnel junc- tion inseries with a voltagesource Uandacapacitor C, (see Fig. I),abasic Coulomb blockade circuit whose normal-normal junction version has beennicknamed the electron "box" [5,6]. The superconducting electrode which iscommon toboththe junction andthecapacitor is surrounded everywhere by insulating material. Whenthe junction tunnel resistance R,issuchthat R,))Rz=h/e,

the number nof excess electrons on this"island" isa good quantum number [3,7].The n-dependent part of theground-state energyof the circuit,including thework done by thesource U, is given byE„=E,(n—_{C, U/e)} +8„,whereE, =e/2C& istheelectrostaticenergyofone excess electron onthe island, C~ the total capacitance of theisland, and8„isthenonelectrostatic part of the ener- gyof theisland. Fora normal island 6'„=0[Fig.2(a)], whereas fora superconducting island, one hasD„=Dop„

where Do istheenergy diAerence betweentheodd-n and even-n island ground states, andp„=n^mod2[Fig.2(c)].

The BCStheory yields Do=6,where 4, isthesupercon- ducting gapof theisland. In equilibrium at zero temper- ature,nwillbe determined bythelowestE„andisthere- foregiven by astaircase function ofU[Figs.2(b)and 2(d)]. Inthe normal case,the stepsareofequal size,

whereas in the superconducting caseeven-n steps are longer than odd-n steps. ForDo&E„the odd-n steps disappear, while for Do~F„theratio p^between the length ofthe odd- and even-n steps isrelated toDo through Do/E,=(1—_{p)/(I+p). Thus,from ameasure-}

ment ofthe equilibrium value ofnas a function ofU, which canbe done by weakly coupling theisland toa Coulomb blockadeelectrometer [5,6,8],as shown inFig.

1, one can in principle infer the valueof Do.

Inpractice, the measurements areperformed atfinite temperature andthecurrent intheelectrometer isdirect- lyrelated ton,thetemporal average ofnwhich we sup- pose equal to (n),the thermal ensemble average ofn.

Theabove analysis must be refined totakeintoaccount the thermal population of^allthepossible statesof the cir- cuit. These states are characterized not only bythenum- bernof excess electrons intheisland, but also by the fillingfactorsof thevarious quasiparticle statesof theis- land. Onefindsthat theaverage valueofnis given by

(ug BOX

Cs

c cc

III

ELECTROMETER

c, cg

I11IIIIIIIIII

II I II I

I

II I

IIIIIIIIIIIIIIIIIII

I I II

IIIIIIIIIIII1

FIG.1. Circuit diagram of the experiment. The rectangular symbols represent SNtunnel junctions. The V-shaped marks denote superconducting electrodes. Thesymbol ndenotes the number of electrons intheisland of the box (marked byafull dot). The variations ofitsaverage nwith the voltage U are detected bymonitoring thecurrent Ithrough theSNSelec- trometer which iscoupledto the box through the capacitor C,. The bias voltage Vandthe gate voltage Uosettheworking pointof the electrometer.

994 P. Lafarge, P. Joyez, D. Estève,

C. Urbina, and M. H. Devoret, PRL 70, 994 (1993).

Electrometer = SET (see what follows)

VOLUME70,NUMBER7 PHYSICAL REVIEW LETTERS 15FEBRUARY1993

2 0.5^—0

0- -0.5

I

-0.3 0 0.3

2h,/e

CO

ic

I

ic 0—

.~ I

pc

I I I I I I I I I

-0.6 -0.3 0 0.3 0.6 ^{I I}

-2

I

3

+~e

I I I

-4 -3 0 1 2 4 5

V (mV)

FIG. 3. 1(V)curvesfor theSNSelectrometer atT =25mK, andzero magnetic field, for three valuesof the gate voltageUo corresponding to maximum, intermediate, andminimum gap.

Theminimum gap corresponds to thebare superconducting gap 2hof two NS junctionsinseries. The dot indicates the optimal bias point for maximum sensitivity. Inset:1(V)curvefor a sin- gle SN junction under the same conditions.

C,Uie

FIG. 4.Variations of the average valuenof the number of extra electrons inthe box as a function of the polarization C,U/e,atT =25mK.TraceiV:normal island. Trace5:super- conducting island. For clarity, traceShas been oAset vertically by4 units.

calculations of the box parameters which gave C~

=0.2+0.05fF, C,=25+5aF,andC,=11~2aF.The experiments were done with thesample mounted in a shielded copper boxthermally anchored to the mixing chamber ofa dilution refrigerator. All voltage and current lines were carefully filtered [14].When neces- sary, thesample was put in its normal stateby a ¹T magnetic fieldproduced byasuperconducting coil.

Toperform themeasurements ofn vsUthebias and gatevoltages V and Uoof the electrometer werefirstad- justed tomaximize r)I/BUo(dotinFig.3).The electrom- eter current Iwasthen recorded asafunction ofU.The resulting sawtooth signal is a measurement, apartfrom a gainfactor,of thesecondtermof Eq.(I).We obtained n by adding tothis sawtooth signal a linear term whose coemcient wasadjusted tonullout theslopeof the teeth.

InFig. 4weshowthemeasured equilibrium valuenas a function of the polarization C,U/eforthe sample inboth the normal and the superconducting state, at 20 mK.

Theeven-odd symmetry of thesteps inthenormal stateis clearly broken inthesuperconducting state. Note that the middle of thesteps inthesuperconducting state coin- cideswiththe middle ofthe steps inthenormal state, as predicted by theory [see Figs.2(b)and2(d)]inthecase D(T)&E,. Our previous experiments onaboxwith an SSjunction never showed any even-odd asymmetry [5].

Webelieve that this was dueto the presence ofa few long-lived, out-of-equilibrium quasiparticles which inthe present experiment are "purged" _bythe normal metal lead.

Becauseof theunavoidable electrostatic cross talk be- tween theU voltage andtheelectrometer island, which was only partially corrected forinour setup, the gainof theelectrometer depends ontheUvoltage. Thisleadsto thenoticeable stepheight variations as Udeparts from

zero. Nevertheless, thesevertical scale distortions do not aAectthe conclusions wedraw fromourdata,which are based only on the length of thesteps alongthehorizontal axis. Thescalingfactor usedforthisaxis corresponds to C,=21~0.⁵aF,ingoodagreement withournumerical estimates. Whenthe temperature wasincreased thesteps becamegradually rounded (data notshown). Fromafit ofthetemperature dependence of the data inthenormal stateusing Eq.(I)weobtained adirect measurement of C~=0.20~0.05fF,also in goodagreement withour nu- mericalestimates.

%'e_havemeasured the odd-even step length ratio_{p as} a function of temperature, thereby obtaining D(T)/E,. The experimental results areshown inFig.5together with thetheoretical predictions inthecaseof a continu- ous BCSdensity of states (dashed line). Since N~ is known from thesample dimensions, the only adjustable parameters are Cq'=0.19fFandd,"'/e=195pV. The parameter C~'is intheerrorrangeof C~whiletheuncer- tainty rangeforh."'isadjacent to the errorrangeof6de- duced from the electrometer 1(V). Apart from this minor discrepancy which maybedueto thefact that the island, contrary to theSleadsoftheelectrometer, isnot covered byanormal layer, there is goodagreement be- tween theory and experiment for temperatures higher than 50 mK. At lower temperatures, thedata deviate significantly fromtheory, ina manner which couldbeex- plained by afailure of the box tofollow thetemperature ofthe thermometer. However, we find thisexplanation unlikely. Inaprevious run on aNN boxwithparameters adapted to calibration purposes, thestaircase sharpness precisely followed thetemperature down to35 mK, A more likely explanation isthat thedensity of states ofthe island may not be astrictly smooth BCSone. Toillus- trate this point, we show inFig.5a complete fitofthe data(fullline) using a minimal model: Inaddition to the 996

A single electron box

With a superconducting island, extra energy ∆ for odd number of e-.

1.4 The SET and the turnstile

The Single Electron Transistor (SET)

Consider the configuration with two tunnel junctions:

The electro-static energy is identical to the box case, Q = - ne:

where C_Σ = 2C + C_g

The current through the turnstile depends on the gate voltage:

Single Electron Transistor = SET and also on temperature.

C C

C_g

V/2 -V/2

V_g

a b

E_c

( )

n ⁼

(

^−ne−Qg

)

²

2C_Σ

The Single Electron Transistor (SET)

Consider the configuration with two tunnel junctions:

Energy changes when one electron tunnels from left to central island: n-1->n, work done by the source eV/2 taken into account:

Similar expression for tunneling from center to right: n->n-1.

C C

C_g

V/2 -V/2

V_g

a b

ΔE_a=eV

2 +E_c

( )

n ^−Ec

(

ⁿ⁻¹

)

^=eV₂ ^{+e C#}$ ^gVg+

(

^{n−1 2}

)

^e%

&

C_Σ

ΔE_b =eV

2 +E_c

(

n−1

)

^−Ec

( )

^{n =eV}₂ ^{−e C#}$ ^gVg+

(

^{n−1 2}

)

^e%

&

C_Σ

∆E_a ∆E_b

Opposite transitions: - ∆E_a,b e- e-