gle Electron Tunneling Artificial Atoms een an axiom of mine tle things are infinitely the most impo Conan Doyle Danny Porath 003 (K
ased on works by. ian Shönenberger - present oph Wasshuber Ph.D. Th ntin Likharev IEEE 87, 60 Meirave and E.B. Fo ductor Science and Technology, ouwenhoven Reviews (see si Millo - presentation
ooks and Internet Sites ransport in Mesoscopic Systems ge Tunneling, H. Grabert & M ction to Mesoscopic Physics, Y.iue.tuwien.ac.at/publications/PhD%0Theses/w p://vortex.tn.tudelft.nl/grkouwen/reviewpub.htm ww.aip.org/web/aiphome/pt/vol-54/iss-5/archiv ton.edu/~chouweb/newproject/research/qdt/si http://qt.tn.tudelft.nl/publi/papers.html www.unibas.ch/phys-meso...
r: Homework 7 ms, Mark Kastner, Physics Today, Janu r: ts, L. Kouwenhoven and C. Marcus, Ph 35. r: he Coulomb staircase in a two-junction tron charge, a and M. Tinkham, Physical Review B,
Outline SET: sic concept thodox theory eneral treatment uble barrier tunnel junction ulomb staircase gle electron transistor ability diagram
ices (SET Devices), based on the single electrons between smal ve already enabled several importa. Several other applications of an vices in unique scientific instrume eem quite feasible. On the othe silicon transistors being replaced ices in integrated digital circuits nd remains uncertain. Neverthele ment does not happen, single-ele play an important role by shedding size limitations of new electro recent research in this field ha ng by-product ideas which may ss-memory and digital-d
Law For many Elec - - - - + + + + V=IR I V
ic Concept of Single Ele e After of a single uncompensated electron c field E which may prevent the add
The Charging Energy g energy: E C = e C rticle or the quantum dot (QD) size i ron De-Broglie wavelength, energy tantial on addition energy: a = E + k E C E k - e Kinetic E k = g(ef )V V- Volume g(e F ) DOS @
ulomb Blockade and Sta nnel TJ or tion Capacitor CS QD V thresh CB blockade b staircase ts can be observed if: e h K T > B RT
e Motion and Quantiza
Various Capacitors
The Charging Energy round 3D ball with a free, degenera n=0 cm -3, electron effective mass m matrix (dielectric constant ε=4), with 0
The Charging Energy => Molecules SET effects at RT QD size -0 nm (=> E a ~ ev - 50 mev hermally induced tunneling E a s ~0-00 K B T quires: E a ~ ev => ~ nm e range there is a strong depe ape.
ergy Change Upon e Ad h 3 ( 3π n) n- carrier con
rgy Change Upon e Add Si and Al
Realizations of Quantu tor DEG:
dox Theory Major Ass Spectrum - the electr n inside the conductors is ignor ergy spectrum is treated as eaking this assumption is va but it frequently gives an of observations as soon as E k << us tunneling - the time τ t hrough the barrier is assum mall in comparison with other he interval between neighbori is assumption is valid for tun gle-electron devices of practic
dox Theory Major Ass quantum processes consisting s tunneling events ("co-tunn is assumption is valid if the r tunnel barriers of the syste the quantum unit of resistance R T >>R Q, R Q =h/4e 6.5 k and shape are ignored 0D Q us charge redistribution after ctra at the leads are continues limitations, the orthodox theo D s and most of the data for SC
The Orthodox Theory ng of a single electron through rier is always a random eve e Γ (i.e. probability per unit lely on the reduction W tic) energy of the system as ng event. = e I W e o W k T B W) e "seed dc I-V curve of the tunnel ingle-electron charging effects.
The Orthodox Theory following general expressions are rath Or: W = W = e(v e(v e i + i V ) voltage drops across the barrier be event. iprocal capacitance matrix of the sys f V ) [ t e(c ) kl (C ) kk + (C t
eling Rate vs. The Ener only tunneling events decreasing the dissipating the difference) are possib
Diagram of a Tunnel Ju explains why Γ W at W>>k B T proportional to the number of occ e electron source, which contribute
neral Treatment of The C N q = ϕ (i =,,..., N i C ij j= j c = N i= qϕ = i i N N i= j= (C - ) ij q i er of islands capacitances, C ij the inter-islands c iprocal capacitance matrix of the syst ions contact potentials bers of the islands separated by the
etailed Treatment - Tran robability amplitudes of the electron sion probability can be derived from EΨ. robability: 3 B B T = k k k k 8 k k k k + CO robability for a rectangular barrier: m * E m * (E k h = B
ransmission Probabilitie m *E 0 α-decay Atom ionization
The Tunnel Rate e from initial state i to a final state f f π h ( F) = T δ( E E k,k e difference in initial and final free e el rate: i f e E E k T B F T k,k i f(e ) rmi-dirac distribution f i i f ( -f(e )) δ( -f(e) = i f f E f(-e)
ct of the Fermi Function ape at the Relevant Tem
The Tunnel Barrier rs we can approximate T kikf ~ T : T i f f(e ) i ( -f(e )) δ( f E E c.i de i E c,f de D (E )D (E f i i f )f(e )( -f(e ) initial and final states of the conduc The DOS on the initial and final side tant D s and δ function reduction: π = h ) T DD def(e) ( -f(e- i f E c f i f
he Tunneling Resistance of charging we get a straight line (O ntroduce the tunneling resistance R T istance incorporates the transmission istance: get: - F e F k T B R T h = πe T T=0 DD At T=0: f Γ( F) = i
m Tunnel Resistance fo Electron Charging anically the electrons are not localiz er than N electrons on the island. -<N> uncertainty principle: E t > h R T C E = R T > h e = R Q 6kΩ
ouble Barrier Tunnel Ju, q =C V, q=(q -q +q 0 )=ne q 0 =(C Φ -C Φ )/e charge, induced by stray capacitances
ouble Barrier Tunnel Ju drop: b + ne q0 CV b V = CΣ Σ = C + C b = V V + V energy: q C + q C = C C V b + (ne C Σ q charge, induced by stray capacitances
Change in The Free Ene ne by the voltage source in tun nevb C C ergy of the complete circuit is: n W = Σ C Σ ( C C V + (ne q ) ) b W = nevb C C 0 Σ + ev in the free energy upon tunnelin e e ±,n ) F(n,n ) = ± ( Vb C C Σ n,n e e ± ) F(n,n ) = ± ( Vb C C + +
Change in The Free Ene erm in the above equations causes voltage V b exceeds a threshold voltag capacitance min(c,c ) V > b e C Σ This is the Coulomb blockade q 0 =0
The Coulomb Staircase ectron tunnels in/out of the island t s, it is favorable for another elect ther junction. number of electrons on the island a given voltage drop.
c Junctions - The Coulom <R T =R T Staircase appear: The coulomb staircase
ngle Electron Transistor ctrode the background charge q can b ing a third gate electrode we g single electron transistor trode controls the source-drain curre
ation of the Previous Eq C = C + C + C is: so: drop: = V q = ne + q0 + Cg (Vg V ( C + C ) = q g C V b V + b C g C C C Σ V g g Σ ( V ) 0 q0 + Cg g V V g + ne ne + q 0 q 0
ification of the Free En in the free energy upon tunnelin (( ) + ± = b g Σ V C C e C e ) F(n,n ) ( + ± = ± g b Σ C V V C e C e ) F(n,n )
Example
The Stability Diagram onditions enable to draw a stabi oscillations:
Summary effects are dominant nic behavior of small objec portant effects: CB & CS thodox theory uble barrier tunnel junction electron transistors y diagrams describe the pe of these small systems