Lecture 2: “Single-‐molecule transistors: Coulomb blockade and

Lecture 2: “Single-­‐molecule transistors: Coulomb blockade and Kondo physics” References
Chapter 15 Molecular Electronics: An Introduc6on to Theory and Experiment, J.C. Cuevas and E. Scheer, (World ScienTfic, Singapore, 2010). See also: 1) “Single-­‐molecule electrical junc6ons”, Y. Selzer and D.L. Allara, Annu. Rev. Phys. Chem. 57, 593 (2006). 2) “Charge transport and single-­‐electron effects in nanoscale systems”, J.M. Thijssen and H.S.J. Van der Zant, Phys. Stat. Sol. (b) 245, 1455 (2008). €
ΔE = | ε 0 − E F | = injection energy
Γ = ΓL + ΓR = level width
 Traversal 4me: τ =  / ΔE 2 + Γ2
 Energy scale of the Coulomb interac4on: U
€In this lecture we focus on situaTons in which: τ >>  /U
€
and therefore, the transport is dominated by the Coulomb repulsion of the electrons inside the molecule. €
Specially, this situaTon is realized in resonant situaTons when the metal-­‐molecule coupling is relaTvely weak. How small and how cold should a conductor be so that adding or
subtracting a single electron has a measurable effect? 1) The capacitance C of the island (or dot) has to be such that the charging energy (e2/C) be smaller than the thermal energy (kBT): e 2 /C >> k B T
2) The barriers have to be sufficiently opaque such that the electrons are located in the dot: ΔEΔt = (e 2 /C)(Rt C) > h ⇒ Rt >> h /e 2
€
In m€
olecular transistors these two requirements can be easily met!!!  To resolve the discrete levels of a quantum dot: ΔE >> kB T
 The level spacing at the Fermi energy for a box of size L depends on the dimensionality: ⎧ N /4
(1D)
2 2
 π ⎪
€ ⎨
ΔE =
(2D)
2 × 1/π
mL ⎪
2
1/ 3
(1/3
π
N)
(3D)
⎩
 The level spacing of a 100 nm 2D dot is around 0.03 meV, which is large enough to be observable at diluTon refrigerator temperatures (100 mK  0.0086 meV).  Using 3€
D metals to form a dot one needs to make dots as small as 5 nm in order to see atom-­‐like properTes.  In the case of molecular juncTons, the level space is essenTally the HOMO-­‐LUMO gap and it is typically several electronvolts. Therefore, the level quanTzaTon is easily observable in molecular transistor even at room temperature. Carbon nanotube junction: S.J. Tans et al., Nature 386, 474 (1997)
N
E dot (N) = U(N) + ∑ E p
p =1
E p ( p = 1,2,...) = single - electron energy levels
U(N) = (Ne) 2 /2C − NeVext
C = CS + CD + CG
€
€
ΓL( p ), ΓR( p ) ⇒ tunneling rates
€
Vext = (CSVS + CGVG + CDVD ) /C
€
kB T,ΔE >> h(ΓL( p ) + ΓR( p ) )
(weak coupling) " Periodicity of the oscilla4ons:  Dot chemical potenTal: ⎛
1 ⎞ e 2
µdot (N) = E dot (N) − E dot (N −1) = ⎜ N − ⎟ − eVext + E N
⎝
2 ⎠ C
 Electrons can flow from leg to right when: µL ≥ µdot ≥ µR
€
€
 For small bias voltages, VSD ~ 0: ⎛
1 ⎞ e 2
µdot (N) = ⎜ N − ⎟ − eαVG + E N ;
⎝
2 ⎠ C
(α = CG /C = gate coupling)
 Thus, the addiTon energy is given by: €
e2
e2
Δµ(N) = µdot (N +1) − µdot (N) = + E N +1 − E N = + ΔE
C
C
 In the absence of charging effects, the addiTon energy is determined by the irregular spacing ΔE of the single-­‐electron levels. The charging energy e2/C regulates the spacing and € when it is much larger than the level spacing (as in metallic islands), it determines the periodicity of the Coulomb oscillaTons.  From an experimental point of view, the Coulomb oscillaTons are measured as a funcTon of the gate voltage and the peak spacing is given by: ΔVG = Δµ(N) /eα = (e 2 /C + ΔE) /eα
N
2
while the condiTon e α
V
G = (N
−1/2)e
/C
+
E N gives the gate voltage of the N-­‐th Coulomb peak. €
 Different tunneling processes (energy conserva4on): state p in the dot (N electrons) - - > left lead at energy E pf ,l (N) :
E pf ,l (N) = E p + U(N) − U(N −1) − (1 − η)eV
left lead at energy E pi,l (N) - - > state p in the dot (N electrons) :
E pi,l (N) = E p + U(N +1) − U(N) − (1 − η)eV
state p in the dot (N electrons) - -- > right lead at energy E f ,r (N) :
E f ,r (N) = E p + U(N) − U(N −1) + ηeV
right lead at energy E pi,r (N) - -- > state p in the dot (N electrons) :
E pi,r (N) = E p + U(N +1) − U(N) + ηeV
 Sta4onary current through the leN barrier: e ∞
I = ∑ ∑ ΓL( p )P({n i }) δ n p ,0 f (E pi,l (N) − E F ) − δ n p ,1[1 − f (E pf ,l (N) − E F )]
 p =1 {n i }
(
)
In equilibrium the probability distribuTon P({ni}) is given by the Gibbs distribuTon in the grand canonical ensemble: ⎡ 1 ⎛ ∞
⎞ ⎤
1
Peq ({n i }) = exp ⎢−
⎜∑ E i n i + U(N) − NE F ⎟ ⎥;
Z
⎠ ⎦
⎣ k B T ⎝ i=1
Z = partition function
The non-­‐equilibrium probability distribuTon P is a staTonary soluTon of the kineTc equaTon: €
∂
P({n i }) = 0 = −∑ P({n i })δ n p ,0 [ΓL( p ) f (E i,l (N) − E F ) +ΓR( p ) f (E i,r (N) − E F )]
∂t
p
−∑ P({n i })δ n p ,1[ΓL( p ) (1 − f (E f ,l (N) − E F )) +ΓR( p ) (1 − f (E f ,r (N) − E F ))]
p
+∑ P(n1,…,n p −1,1,n p +1,…)δ n p ,0 [ΓL( p ) (1 − f (E f ,l (N +1) − E F )) +ΓR( p ) (1 − f (E f ,r (N +1) − E F ))]
p
+∑ P(n1,…,n p −1,0,n p +1,…)δ n p ,1[ΓL( p ) f (E i,l (N −1) − E F ) +ΓR( p ) f (E i,r (N −1) − E F )]
p
" Linear response theory: ⎛ eV
⎞
P({n i }) ≡ Peq ({n i })⎜1+
Ψ({n i })⎟
k
T
⎝
⎠
B
The joint probability that the quantum dot contains N electrons and that the level is occupied is: €Peq (N,n p = 1) = ∑ Peq ({n i })δ N ,∑i n i δ n p ,1
{n i }
In terms of this probability the conductance is given by: e 2 ∞ €∞ ΓR( p )ΓR( p )
G=
∑
∑
( p)
( p ) Peq (N,n p = 1) 1 − f (E p + U(N) − U(N −1) − E F )
kB T p =1 N =1 ΓL + ΓR
[
 Limit: €
G(VG ,T) /Gmax
kB T << e 2 /C,ΔE
Gmax
€
]
⎛ eα (VG − V0 ) ⎞
= cosh ⎜
⎟
⎝ 2kB T ⎠
−2
⎛ e 2 ⎞ π
ΓL(N 0 )ΓR(N 0 )
= ⎜ ⎟
(N )
(N )
⎝ h ⎠ 2k B T ΓL 0 + ΓR 0
E1 − E F = 50 meV; E 2 − E F = 80 meV
ΔE = 30 meV; e 2 /C = 100 meV
T = 30 K; ΓL( p ) = ΓR( p ) = 1 meV; η = 0.6
 Elas4c cotunneling process:  Inelas4c cotunneling process:  Spin-­‐flip cotunneling processes can change the spectrum of the dot leading to the screening of the localized spin and to the appearance of the so-­‐called Kondo resonance.  The Kondo resonance lies exactly at the Fermi energy, no maler the posiTon of the original level. For this reason, the Kondo effect leads to an enhancement of the conductance . The only requirement for this effect to occur is that the temperature is below the Kondo temperature (see below).  The width of the Kondo resonance is proporTonal to the characterisTc energy scale for Kondo physics, the so-­‐called Kondo temperature: ⎛ πε 0 (ε 0 + U) ⎞
ΓU
kB TK =
exp⎜
⎟
⎝
⎠
2
ΓU
Transport signatures of the Kondo effect: Zero-­‐bias line in the stability diagram for odd number of electrons in the dot. Peak in the low-­‐bias conductance at low temperatures with width equal to the Kondo temperature. CharacterisTc temperature dependence of the linear conductance. Park et al., Nature 407, 57 (2000)
Park et al., Nature 417, 722 (2002)
S. Kubatkin et al., Nature 425, 698 (2003).
(OPV5) Park et al., Nature 417, 722 (2002)
Molecules: Co-­‐ion compounds; TK = 10 -­‐25 K. Liang et al., Nature 417, 725 (2002)
Molecules: divanadium compounds.