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```Kondo Effect in Quantum Dots
Seminar: Nanoscopic Systems
29. 7. 2005
Gesa von Bornstaedt
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Outline
1. What is the Kondo effect?
2. Calculation of the Kondo effect for a single magnetic impurity
3. ”Poor Man’s Scaling” - A renormalization group approach
4. Investigation of the Kondo effect in quantum dots
5. Conclusion
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
1.
What is the Kondo effect?
• Resistivity is dominated by phonon scattering for T 0
• In most metals the resistivity decreases monotonically for T → 0
• But: resistance minimum in some metals for T → 0
• Kondo effect: Resonant spin flip scattering at local magnetic impurities
• Influences transport properties
Figure 1: Minimum in the resistivity of Au (de Haas, de Boer and van den Berg, 1934)
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
2.
Calculation for a single local magnetic impurity
Hamiltonian
Kondo’s assumption:
a local magnetic moment associated with a spin S
which is coupled via an exchange interaction with the conduction band electron.
H
=
1
∑ k c†kσ ckσ + 2 J (ω)
S+ s− + S− s+ + Jz (ω) S z s z
kσ
=
1
∑ k c†kσ ckσ + 2 ∑0 J (ω)
kσ
S = local impurity spin
k,k
S+ c†k↓ ck0 ↑ + S− c†k↑ ck0 ↓ + Jz (ω) S z c†k↑ ck0 ↑ − c†k↓ ck0 ↓
s = conduction electron spin
Figure 2: Spin flip scattering of a localized impurity moment by a conduction electron
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Perturbative approach
• T > 0 =⇒ expectation values can’t be taken with respect to the ground state
• Appropriate: Matsubara Method
Starting point: H = H0 + V (t) with
V (t) =
1
J1 (ω) S+ s− + S− s+ + Jz (ω) S z s z
2
Generalized Wick Theorem for n particle Green’s function:
h Tτ (UVW...XYZ )i(0) = { f ully contracted terms}
Problem:
• Wick’s theorem only valid for operators, whose commutator is a c-number
• Not the case for the impurity spin,
obeying the standard commutation relations of angular momentum
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Pseudo fermion method
Representation of S± , S z through canonical field operators
~S = ∑σσ 0 fσ† ~Sσσ 0 fσ 0
In formulas:
for example:
S z | ↑i = 12 | ↑i
Problem:
=⇒
corresponds to
1
2
f ↑† f ↑ − f ↓† f ↓ | ↑i = 12 | ↑i
artificially enlarged Fock space
project out the unphysical part with the following constraint
Q=
∑ fσ† fσ = 1
σ
All physical relevant quantities depend on the partition function :
h
i
Zc = lim tr Q e−β( H +λ (Q−1))
λ →∞
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Evaluation of Wick’s theorem
Two particle interaction using the pseudo fermions:
V (t)
=
∑
klmn
1
v(kl; nm) f k† f l c†m cn
2
k, l, m, n ∈ {↑, ↓}
Two particle correlation function looks like:
D E
b f † f l c†m cn V
b f † f l c†m cn f † f l c†m cn
∝
Tτ V
1
2 k
1
2
k 1
k 2
1
2
Remember:
• any contraction between a pseudo and a fermion operator = 0
• contraction( f k f l† ) ∝ δkl
=⇒ just two fully contracted terms survive,
corresponding to two diagrams
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Diagrams
The scattering matrix is defined as the sum of diagram (I) and (II)
• Solid lines: bare electron operators
• Dashed lines: impurity pseudo fermion operators
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Evaluation of the diagrams
Vertex part of the diagrams:
(I) = −
11
β 4 σ,τ∑
,ω
J1 hτ | S+ |σ2 ihσ |s− |σ1 i + hτ | S− |σ2 ihσ |s+ |σ1 i + 2Jz hτ | S z |σ2 ihσ |s z |σ1 i
n ,k
· J1 hσ4 | S+ |τ ihσ3 |s− |σ i + hσ4 | S− |τ ihσ3 |s+ |σ i + 2Jz hσ4 | S z |τ ihσ3 |s z |σ i
· Gkσ (iωn ) gτ (iω − iωn )
(I I) = −
11
β 4 σ,τ∑
,ω
J1 hτ | S+ |σ2 ihσ3 |s− |σ i + hτ | S− |σ2 ihσ3 |s+ |σ i + 2Jz hτ | S z |σ2 ihσ3 |s z |σ i
n ,k
· J1 hσ4 | S+ |τ ihσ |s− |σ1 i + hσ4 | S− |τ ihσ |s+ |σ1 i + 2Jz hσ4 | S z |τ ihσ |s z |σ1 i
· Gkσ (iωn ) gτ (−iω + iωn )
where the free Matsubara Green’s functions for electrons and pseudo fermions are :
Gkν (iωn ) =
Gesa von Bornstaedt
1
iωn −k
gτ (iνn ) =
1
iνn −λ
Kondo Effect in Quantum Dots
Simplifications
• Sum over Matsubara Frequencies
Fermionic Matsubara Frequencies correspond to singularities of Fermi distribution
=⇒ The sum can be replaced by a contour integral:
(I) ∝ −
1
1
1
∑ iωn − k · iω − iωn − λ =
βω
n
Figure 3:
Gesa von Bornstaedt
1
β
I
dz
1
f (−λ ) − f (k )
1
f ( z)
=
·
2πi
z − k iω − z − λ
iω − k − λ
∑n F (iωn ) = −
H dz
2πi F ( z ) f ( z )
Kondo Effect in Quantum Dots
Further simplifications
• Pseudo fermion energy is gauged to get rid of the λ in the denominator
Constraint Q allows a U(1) gauge symmetry for this energy
f 0 = eiα f ⇒ f 0† f 0 = f † f
iω → ω + λ
iω → ω − λ
• Project out unphysical part by λ → ∞ , using limλ →∞ f (−λ ) = 1
( I ) −→
∑
k
( I I ) −→
1 − f (k )
ω − k
f ( )
∑ ω −kk
k
• Evaluation of the spin sum using
– Completeness relations
– Spin algebra
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Second order term of the scattering matrix
t(2) =
1 2
1 − 2 f (k )
3 2
1
J hσ4 |~S|σ2 ihσ3 |~s|σ1 i ∑
+
J hσ4 |σ2 ihσ3 |σ1 i ∑
2
ω − k
16
ω − k
k
k
• Evaluation of the k-sum
(assuming a constant density N(0) for the conduction electrons):
−∑
k
1 − 2 f (k )
ω − k
=
N (0)
Z D
−D
∼
= 2N (0) ln
d
tanh(/2T )
−ω
D
max(|ω| , T )
For electrons near the Fermi surface (ω = 0) and small temperatures T the result is:
t(2) = N (0) J 2 hσ4 |~S|σ2 ihσ3 |~s|σ1 iln
Gesa von Bornstaedt
D
T
Kondo Effect in Quantum Dots
Scattering matrix and resistivity
• The resistivity is connected to the scattering matrix via:
ρ( T ) ∝ |t|2
Scattering matrix:
D
~
t = J S~s 1 + N (0) J ln
T
D
=⇒ ρ( T ) ∝
1 + 2N (0) J ln
T
• Problem:
This term for the resistivity is divergent for T → 0
• Perturbation theory breaks down,
when the 2nd order term becomes ' 1st order term
• This limit defines the so called Kondo temperature:
Tk = De
Gesa von Bornstaedt
− 2N1(0) J
Kondo Effect in Quantum Dots
3.
”Poor Man’s Scaling” - A renormalization group approach
• Anderson’s idea:
Use scaling to continue the perturbative approach to small temperatures ( T < TK )
• Idea: reduce the bandwidth D of the conduction band
• Constraint: The underlying physical quantity, the scattering matrix t, stays invariant
• =⇒ Coupling constants (J1 , Jz ) change with bandwidth D
t
⇒t
= V + VPdD G0 t + V (1 − PdD ) G0 t
= V 0 + V 0 (1 − PdD ) G0 t
Gesa von Bornstaedt
with V 0 = V + PdD G0 V = V + dV
Kondo Effect in Quantum Dots
Renormalization
Evaluating the integral for the reduced bandwidth:
Z
int
d
tanh(/2T )
−ω
=
Z D
−D
d
tanh(/2T )
−
−ω
Z D−dD
−( D −dD )
tanh(/2T )
= 2d(lnD )
−ω
d
Change of the potential term
dV = −2N (0)
1
Jz J1 S+ s− + S− s+ + J12 S z s z
2
d(lnD )
Renormalization Group Equations:
dJz
d lnD
dJ1
= −2N (0) Jz J1
d lnD
= −2N (0) J12
In the physical state ( J1 = Jz ) this leads to one analytically soluble differential equation:
J (D) =
1
J0
1
+ 2N (0)ln( DD0 )
→ Divergent for T = TK
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Physical consequences
• Low temperature regime corresponds to diverging coupling constant
• Since J → ∞ the impurity and conduction electron spin have to allign
antiparallel to minimize the total energy
• For T → 0 local impurity spin forms a singlet with a conduction electron
• Local impurity spin is screened by the conduction electrons
• Behaves like a non-magnetic one
• Saturation of the resistivity becomes explicable
Figure 4: Sketch of the coupling constant and the resistance minimum versus temperature
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
4.
Investigation of the Kondo effect in quantum dots
Quantum Dot
• Small solid-state device
• Single electron transistor (SET)
• Contains confined ’droplet’ of electrons
• Integer number of electrons N
• Artificial atom
Figure 5: Scanning electron microscope image of a SET
(Picture taken from: D. Goldhaber-Gordon, H. Shtrikman et al.: ”Kondo effect in a single-electron transistor”,
Nature vol 391, 156 (1998).)
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Transport in detail
• Tunable tunnel coupling to the leads
• Has a single spin-degenerate energy state 0
• Occupied by one electron with spin up or down
• First order tunneling is blocked by the coulomb blockade
• Virtual tunnel events with an effective spin-flip are possible
• Successive spin-flip processes screen the local spin on the dot
Figure 6: Possible virtual tunnel event
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Why is a quantum dot a Kondo system?
• For an odd number of electrons the dot has a net spin magnetic moment
For an even number of electrons it is no Kondo system
• Individual artificial magnetic impurity
• Spin singlet state is formed between the unpaired localized electron and delocalized
• Quantum dot has a narrow-resonance in the density-of-states (DOS)
• This Kondo resonance gives rise to enhanced conductance through the dot
Figure 7: Kondo resonance (Lower energy bump: broadened single-particle state 0 )
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
Minimum in the conductance
• Minimum in δG strongly resembles the resistance minimum for magnetic impurities
• 3, 5, 7 show Kondo behavior (odd number of electrons)
• 2, 4, 6 are no Kondo systems (even number of electrons)
Figure 8: Conductance versus temperature
(Data taken from: Sara M. Cronenwett, T. Oosterkamp, L.Kouwenhoven: ”A Tunable Kondo effect in Quantum
Dots”, Science vol 281, 540 (1998))
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
5.
Conclusion
• Kondo systems show unusual transport properties for low temperatures
• Some metals have a minimum in the resistivity
• Quantum dots have a minimum in the conductance
• This behavior is due to the same physical effect of resonant spin flip scattering
• For even smaller temperatures the localized spin is screened
Advantages of investigating Kondo physics in quantum dots
• The Kondo system can be switched on and off
• A single localized state can be studied instead of a statistical distribution
• TK can be tuned to experimentally accessible temperatures
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
References
[1] P. Coleman: ”Monogram”, (2004).
[2] Sara M. Cronenwett, T. Oosterkamp, L.Kouwenhoven:
”A Tunable Kondo effect in Quantum Dots”, SciencVolol 281, 540 (1998).
[3] D. Goldhaber-Gordon, H. Shtrikman et al.:”Kondo effect in a single-electron transistor”,
NaturVolol 391, 156 (1998).
[4] A.C. Hewson: ”The Kondo Problem to Heavy Fermions”, Cambridge (1993).
[5] Y.Meir, N. Wingreen: ”Landauer Formula for the Current through an Interacting Electron Region”, Phys. Rev. lettersVolol 168, no 16 (1992).
[6] W. Nolting: Vielteilchentheorie, Braunschweig (1997).
Gesa von Bornstaedt
Kondo Effect in Quantum Dots
```
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