THE KONDO MODEL

Jens Schalkowski, 08.07.2010

Content

History

Resistivity of a metal

Kondo effect

The Kondo model

perturbation method

PMS

HistoryResistivity of a metal

determined by different scattering mechanisms:

between conduction elelctrons and lattice distortions

electron-electron

between electrons and static impurities

ρelPhonon ∝ T 5

ρelel−el ∝ T 2

monotonic temperature dependence

saturation in the limit

HistoryResistivity of a metal

T → 0

ρel(T ) = acimpρel0 + bT 2 + cT 5

limT→0ρel(T ) = acimpρ

el0

HistoryKondo effect

1934 de Haas, de Boer and van den Berg

electrical resistivity of Au

unexpected local minimum

HistoryKondo effect

1964 solved by J. Kondo

minimum associated with magnetic impurities

novel scattering mechanism

spin-flip scattering

temperature dependant

HistoryKondo effect

new energy scale

Kondo temperature TK

Kondo effect dominates for temperature near TK

characteristic resistivity ρel1

ρel(T ) = acimpρel0 + bT 2 + cT 5 + cimpρ

el1 ln

�TK

T

�

The Kondo modelperturbation method

scattering of conduction electrons from a localized magnetic impurity

HK = Hl +�

kk�

J

�S

z�α†

sk↑αsk�↑ − α†sk↓αsk�↓

�+ S

+α†sk↓αsk�↑ + S

−α†sk↑αsk�↓

�

HK = Hl + H�

H� = 2JS · s0

The Kondo modelperturbation method

Schrödinger equation

formal solution

��− Hl

�| Ψ� = H

� | Ψ�

| Ψ� =| Ψ0�+1

� + i0+ − Hl

H� | Ψ�

T = H� + H

� 1� + i0+ − Hl

H� + H

� 1� + i0+ − Hl

H� 1� + i0+ − Hl

H� + ...

The Kondo modelperturbation method

The Kondo modelperturbation method

first order contributions

�k� ↑| T (1) | k ↑� = JSz

�k� ↓| T (1) | k ↓� = −JSz

�k� ↑| T (1) | k ↓� = JS−

�k� ↓| T (1) | k ↑� = JS+

The Kondo modelperturbation method

second order in J

scattering propability

number of magnetic impurities

Wkk� =2πNimp

� �T (1)kk��2 = |J |2 2πNimp

� S(S + 1)

S = �Sz�

Nimp

The Kondo modelperturbation method

ρelimp =

m

ne2τ (kF )

[τ (kF )]−1 =�

−→k �

W−→k−→k � (1− cos θ�) δ

��−→

k− �−→

k �

�

[τ (kF )]−1 =3πJ2S(S + 1)cimpn

2�F �

The Kondo modelperturbation method

second order contribution to the resistivity

temperature independent

ρel,(2)imp =

3πmJ2S(S + 1)cimp

2e2�F �

The Kondo modelperturbation method

second order processes (c)

The Kondo modelperturbation method

�k� ↑| T (2) | k ↑� =�

ki

J2�k� ↑| α†

sk�↑αski↓S− 1

� + i0+ − Hl

S+α†

ski↓αsk↑ | k ↑�

�k� ↑| T (2) | k ↑� =�

ki

J2 S−S+ [1 − f(�ki ]� − �ki + i0+

transport relaxation time

The Kondo modelperturbation method

[τ (kF )]−1 =3πJ2S(S + 1)cimpn

2�F �

�1 + 4Jρ(0) ln

D

max (|�|, kT )

�

ρel,(3)imp =

3πmJ2S(S + 1)cimp

2e2�F �

�1− 4Jρ(0) ln

�kT

D

��

resistivity:

unphysical, divergent resistivity

perturbative method fails for sufficiently small temperatures

The Kondo modelperturbation method

ρel(T ) = acimpρel0 + bT 2 + cT 5 + cimpρ

el1 ln

�TK

T

�

Poor man‘s scaling (PMS)

1970 P.W. Anderson

effective Hamiltonian that captures the low-energy properties of a given system

scaling equation of the Kondo

The Kondo modelPMS

dJ

d lnD= −2ρJ2

Summary

The non-trivial physics associated with the presence of magnetic impurities in a solid is referred to as the Kondo effect

References

Introduction to the Kondo effect

Kondo Effekt in Supraleitender UmgebungDiplomarbeit von Julia Sabelin