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THE KONDO MODEL
Jens Schalkowski, 08.07.2010
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Content
History
Resistivity of a metal
Kondo effect
The Kondo model
perturbation method
PMS
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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
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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
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HistoryKondo effect
1934 de Haas, de Boer and van den Berg
electrical resistivity of Au
unexpected local minimum
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HistoryKondo effect
1964 solved by J. Kondo
minimum associated with magnetic impurities
novel scattering mechanism
spin-flip scattering
temperature dependant
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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
�
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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
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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� + ...
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The Kondo modelperturbation method
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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+
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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
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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 �
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The Kondo modelperturbation method
second order contribution to the resistivity
temperature independent
ρel,(2)imp =
3πmJ2S(S + 1)cimp
2e2�F �
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The Kondo modelperturbation method
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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+
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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
��
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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
�
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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
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Summary
The non-trivial physics associated with the presence of magnetic impurities in a solid is referred to as the Kondo effect
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References
Introduction to the Kondo effect
Kondo Effekt in Supraleitender UmgebungDiplomarbeit von Julia Sabelin