Transcript Heavy Fermions - University of Tennessee

Heavy Fermions
Student: Leland Harriger
Professor: Elbio Dagotto
Class: Solid State II, UTK
Date: April 23, 2009
Structure of Presentation
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Fermi Gas
Modifications to Fermi Gas
Examples and Properties of Heavy Fermions
Interactions Important to Heavy Fermions
Common Features within Heavy Fermions
Fermi Gas Theory
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The simplest model: Particle in a Box
The Equation
The Solution
K-space
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Fermi Surface
4
  k F3
N  2 3
3
 2 


 L 
F 
 2 k F2
F 
2m
  3 N 


2m  V 
2
2
2
3
Density of States and Fermi-Dirac
Distribution
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Note that the systems energy is directly related to the
number of orbitals:
  3 N 



2m  V 
2
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2
2
3
3
2
dN
V  2m 
 D( ) 

 2   2
2 
d 2   
1
Gives us the number of orbitals per unit energy.
Combine this with the probability of occupation:
1
f ( ) 
e
(   )
k bT
1
X   d  f ( )  D ( )  X ( )
Heat Capacity
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How reliable is this model?
Classical particles in a box (Ideal Gas)
2 too big
3
~10
C  Nk
2
Quantum particles in a box (Fermi Gas)
 Nk T
of
same
order
C 
2T
el
b
2
B
el
F
Experimental Agreement
C  T  AT
ᵞγ(exp)
Metal
Ag
Cu
Rb
Li
0.646
0.695
2.41
1.63
Source: N.E. Phillips
 m
3
γ0 (free
0.65
0.50
1.97
0.75
γ/γ0
electron)
1.00
1.39
1.22
2.17
mth*


m 0
Refining the model
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Take into account the ion cores
V ( x  T )  V ( x) 
 ( x  T )  e  ( x)
iKT
N 1
V ( x)    ( x  jT )
j 0
Interaction with the cores


dk
F 
dt
1 d
vg 
 dk
 2 dvg
F 2 2
d  dk dt
m* 
Electron-Electron Interactions
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For Metals:
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Conduction electrons are 2Å apart.
Mean free paths are >104Å at room temp.
Why:
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Coulomb Screening
Exclusion Principle
Fermi Fluid
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Takes into account electron-electron
interactions
Complicated interactions treated as noninteracting quasiparticles above an inert
Fermi-sea.
Formulation:
H    k ck, ck , 
k ,

k , k ' ,q , , '
Vk ,k ' ,q ckq, ck' q, ' ck , ck ' , '
Heavy Fermions
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Begin by example:
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f-electron system CeAl3
Specific Heat is linear in T
~ 1000 times larger than expected by Fermi
Gas Theory
Implies m* ~ 1000 times larger
Interesting Properties:
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Heavy Fermion Systems were the first display
NFL behavior.
They also are an example of “exotic
superconductivity”
Rich Phase Diagrams Exhibiting both
NFL behavior and superconductivity.
Source: Sanchez
Heat Capacity
Conductivity
Magnetic
Susceptibility
Y1-xUxPd
C ~ -Tln(T)
 ~ 0 + AT1.1
m ~  - T1/2
Fermi Liquid
C = T
 = 0 + AT2
m = 
Source: Seaman et al.
Phases and properties
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Heavy Fermion is NOT
synonymous with Non-Fermi
Liquid.
However, in the Fermi Liquid
phase heavy fermions have
anonymously large electronic
specific heat coefficient and
Sucseptibility.
(2-4 orders of magnitude larger
than Cu)
Kondo Effect

 (T )  0  AT 2  BT 5  cm ln 
T 
RKKY Interaction
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Magnetic impurities
replaced by
magnetic lattice.
Indirect exchange
coupling
established
between magnetic
ions.
Competition between interactions.
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Two different energy
scales:
TRKKY  J 
2
TK   e
1

1
J 
Coherence and Delocalization
U
C
 T
T
U
S 
T
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T*
S   dT  R ln( 2)
0
T* = coherence temperature
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We see: reduced resistivity, modified spin
sucseptibility, observed Knight shift, sudden
entropy change, and more.
Why: delocalization of the f-electrons.
Attempting a Universal Model
TK   1e

1
J 
J  [ln(TK  )] 1  c 1T * 
T  cJ 
*
2
Estimate

3
2
NFL and QCP Scaling
References
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Z. Fisk, et. al. PNAS 92, 6663 (1995).
Yi-feng Yang, et. al. Nature 454, 611 (2007).
V.V. Krishnamurthy, et. al. PRB 78 024413 (2008).
J.P. Sanchez ESRF
http://www.esrf.eu/UsersAndScience/Publications/Highlights/2002/HRRS/H
RRS1
http://en.wikipedia.org/wiki/Kondo_effect
Kittel Solid State Physics