Heavy FermionsHeavy Fermions

Student: Leland HarrigerStudent: Leland HarrigerProfessor: Elbio DagottoProfessor: Elbio DagottoClass: Solid State II, UTKClass: Solid State II, UTK

Date: April 23, 2009Date: April 23, 2009

Structure of Presentation

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

The simplest model: Particle in a Box

The Equation

The Solution

K-space

Fermi Surface

3

3

23

4

2

L

kN

F

m

kFF 2

22

3

222 3

2

V

N

mF

Density of States and Fermi-Dirac Distribution Note that the systems energy is directly related to the

number of orbitals:

Gives us the number of orbitals per unit energy. Combine this with the probability of occupation:

3

222 3

2

V

N

m

2

12

3

22

2

2)(

mV

d

dND

1

1)(

)(

Tkbe

f )()()( XDfdX

Heat Capacity

How reliable is this model? Classical particles in a box (Ideal Gas) ~102 too big Quantum particles in a box (Fermi Gas) of same order

bel NkC2

3

F

Bel T

TNkC

2

2

Experimental Agreement

3ATTC

Metal ᵞγ(exp) γ0 (free electron)

γ/γ0

Ag 0.646 0.65 1.00Cu 0.695 0.50 1.39Rb 2.41 1.97 1.22Li 1.63 0.75 2.17

Source: N.E. Phillips

m0

m

mth*

Refining the model

Take into account the ion cores

)()(

)()(

xeTx

xVTxViKT

1

0

)()(N

j

jTxxV

Interaction with the cores

dt

kdF

dk

dvg

1

22

2

dkd

22

2

dkd

Fdt

dvg

*m

Electron-Electron Interactions For Metals:

Conduction electrons are 2Å apart. Mean free paths are >104Å at room temp.

Why: Coulomb Screening Exclusion Principle

Fermi Fluid

Takes into account electron-electron interactions

Complicated interactions treated as non-interacting quasiparticles above an inert Fermi-sea.

Formulation:

,

,,k

kkk ccH ''

''

''' ,,,,,,

,,,,

kkqkk

qkqkqkkccccV

Heavy Fermions Begin by example:

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: 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.

Y1-xUxPd Fermi Liquid

Heat Capacity C ~ -Tln(T) C = TConductivity ~ 0 + AT1.1 = 0 + AT2

Magnetic Susceptibility

m ~ - T1/2 m = Source: Seaman et al.

Source: Sanchez

Phases and properties

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

0)( T 2AT 5BT

T

cm

ln

RKKY Interaction

Magnetic impurities replaced by magnetic lattice.

Indirect exchange coupling established between magnetic ions.

Competition between interactions.

2JTRKKY

JK eT

11

Two different energy scales:

Coherence and Delocalization

T* = coherence temperature We see: reduced resistivity, modified spin

sucseptibility, observed Knight shift, sudden entropy change, and more.

Why: delocalization of the f-electrons.

TT

UC

T

US

)2ln(

*

0

RdTST

Attempting a Universal Model

2* cJT

JK eT

11

*11)][ln( TcTJ K

Estimate

2

3

NFL and QCP Scaling

References

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/HRRS1

http://en.wikipedia.org/wiki/Kondo_effect Kittel Solid State Physics