1. Ferromagnetism and the quantum critical point in Zr1-xNbxZn2
D. Sokolov, W. Gannon, and M. C. Aronson
*University of Michigan
Z. Fisk
Florida State University
* Supported by the U. S. National Science Foundation

2. Quantum Critical Points and Superconductivity
Heavy Fermion Intermetallics
Complex Oxides
CePd2Si2 (Mathur 1998)

3. Quantum Criticality in Itinerant Magnets
Antiferromagnet
Ferromagnet
Cr-V (Yeh 2002)
UGe2 (Saxena 2000)

4. ZrZn2: Itinerant Ferromagnet
ZrZn2 spin density (4.2 K)
C15 Laves Phase
Pickart 1964

5. Ferromagnetic Quantum Critical Point in ZrZn2 : Pressure
(Grosche 1995)
10.5 kbar
Pc = 8 kbar
0.4 kbar

6. Zr1-xNbxZn2 Polycrystalline samples m(H=0,T) = m0,0(x)(1-T/TC)b b=0.5
TC and m(H=0,T=0) are suppressed by Nb doping x

7. Quantum Critical Point in Zr1-xNbxZn2 (x=xC=0.085)TC
% Nb (x)
Three dimensional Heisenberg ferromagnet (xC=0.085)
TC(d+n)/z Q (x-xC) (d=3, z=2+n=3) TC4/3 Q (x-xC)

8. ZrZn2: Stoner Ferromagnet
Huang 1988
states/Ry-cell
E (Ry) xC [
Stoner ferromagnet (+ magnetic fluctuations)
a=I N(Ef) m0 Q (a-1)1/2 TC Q (a-1)1/ mo Q TC

9. The Initial Susceptibility c
xC=8.5%
x<xC c-1=co-1t g t=(T-TC)/TC
x>xC c-1=co-1+CTg
x=xC c-1=CTg

10. The Critical Susceptibility (x<xC)
x=0 c= cot-g g=1.08+/ mean field <t<100
x= c=cot-g g=1.33+/ Heisenberg ferromagnet 0.01<t<100
TC (K) g reduced temperatures
Fe 1044 K 1.33+/ <t<10-2
Co 1388 K 1.21+/ <t<10-2
Ni 627 K 1.35+/ <t<10-2
3d Heisenberg model
Suppression of mean field behavior near quantum critical point

11. Mean Field – Heisenberg Crossover
slope=-1
slope=-4/3
x<<xC, large t: c~t-1 x~xC, small t: c~t-4/3/(x-xC)
Crossover function: c~t-4/3 f(t1/3/(x-xC)) ~t-4/3f(y)
large y: f(y)~y-4/3 small y: f(y)~y-1

12. The Ordered Phase T L>>x L<<x x xC
equal reduced
temperatures x xC
Interactions become more local as xYxC, : L<<x
increasing importance of fluctuations
Matrix more highly polarizable as xYxC: divergence of co

13. Paramagnetic Phase (x>xC)
0.14
0.12
0.09
T=0
c-1=co-1 + CTg
xYxC: co-1Y0
x=0.08
x=0.12
x=0.14
Stoner ferromagnet: c(T,x) = cpauli/(1-a(x))

14. The T=0 Disordered StateFM
x>xC: c(T)-1=c0-1+CT4/3 [
x=xC c(T)=C/Tg g=d+n/z = 4/3 (3d Heisenberg FM)
g=1/2n=1 (n=1/d-1) (clean QC FM)
g=1/n=1 (n=1/d-2) (dirty QC FM)
g=1-l (Griffiths Phase)

15. Criticality in Zr1-xNbxZn2FM

16. Superconductivity near a Ferromagnetic QCP
g2co/t 60 30 20 10 5
x=xC
x<<xC
c(q,w) = coko2/[k2+q2-iw/h(q)]
Monthoux and Lonzarich 2001
k2=x(T=0)-1~ (x-xC)