Slides

1
Index 3. Pre- critical region

• Slides
• 0 Résumé of figures
• 1-2 Title and Introduction
• 3 Magnetic Phase Diagram of four
  Ce-exemplary systems
• 6-10 Analysis an Antiferromagnetic exemplary
  system CeIn3-xSnx
• 11-21 Pre-critical region identification
• 22-23 Scaling of the Cp/T ? Log (T/T0 )
  dependence in Ce systems
• 24-26 The Maelstrom of the Entropy
• 27-31 The Third law of themodynamics and the
  Quantum Critial Point
• 32-35 Divertimento Dimensionality of
  magnetic correlations
• 36 Conclusions

2
Résumé of figures
3

3. Pre- critical region
Physical properties of systems included in Type
I behavior
Experimental criteria for QCP candidates
4

Not all magnetic phase boundaries end in a
QCP because only 10 of them have their MO -
phase boundary experimentally traced in more than
one decade TN,C / TN,C (0) lt 0.1 (labeled as
Type - I - )
In the following figure we compare the magnetic
phase diagrams of four Ce-systems whose magnetic
phase boundaries are proved to be traced for more
than one dacade in temperature
5
Magnetic Phase Diagram of four Ce-exemplary
systems belonging to Group -I-, vs. normalized
control parameters
(1)
(2) Ferro
(3)
(4)
TN ,C ? ? ? xcr- x
?
Refs (1) Pedrazzini et al., Acta Phys. Pol. 34
(2003) 363 (2) Sereni, Physica B 354
(2004) 331 (3) v.Löhneysen, JMMM 108 (1992)
45 (4) Umeo et al., Physica B 259 (1999) 407.
6
Analysis an Antiferromagnetic exemplary system
CeIn3-xSnx Pedrazzini et al., Euro Phys J B 38
(2004) 445 Sereni, Physica B 354 (2004) 331
Thanks to P. Pedrazzini Centro Atomico
Bariloche C. Geibel Max-Planck-Institut
for R Küchler Chem. Physics of Solids
H. Wilhelm Dresden
7
Crystalline Structure

• cubic AuCu3-type
• ?7 ground state, ?8

x
0
1
3
2
4.73
CeIn3-xSnx
4.72

4.71

• Clear deviation from Vegards law

293 K
4.70
a Å
4.69
77 K

4.68
J. Sakurai et al., Sol. Stat.
Comm. 50 (1984) 71
4.67
8
Magnetic Susceptibility

• AF transition in the x 0.15 sample. The
  transition broadens for x0.3.
• A bump develops around 40 K for x gt 0.45.
• The samples with 0.8 lt x lt 3 can be described
  with a single energy scale Tsf

9
Inverse magnetic susceptibility crystal field
splitting evaluation
10
Normalized electrical resistivity T(?max)
evolution
Pedrazzini et al., Euro Phys J B 38 (2004) 445
11
Low temperature properties Pre-critical region
identification
12
Phase Diagram around the magnetic instability
region
Pedrazzini et al., Euro Phys J B 38 (2004) 445
13
Evolution from MF-type behavior to pre-critical
regime comparison of two exemplary systems
14

• Within this pre-critical region x lt x lt xcr
  peculiar features are observed
• linear TN ,C vs. ? dependence
• specific heat (Cm /T ) properties scaling
• analysis of the entropy gain
• dimensionality of magnetic excitations
• spin dynamics (F)

Pre-critical fluctuations were already considered
by Schlottmann, Acta Phys. Pol. 34 (2003) 391
15
1st. order transit. ?H/T ? 3 RLn2
Magnetic contribution to specific heat
Cm / Tmax ? constant
A long tail developes at T gtTN (QP region)
involv- ing significant fraction of degrees of
freedom
There is a tendency to ? -Ln T dependence
See also CeCu5xAu1-x Löhneysen, JMMM 200
(1999) 532

Sereni, Physica B (2004) to be published
16
Comparison of Cm /Tmax for four Ce-exemplary
systems vs. normalized conc. (press)
X
X cr
X
(? x xcr - x )
There are further examples of constant Cm / Tmax
Ce (Pd, Rh)2 Si2 and Ce (Cu,Co)2 Ge2
17

From Landau theory for phase transitions Cm / T
(at TN,C ) ao2 / 2bo G(?,T) G0 (T) ao (T -
Tcr ) ? 2 bo ? 4 This relation indicates that,
within the pre-critical region G (?) is "locked",
even if it keeps changing with temperature.
Once established that Cm / Tmax const. , we
proceed to analyze the Cm / T tails above TN
in CeIn3-xSnx alloys.

18
Evolution of Cm / T within the xlt x lt xcr region
19

Despite the tendency to a -Ln T dependence above
TN , Cm / T (x) does not scale with a reduced
temperature ( like t T / TK ) but with ? T
T TN.
20
Scaling of Cm / T with ? T
0.41 ?
S / RLn2 ?
?
0.80

Data from Sereni, Physica B 320 (2002) 376
21
Comparison between two exemplary systems
Sereni, Physica B 320 (2002) 376
22
Scaling of the Cp/T ? Log (T/T0 ) dependence in
Ce systems
It is possible to scale Ce systems with a
universal function of the form Cp/t - D Log
(t )E T0 , where D7.2 J/molK, tT/ T0 (with T0
a characteristic energy scale) and E accounting
for high temperature contributions Sereni et al,
Physica B 230-232 (1997) 580. Because there are
only two independent parameters, the D value was
fixed in 7.2 J/molK by taking as reference the
system CeCu5.7Au0.3 under p8.2kbar Bogenberger
et al. Phys. Rev. Lett. 74 (1995) 1016 which is
fitted with E0. In the following figure we show
an example of such a scaling applied to three
different Ce compounds (binaries and ternary)
with different crystalline structures, all of
them close to their respective quantun critical
points. Though T0 cannot directly identified with
TK because the system does not behave as a Fermi
Liquid, its absolute value is very close to the
extracted from the temperature of resistivity
maximum for example.
23
Scaling of different exemplary compounds
24
The Maelstrom of the Entropy
The parameter D determines the amount of entropy
(?S) and at t1 (i.e. TT0 ) one finds that ? S
amounts to 0.54RLn2 (D/Ln10). Since Cp/t - D
Log (t ) is a universal function for systems
close to their quantum critical points, such a
lack of entropy appears as a physical
characteristic. In the following figures we
analyze this peculiar finding on the simple
binary-cubic CeIn3-xSnx system
25
Comparison of Entropy contributions between the
ordered MO and quasi-paramagnetic QP phase
2/3 ?? ?SQP
T TN
xcr0.67
T0
?SMO 1/3 ??
?
x 0.30
? xcr - x
26

• From this representation we see that, while ?
  SMO (x) ? 0, no degrees of freedom are
  transferred to ? SQP .
• Then, at x xcr about 0.4 of the total RLn2
  entropy seems to be "missed" within the negative
  range of ? T used for the Cm / T ? 0
  extrapolation.
• Unless this "missed" fraction of entropy is
  released below the experi-mental range (now
  extended to 40 mK ? 2 decades in T), an unusual
  zero point entropy S0 ? 0 occurs.
• Fractional (R/2 Ln2) entropy gain was reported
  in (U,Y)Pd3 Seaman et al., PRL 67 (1991) 2882
  and predicted by theory in the multi-channel
  Kondo problem Tzvelik et al., J. Phys. C 18
  (1985) 159 Schlottmann et al., Phys.Lett. A
  142 (1989) 245.

27
The Third law of themodynamics and the Quantum
Critial Point
Nernsts theorem states that at the absolute zero
of temperature the entropy of any body is zero,
this implies that if a body has no disposable
energy it would have only one possible state and
thus S RLn(1) 0. This law implies that the
system is in full equilibrium respect to all
its variables, and therefore there are no
frozen-in configurations. This condiction fails
when a metastable configurations occur, like in
solid solutions for example. In such a case at
least one thermodynamical variable (h) is
constrained in a fixed value other than its
equilibrium, and the system attains a constrained
equilibrium Ref. 1Abriata Laughlin, Prog.
Mat. Sci. 49 (2004) 367. Then, even if the
internal energy (U) attains its lowest value and
dS/dUb1/T is infinite, i.e. T0, constrained h
does not allow S to be in its lowest possible
value Fig. 2, Ref 1.
28
Application to the CeIn3-xSnx system The first
question is whether a constrained equilibrium can
be recognized. 1) As seen in the Phase Diagram
the magnetic phase boundary Tn(x) turns its
classical negative curvature in a linear x
dependence within the pre-critical region. 2)
The resistivity maximum is locked at 20K within
this range on x 3) The maximum of Cm/T is also
independent of x (see discussion on Free Energy
locked dependence on the order parameter
above) 4) The logarithmic tail of Cm/T (at T gt
Tn) is scaled by Cm/T-A log (T-Tn) Notice that
? T T-Tn xcr-x is the distance d to the QCP
29

• Entropy
• Contrary to the structural contribution to the
  entropy in solids, where its absolute value has
  to be referred to the T0 value, in Rare Earths
  magnetic compunds the magnetic entropy Sm can be
  known a priori from the crystal field (CF)
  ground state (GS) degeneracy. In the case of Ce
  compunds, with J5/2, a doublet GS is guaranteed
  (with a few CsCl-cubic exceptions). This allows
  to have a fixed value of Sm RLn2 at high enough
  temperature skiping the constrain of a Sm (T0)
  reference.
• Since we have determined that for CeIn3-xSnx
• at x xcr about 0.4 of the total RLn2 entropy
  seems to be "missed" within the negative range of
  ? T used for the Cm / T ? 0 extrapolation,
  thus unless this "missed" fraction of entropy is
  released below the experimental range an unusual
  zero point entropy S0 ? 0 occurs, and
• while ? SMO (x) ? 0, no degrees of freedom are
  transferred to ? SQP ,
• we have to search for an alternative component to
  the system with an increasing contribution along
  the pre-critical range.
• The quantum character of the critical point
  provides the possibility for the existence of a
  (at least) two levels GS between which quantum
  mechanics allows a tunneling process Kirkpatrik
  Belitz PRB 67 (2003) 024419 (see next
  figure).

30
Schematic representation of the duoblet ground
state, splitted by the (classical) magnetic
transition and the access to a quantum two
levels scheme connected by quantum tunneling.
Paramagnetic doublet
Antiferromagnetic ground state
Quantum tunneling between two levels
31
Thus, the fraction of entropy corresponding to
the MO phase is progressively shared in the
quantum degenerate GS. So, at the critical
concentration the GS level is splitted due to the
uncertainty principle that gives a finite
probability to the particles to be in more than
one quantum level. In such a case, the entropy
gain in the quantum region will be SRLn(g1 /g0
)RLn(3/2) that is 0.6RLn2, extremely close to
the measured value. Then the missed entropy if
due to the modification of the GS level scheme
under quantum effect.
32

Divertimento
Dimensionality of magnetic correlations Neutron
scattering measurements on CeCu5xAu1-x for x ?
xcr tested the low dimensionality of
fluctuations related to the QCP Schröder et
al., Phys. Rev. Lett. 80 (1998) 5623.
Similar study is not possible in CeIn3-xSnx
because of the strong neutron absorption of In
nuclei.
As alternative, one may check whether any
evidence about the dimensionality of magnetic
correlations can be extracted from
thermodynamical results. For that
purpose we have evaluated the internal energy Um
(x), together with Sm(x), in samples within the
pre-critical concentration and compared them with
Ising and Heisenberg model predictions for 1, 2
and 3 dimensional systems with different lattice
structures (i.e. coordination number).
33
Predictions for 1, 2 and 3D systems in Ising and
Heisenberg models
simple quadratic
0.88
Experimental
UTot /RTcr
0.68
0.39
Ucr-? /RTcr
Scr-? /R
34

• These results nicely fit into the predicted
  values for a 2D- Ising quadratic layer.
• This is consistent with the Cm ? T 2 dependence
  used for the ?SMO analysis and
• with a possible layer-reminiscence of the CeIn3
  AF structure - 3D-cubic - (Beniot et al.,
  Sol.State Commun. 34 (1980) 293)

Certainly these results are not conclusive like
those from neutron scattering

35
Ce - crystalline position and Magnetic Structure
of CeIn3-xSnx
Ce
Ce
. Propagation vector (1/2,1/2,1/2) Beniot et
al., Sol.State Commun. 34 (1980) 293
36

Conclusions Apart from the known Cm / T ? -Ln
T dependence, other QCP- related effects arise in
thermal and magnetic properties of Ce-lattice
systems within their Pre-critical region (x lt x
lt xcr ) and in the range of 0.1 TN,C (0)
(Group-I )

• Change from the negative curvature of TN,C ( x)
  to a linear x dependence

• Nearly constant value of Cm / T at T TN,C ( x )

• G(T, ? ) is "locked" in its dependence on x

• Scaling of Cm / T with ? T T - TN ? ?
  (above and below TN )

• Up to 0.4RLn2 degrees of freedom are not
  transferred to the Q-P phase absobed into S0 (x
  ? xcr ) where a two quantum levels GS forms?

• Signs of low dimensional magnetic excitations are
  detected