We have measured the specific heat, C, of single-crystal UPt_3 in the superconducting regime as a function of temperature, T, and magnetic field, H, parallel to the c axis. We find that C(T) at fixed H<H_(c2) shows no evidence for different superconducting states. In contrast, our field-sweep data, C(H) at fixed T, have sharp changes in slope at H≊H_(c2)/2. The phase diagram deduced from these features agrees with neutron-scattering and torsional-oscillator results on the same samples. These thermodynamic measurements as a function of magnetic field constrain theories of exotic superconductivity in UPt_3

Bucher, E.

Ellman, B.

Rosenbaum, T. F.

Yang, J.

Caltech Authors - Main

VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 1990
Field-Dependent Specific Heat and Multiple Superconducting Phases in UPt3
B. Ellman, J. Yang, and T. F. Rosenbaum
The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Iiiinois 60637
E. Bucher
ATd'c T Bell Laboratories, Murray Hill, New Jersey 07974
and University of Konstanz, Konstanz 7750, West Germany
(Received 4 December 1989)
We have measured the specific heat, C, of single-crystal Upt3 in the superconducting regime as a
function of temperature, T, and magnetic field, H, parallel to the c axis. We find that C(T) at fixed
H (H, i shows no evidence for different superconducting states. In contrast, our field-sweep data, C(H)
at fixed T, have sharp changes in slope at H =H, i/2. The phase diagrain deduced from these features
agrees with neutron-scattering and torsional-oscillator results on the same samples. These thermo-
dynamic measurements as a function of magnetic-field constrain theories of exotic superconductivity in
UPt3.
PACS numbers: 74.70.Tx, 74.30.Ek
While many questions about the heavy-fermion system
UPti remain unanswered, there is a growing consensus
that the superconducting state is non-BCS, both from
the standpoint of the source of the attractive interaction
and in the nature of the resultant pairing. Early work'
suggested that magnetic fluctuations rather than pho-
nons supplied the effective attraction. This has been
borne out by recent inelastic neutron-scattering experi-
ments which reveal the existence of antiferromagnetic
(AFM) fiuctuations with energies comparable to the su-
perconducting transition temperature, T, =0.5 K. Fur-
thermore, elastic neutron and muon-spin-resonance
data' have shown that UPts is an antiferromagnet well
before it goes superconducting. This is particularly im-
portant since a wide variety of measurements (e.g. , ul-
trasound, torsional oscillator, and specific heat ) have
hinted at a rich phase diagram in the H-T plane for
T( T, . These phases are usually understood by postu-
lating that UPts undergoes d-wave pairing, giving a vec-
tor rather than a scalar order parameter, which, in turn,
can nontrivially couple to the AFM order parameter.
Thus, while the details are still uncertain, it is clear that
UPts supports a most unusual low-temperature quantum
state.
In an effort to better understand this state, we have
performed extensive magnetic-field and temperature-
dependent specific-heat measurements on a single-crystal
sample of UPts with the field parallel to the c axis.
While our zero-field temperature-sweep data show only
one, fairly broad transition, rather than two as in Ref. 7,
our fixed-temperature field sweeps display sharp features
located at field values Hi =H, 2/2. Interpretation of
these features as signatures of a thermodynamic phase
transition leads to an H-T phase diagram consistent
with those obtained by various nonthermodynamic
means, ' ' but in contrast to fixed-field, variable-
temperature calorimetry. ' The advantages of field
sweeps for locating the phase boundary prevail in a
variety of measurements and are discussed in greater de-
tail below.
The experiments were performed using standard
heat-pulse techniques in a helium dilution refrigerator
for 0.1~ T(0.6 K and O~H ~30 kOe. The ther-
mometer was a carbon chip whose magnetoresistance
made it necessary to recalibrate it at every magnetic-
field point during field sweeps. The heater was made of
Au/Cr and was field independent to better than 0.02%
up to 30 kOe. Our sample was a 0.15-g single crystal of
UPti oriented such that Hllc. Previous characterization
using microprobe and mass spectrometry revealed no
contamination greater than 10 ppm by weight. It is no-
table that our sample is a piece of one of the crystals
used in the neutron-scattering work of Ref. 9, making
direct magnetic and thermodynamic comparisons possi-
ble. For example, the data demonstrate that the evolu-
tion of the AFM order parameter seen in Ref. 9 can
occur in a sample with a single transition in the zero-
field specific heat.
We plot in Fig. 1 specific heat as a function of temper-
ature for several fixed fields. That the zero-field data
shows one transition can be seen best in a plot of C/T vs
T as in the inset of Fig. 1. The data in the normal state
give a Sommerfeld constant y 0.46+.0.005 3/molK2,
consistent with previous measurements. Interestingly,
while our broad transition may indicate an imperfect
sample, our residual linear specific heat yo C/T ) T-ti at
zero field (listed in Table I and considered a measure of
sample quality) is smaller than that of the samples in
Ref. 7, even though the latter samples have narrower
transitions. The increase in yo as H is increased is
presumably due to the presence of vortex-core quasipar-
ticles at zero temperature. Values of the entropy for
several field values are also tabulated. They are calcu-
lated using a C-T extrapolation at low temperatures,
1990 The American Physical Society 1569
VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 1990
0.3
0.7—
05
O
03
0.2
0
(D
0
3
0.1—
0.0 0.0
000 025
T (K)
V
vv 0.0
50
~ 7 5
v 30.0
0.2
T (K)
kj pp
p
t yp
g GQQ ooll)IP
J p
0 50 oe+
H (koe)
C3
C
C3
-=- 0.350 K
0
0+0 ~ OQO(g
(p0.3— ~ pe ~0.&00 K
0 ~
O O~0
OO
~V'g
~&@' VV~
p ~V
V'
'7
0.0 ~~g p'V
0.205 K
UPt3
H
I
c
FIG. 1. Specific heat, C, of UPt3 as a function of tempera-
ture, T, for various fixed magnetic fields, Hllc. Inset: C/T for
H 0 with a T extrapolation to T 0. No features indicative
of a phase-boundary crossing are visible for the temperature
sweeps in the superconducting regime (H ( 30 kOe).
—0.3 ~—— ——
0.0
0.140 K
H/Hc2
1.0
TABLE I. Summary of sample parameters derived from
specific-heat measurements C(T) in different magnetic fields.
H
(kOe)
Tc $0
(K) (3/mol K ')
TcS„(C/T)dT 5 yT
3/mol K) (3/mol K)
0.0 0.485
2.5 0.444
5.0 0.410
6.25 0.400
7.5 0.381
0. 1 1
0.14
0.19
0.17
0.22
0.214
0.188
0.174
0.163
0.158
0.226
0.202
0.190
0.182
0.176
and compared with the expected normal-state entropy
yT, (H). We find the zero-field entropy to be well con-
served. (The normal-state entropy, S„,is within 6% of
the superconducting value, S„.) With increasing field,
however, the agreement is less satisfactory, with the su-
perconducting entropy falling substantially below its
normal-state counterpart. This may reflect a field-
dependent variation in the power law describing the
low-temperature specific heat. It is difficult to quantify
this effect, however, due to the limited temperature
range over which it is possible to take data.
Phase diagrams for superconducting UPt3 derived
from ultrasonic attenuation and other data indicate that
our specific-heat temperature sweeps for H(30 kOe
cross a phase boundary at roughly H H, 2/2. However,
we observe no features indicative of such a crossing in
the data of Fig. 1. We therefore took data cutting the
H-T plane orthogonal to the temperature sweeps by
sweeping field at fixed T. A similar approach is neces-
sary in, for example, ultrasonic-attenuation measure-
ments, the reason probably being that the phase bound-
ary is more nearly parallel to the T axis in the tempera-
ture range of interest (see, for example, Ref. 5 and our
Fig. 4). Thus, the transition may be undetectable in a
FIG. 2. Specific heat of UPt3 as a function of magnetic field
at fixed temperatures. The specific heat and magnetic field
have been normalized by the normal-state specific heat, C„
and the upper critical field, 0,2, respectively. Distinct features
in the form of slope changes and maxima occur at field values
H=H, 2/2 at all T.
temperature sweep, but it would remain sharp and accu-
rately defined in a field sweep.
This expectation is borne out by the field-sweep data
of Fig. 2 in which the normalized specific heat
[C(H) —C„]/C„is plotted as a function of H/H, q, where
C„is the normal-state specific heat at T, (H) and H, 2 is
the calorimetric upper critical field, defined as the field
at which C(H) becomes constant. Note that, since
C„yTfor all H, the ordinate is essentially C/T
dS/dT. The data display several regimes of behavior.
At T(T„Cdrops monotonically to C„,displaying an
abrupt change or "kink" in the slope at H =H, 2/2. At
lower T (e.g. , our 0.205-K data), C(H) has a sharp
maximum at H, q/2. Finally, at still lower T, C rises and
then suddenly flattens off, again at a field value roughly
half that of the critical field.
In interpreting these data, it is important to realize
that an ideal conventional superconductor can show
somewhat similar qualitative behavior. For example, at
temperatures for which the normal-state specific heat is
below the zero-field value, C will rise with field as vor-
tices supply quasiparticles, then abruptly fall to C„at
H =H, 2. Then, if the transition is for any reason
smeared out, the peak in C is rounded off and the drop to
C„becomes more gradual. Several factors make it like-
ly, ho~ever, that the features observed in our field
sweeps are not due to conventional processes. For exam-
ple, the changes in slope and the maxima in our C(H)
1570
VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARcH 1990
0.100
I
UptG
- 0.17
20
15
0
M 0.095
3
- 0.15
(1)
O 10—
0.090 10 15 0
H (kOe)
I
10
0.13 ' 0.0 0.2 0.4
FIG. 3. Field-sweep specific heats C(H) at T 0.205 and
0.300 K, plotted without normalization. The change in slope at
HI is remarkably sharp in contrast to the broadened transition
in C(T) (Fig. I). Solid lines are linear fits to C(H) above and
below Hl.
are located relatively near H H, Q2 for all tempera-
tures, in comparison with those in conventional systems
which move quickly and smoothly to higher fractions of
H, q at lower T. The most convincing evidence that the
features in the field sweeps are nontrivial, though, is the
acuity of the features themselves. It is very difficult to
reconcile the sharpness of the slope changes and maxima
with any process that substantially smears out the transi-
tion at H, 2. Such a reconciliation is necessary, however,
if a conventional explanation is to account both for the
sharpness (arguing for sudden transitions) and for the
fact that the features occur so far from H, 2, an observa-
tion that argues for a strongly smeared transition. Thus,
we believe that the features in the specific-heat field
sweeps are signatures of the intrinsic superconducting
behavior of UPt3.
The aforementioned sharpness of C(H) at H H, 2 is
more readily visible in Fig. 3, where we plot C(H) at
T 0.205 and 0.300 K on separate scales without nor-
malization. The linear fits are of the form
Ci+IriiH, H (Hi,
C'"' 'C+ H H)H (I)
where Hi, the location of the slope change, was varied
along with Ci, C2, m i, and m2 so as to minimize the to-
tal g . The resulting values of Ht are plotted in the
phase diagram, Fig. 4, along with the calorimetric values
of H, 2 and the neutron results obtained on this sample
from Ref. 9 (region bracketed by the dashed lines). Er-
ror bars signify deviations of Hi sufficient to increase g
by 10%. Our results are seen to be consistent with the
neutron measurements, reinforcing the contention that
the features in C(H) are associated with changes in the
order parameters present in UPt3 below T,. The phase
boundary derived from C(H) agrees within error bars
FIG. 4. Phase diagram in the H-T plane. The circles are
calorimetric H, 2's and phase-boundary field values as deter-
mined from C(H). The dashed lines delineate the phase
boundary between superconducting states as determined from
neutron-scattering studies done on the same sample (Ref. 9).
Solid lines are guides to the eye.
with that determined from torsional-oscillator data for
T) 0.3 K, and lies approximately 20% higher at lower
T. We note that the thermodynamic upper critical field
also agrees with the torsional-oscillator H, 2 at high T,
and is about 10% greater in the low-temperature regime.
Scans of C(T) at fixed H perpendicular to c by Hassel-
bach et al. " also indicate the possibility of multiple su-
perconducting phases with the field in the basal plane.
Just as the neutron data probe the role of magnetism
in the phase diagram, our results clarify the thermo-
dynamic properties of the phases. Substantial work in
recent years has concentrated on calculating a free ener-
gy for UPt3 starting from Landau-Ginzberg-type expan-
sions. Two important examples are the work of Blount,
Varma, and Aeppli and that of the groups listed in Ref.
12. In the former, the antiferromagnetic order parame-
ter is resolved into transverse and longitudinal com-
ponents whose coupling with the superconducting order
parameter plays a central role in the energetics. Within
this framework, the "phase boundary" of Fig. 4 is not a
true thermodynamic phase transition with Hllc; rather, it
is a change in the magnitude of the transverse com-
ponent of the staggered magnetization. In contrast, the
calculations by the groups of Ref. 12 predict a true tran-
sition at Hi, even in the absence of magnetic order. The
role of the AFM order parameter is then to lower the
crystal symmetry in zero field from hexagonal to ortho-
rhombic and therefore to create two transitions.
The sharpness of the features in C(H) presented here
seem surprising in the context of a smooth crossover be-
tween different states and, at a minimum, must strongly
constrain the parameters which enter the Blount, Uarma,
and Aeppli free energy describing the coupling between
1571
VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARcH 1990
the magnetic and superconducting order parameters. If,
instead, the observed boundary marks a transition be-
tween distinct thermodynamic phases, then the question
arises as to the order of the transition. The supposition
is made by Joynt and co-workers' that the transition in
nonzero field is most likely first order. We observed no
evidence for hysteresis within 300 Oe in the peak posi-
tion Hi of C(H) at T 0.205 K when the field was
swept sequentially through Hi in both directions. How-
ever, this does not exclude the possibility of a first-order
transition since hysteresis depends as well on the degree
of nucleation and domain formation in the sample.
Our data also indicate an essentially linear dependence
of C on H at all T and for both H ~ Hi and H) Hi
(but with separate slopes). Although it is difficult to
evaluate physical quantities away from H, 2 and the
smearing of the transition in C(T) makes the problem
that much more complicated, the robust nature of this
result should also provide insight into the form of the ex-
citation spectrum in both regions of the H-T phase dia-
gram.
In conclusion, we have measured the field-dependent
specific heat C(H) of UPts with Hllc at a variety of tem-
peratures below the zero-field superconducting transition
temperature. Although the sample shows smooth behav-
ior in C(T) at fixed H, we find distinct features in C(H)
at fixed T. These thermodynamic features occur at
roughly the same field values where neutron-scattering
studies reveal a qualitative change in the field evolution
of the AFM order parameter.
We are indebted to G. Aeppli both for generously sup-
plying the sample and for valuable advice. We thank D.
Bishop, E. Blount, R. Joynt, and C. Varma for helpful
discussions. The specific-heat measurements, made at
The University of Chicago, were supported by NSF
Grant No. DMR 88-16817. J. Yang acknowledges sup-
port from NSF Grant No. DMR 83-51992. B. Ellman
submits this paper in partial fulfillment of the require-
ments for the Ph. D. degree in the Physics Department of
The University of Chicago.
i J. E. Hirsch, Phys. Rev. Lett. 54, 1317 (1985); K. Miyake
et al. , Phys. Rev. B 34, 6554 (1986); K. Machida and M. Kato,
Phys. Rev. Lett. 58, 1986 (1987); W. Putikka and R. Joynt,
Phys. Rev. B 37, 2372 (1988); C. J. Pethick and D. Pines,
Phys. Rev. Lett. 57, 118 (1986).
2G. Aeppli et al. , Phys. Rev. Lett. 58, 808 (1987).
~G. Aeppli et al. , Phys. Rev. Lett. 60, 615 (1988).
4D. W. Cooke et al. , Hyperfine Interact. 31, 425 (1986); R.
H. Heffner et al. , Phys. Rev. B 39, 11 345 (1989).
sV. Muller et al. , Phys. Rev. Lett. 58, 1224 (1987); A.
Schenstrom et al. , Phys. Rev. Lett. 62, 332 (1989); V. Muller
et al. , Solid State Commun. 57, 319 (1986); B. S. Shivaram et
al. , Phys. Rev. Lett. 56, 1078 (1986); B. S. Shivaram et al. ,
Phys. Rev. B 35, 5372 (1987); Y. J. Qian et al. , Solid State
Commun. 63, 599 (1987); D. J. Bishop et al. , Phys. Rev. Lett.
53, 1009 (1984).
sR. N. Kleiman et al. , Phys. Rev. Lett. 62, 328 (1989).
7R. A. Fisher et al. , Phys. Rev. Lett. 62, 1411 (1989).
sE. I. Blount, C. M. Varma, and G. Aeppli (to be pub-
lished).
9G. Aeppli et al. , Phys. Rev. Lett. 63, 676 (1989).
'oK. Hasselbach, L. Taillefer, and J. Flouquet (to be pub-
lished).
' 'K. Hassselbach et al. , Phys. Rev. Lett. 63, 93 (1989).
' D. W. Hess et aI. , J. Phys. Condensed Matter 1, 8135
(1989); S. K. Sundaram and R. Joynt, Phys. Rev. B 40, 8780
(1989); R. Joynt, Supercond. Sci. Technol. 1, 210 (1988); K.
Machida and M. Ozaki, J. Phys. Soc. Jpn. 58, 2244 (1989); M.
R. Norman (to be published).
1572


English

Field-dependent specific heat and multiple superconducting phases in UPt_3

VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 1990
Field-Dependent Specific Heat and Multiple Superconducting Phases in UPt3
B. Ellman, J. Yang, and T. F. Rosenbaum
The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Iiiinois 60637
E. Bucher
ATd'c T Bell Laboratories, Murray Hill, New Jersey 07974
and University of Konstanz, Konstanz 7750, West Germany
(Received 4 December 1989)
We have measured the specific heat, C, of single-crystal Upt3 in the superconducting regime as a
function of temperature, T, and magnetic field, H, parallel to the c axis. We find that C(T) at fixed
H (H, i shows no evidence for different superconducting states. In contrast, our field-sweep data, C(H)
at fixed T, have sharp changes in slope at H =H, i/2. The phase diagrain deduced from these features
agrees with neutron-scattering and torsional-oscillator results on the same samples. These thermo-
dynamic measurements as a function of magnetic-field constrain theories of exotic superconductivity in
UPt3.
PACS numbers: 74.70.Tx, 74.30.Ek
While many questions about the heavy-fermion system
UPti remain unanswered, there is a growing consensus
that the superconducting state is non-BCS, both from
the standpoint of the source of the attractive interaction
and in the nature of the resultant pairing. Early work'
suggested that magnetic fluctuations rather than pho-
nons supplied the effective attraction. This has been
borne out by recent inelastic neutron-scattering experi-
ments which reveal the existence of antiferromagnetic
(AFM) fiuctuations with energies comparable to the su-
perconducting transition temperature, T, =0.5 K. Fur-
thermore, elastic neutron and muon-spin-resonance
data' have shown that UPts is an antiferromagnet well
before it goes superconducting. This is particularly im-
portant since a wide variety of measurements (e.g. , ul-
trasound, torsional oscillator, and specific heat ) have
hinted at a rich phase diagram in the H-T plane forT( T, . These phases are usually understood by postu-
lating that UPts undergoes d-wave pairing, giving a vec-
tor rather than a scalar order parameter, which, in turn,
can nontrivially couple to the AFM order parameter.
Thus, while the details are still uncertain, it is clear that
UPts supports a most unusual low-temperature quantum
state.
In an effort to better understand this state, we have
performed extensive magnetic-field and temperature-
dependent specific-heat measurements on a single-crystal
sample of UPts with the field parallel to the c axis.
While our zero-field temperature-sweep data show only
one, fairly broad transition, rather than two as in Ref. 7,
our fixed-temperature field sweeps display sharp features
located at field values Hi =H, 2/2. Interpretation of
these features as signatures of a thermodynamic phase
transition leads to an H-T phase diagram consistent
with those obtained by various nonthermodynamic
means, ' ' but in contrast to fixed-field, variable-
temperature calorimetry. ' The advantages of field
sweeps for locating the phase boundary prevail in a
variety of measurements and are discussed in greater de-
tail below.
The experiments were performed using standard
heat-pulse techniques in a helium dilution refrigerator
for 0.1~ T(0.6 K and O~H ~30 kOe. The ther-
mometer was a carbon chip whose magnetoresistance
made it necessary to recalibrate it at every magnetic-
field point during field sweeps. The heater was made of
Au/Cr and was field independent to better than 0.02%
up to 30 kOe. Our sample was a 0.15-g single crystal of
UPti oriented such that Hllc. Previous characterization
using microprobe and mass spectrometry revealed no
contamination greater than 10 ppm by weight. It is no-
table that our sample is a piece of one of the crystals
used in the neutron-scattering work of Ref. 9, making
direct magnetic and thermodynamic comparisons possi-
ble. For example, the data demonstrate that the evolu-
tion of the AFM order parameter seen in Ref. 9 can
occur in a sample with a single transition in the zero-
field specific heat.
We plot in Fig. 1 specific heat as a function of temper-
ature for several fixed fields. That the zero-field data
shows one transition can be seen best in a plot of C/T vs
T as in the inset of Fig. 1. The data in the normal state
give a Sommerfeld constant y 0.46+.0.005 3/molK2,
consistent with previous measurements. Interestingly,
while our broad transition may indicate an imperfect
sample, our residual linear specific heat yo C/T ) T-ti at
zero field (listed in Table I and considered a measure of
sample quality) is smaller than that of the samples in
Ref. 7, even though the latter samples have narrower
transitions. The increase in yo as H is increased is
presumably due to the presence of vortex-core quasipar-
ticles at zero temperature. Values of the entropy for
several field values are also tabulated. They are calcu-
lated using a C-T extrapolation at low temperatures,
1990 The American Physical Society 1569
VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 1990
0.3 0.7—
05
O
030.2 0
(D
0
3 0.1—
0.0 0.0
000 025
T (K)
V
vv 0.0
50
~ 7 5
v 30.0
0.2
T (K)
kj ppp
t yp
g GQQ ooll)IP
J p0 50 oe+
H (koe)
C3
C
C3
-=- 0.350 K
0 0
+0
~ OQO(g (p0.3—
~ pe ~0.&00 K
0
~
O O~0OO
~V'g
~&@' VV~
p ~V
V'
'7
0.0 ~~g p'V
0.205 K
UPt3
H
I
c
FIG. 1. Specific heat, C, of UPt3 as a function of tempera-
ture, T, for various fixed magnetic fields, Hllc. Inset: C/T for
H 0 with a T extrapolation to T 0. No features indicative
of a phase-boundary crossing are visible for the temperature
sweeps in the superconducting regime (H ( 30 kOe). —0.3 ~—— ——0.0
0.140 K
H/Hc2
1.0
TABLE I. Summary of sample parameters derived from
specific-heat measurements C(T) in different magnetic fields.
H
(kOe)
Tc $0
(K) (3/mol K ')
TcS„(C/T)dT 5 yT
3/mol K) (3/mol K)
0.0 0.485
2.5 0.444
5.0 0.410
6.25 0.400
7.5 0.381
0. 1 1
0.14
0.19
0.17
0.22
0.214
0.188
0.174
0.163
0.158
0.226
0.202
0.190
0.182
0.176
and compared with the expected normal-state entropy
yT, (H). We find the zero-field entropy to be well con-
served. (The normal-state entropy, S„,is within 6% of
the superconducting value, S„.) With increasing field,
however, the agreement is less satisfactory, with the su-
perconducting entropy falling substantially below its
normal-state counterpart. This may reflect a field-
dependent variation in the power law describing the
low-temperature specific heat. It is difficult to quantify
this effect, however, due to the limited temperature
range over which it is possible to take data.
Phase diagrams for superconducting UPt3 derived
from ultrasonic attenuation and other data indicate that
our specific-heat temperature sweeps for H(30 kOe
cross a phase boundary at roughly H H, 2/2. However,
we observe no features indicative of such a crossing in
the data of Fig. 1. We therefore took data cutting the
H-T plane orthogonal to the temperature sweeps by
sweeping field at fixed T. A similar approach is neces-
sary in, for example, ultrasonic-attenuation measure-
ments, the reason probably being that the phase bound-
ary is more nearly parallel to the T axis in the tempera-
ture range of interest (see, for example, Ref. 5 and our
Fig. 4). Thus, the transition may be undetectable in a
FIG. 2. Specific heat of UPt3 as a function of magnetic field
at fixed temperatures. The specific heat and magnetic field
have been normalized by the normal-state specific heat, C„
and the upper critical field, 0,2, respectively. Distinct features
in the form of slope changes and maxima occur at field values
H=H, 2/2 at all T.
temperature sweep, but it would remain sharp and accu-
rately defined in a field sweep.
This expectation is borne out by the field-sweep data
of Fig. 2 in which the normalized specific heat
[C(H) —C„]/C„is plotted as a function of H/H, q, where
C„is the normal-state specific heat at T, (H) and H, 2 is
the calorimetric upper critical field, defined as the field
at which C(H) becomes constant. Note that, sinceC„yTfor all H, the ordinate is essentially C/T
dS/dT. The data display several regimes of behavior.
At T(T„Cdrops monotonically to C„,displaying an
abrupt change or "kink" in the slope at H =H, 2/2. At
lower T (e.g. , our 0.205-K data), C(H) has a sharp
maximum at H, q/2. Finally, at still lower T, C rises and
then suddenly flattens off, again at a field value roughly
half that of the critical field.
In interpreting these data, it is important to realize
that an ideal conventional superconductor can show
somewhat similar qualitative behavior. For example, at
temperatures for which the normal-state specific heat is
below the zero-field value, C will rise with field as vor-
tices supply quasiparticles, then abruptly fall to C„at
H =H, 2. Then, if the transition is for any reason
smeared out, the peak in C is rounded off and the drop to
C„becomes more gradual. Several factors make it like-
ly, ho~ever, that the features observed in our field
sweeps are not due to conventional processes. For exam-
ple, the changes in slope and the maxima in our C(H)
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VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARcH 1990
0.100
I
UptG
- 0.17
20
15
0
M 0.095
3
- 0.15
(1)
O 10—
0.090 10 15 0
H (kOe)
I
10
0.13 ' 0.0 0.2 0.4
FIG. 3. Field-sweep specific heats C(H) at T 0.205 and
0.300 K, plotted without normalization. The change in slope at
HI is remarkably sharp in contrast to the broadened transition
in C(T) (Fig. I). Solid lines are linear fits to C(H) above and
below Hl.
are located relatively near H H, Q2 for all tempera-
tures, in comparison with those in conventional systems
which move quickly and smoothly to higher fractions of
H, q at lower T. The most convincing evidence that the
features in the field sweeps are nontrivial, though, is the
acuity of the features themselves. It is very difficult to
reconcile the sharpness of the slope changes and maxima
with any process that substantially smears out the transi-
tion at H, 2. Such a reconciliation is necessary, however,
if a conventional explanation is to account both for the
sharpness (arguing for sudden transitions) and for the
fact that the features occur so far from H, 2, an observa-
tion that argues for a strongly smeared transition. Thus,
we believe that the features in the specific-heat field
sweeps are signatures of the intrinsic superconducting
behavior of UPt3.
The aforementioned sharpness of C(H) at H H, 2 is
more readily visible in Fig. 3, where we plot C(H) at
T 0.205 and 0.300 K on separate scales without nor-
malization. The linear fits are of the form
Ci+IriiH, H (Hi,C'"' 'C+ H H)H (I)
where Hi, the location of the slope change, was varied
along with Ci, C2, m i, and m2 so as to minimize the to-
tal g . The resulting values of Ht are plotted in the
phase diagram, Fig. 4, along with the calorimetric values
of H, 2 and the neutron results obtained on this sample
from Ref. 9 (region bracketed by the dashed lines). Er-
ror bars signify deviations of Hi sufficient to increase g
by 10%. Our results are seen to be consistent with the
neutron measurements, reinforcing the contention that
the features in C(H) are associated with changes in the
order parameters present in UPt3 below T,. The phase
boundary derived from C(H) agrees within error bars
FIG. 4. Phase diagram in the H-T plane. The circles are
calorimetric H, 2's and phase-boundary field values as deter-
mined from C(H). The dashed lines delineate the phase
boundary between superconducting states as determined from
neutron-scattering studies done on the same sample (Ref. 9).
Solid lines are guides to the eye.
with that determined from torsional-oscillator data forT) 0.3 K, and lies approximately 20% higher at lower
T. We note that the thermodynamic upper critical field
also agrees with the torsional-oscillator H, 2 at high T,
and is about 10% greater in the low-temperature regime.
Scans of C(T) at fixed H perpendicular to c by Hassel-
bach et al. " also indicate the possibility of multiple su-
perconducting phases with the field in the basal plane.
Just as the neutron data probe the role of magnetism
in the phase diagram, our results clarify the thermo-
dynamic properties of the phases. Substantial work in
recent years has concentrated on calculating a free ener-
gy for UPt3 starting from Landau-Ginzberg-type expan-
sions. Two important examples are the work of Blount,
Varma, and Aeppli and that of the groups listed in Ref.
12. In the former, the antiferromagnetic order parame-
ter is resolved into transverse and longitudinal com-
ponents whose coupling with the superconducting order
parameter plays a central role in the energetics. Within
this framework, the "phase boundary" of Fig. 4 is not a
true thermodynamic phase transition with Hllc; rather, it
is a change in the magnitude of the transverse com-
ponent of the staggered magnetization. In contrast, the
calculations by the groups of Ref. 12 predict a true tran-
sition at Hi, even in the absence of magnetic order. The
role of the AFM order parameter is then to lower the
crystal symmetry in zero field from hexagonal to ortho-
rhombic and therefore to create two transitions.
The sharpness of the features in C(H) presented here
seem surprising in the context of a smooth crossover be-
tween different states and, at a minimum, must strongly
constrain the parameters which enter the Blount, Uarma,
and Aeppli free energy describing the coupling between
1571
VOLUME 64, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARcH 1990
the magnetic and superconducting order parameters. If,
instead, the observed boundary marks a transition be-
tween distinct thermodynamic phases, then the question
arises as to the order of the transition. The supposition
is made by Joynt and co-workers' that the transition in
nonzero field is most likely first order. We observed no
evidence for hysteresis within 300 Oe in the peak posi-
tion Hi of C(H) at T 0.205 K when the field was
swept sequentially through Hi in both directions. How-
ever, this does not exclude the possibility of a first-order
transition since hysteresis depends as well on the degree
of nucleation and domain formation in the sample.
Our data also indicate an essentially linear dependence
of C on H at all T and for both H ~ Hi and H) Hi
(but with separate slopes). Although it is difficult to
evaluate physical quantities away from H, 2 and the
smearing of the transition in C(T) makes the problem
that much more complicated, the robust nature of this
result should also provide insight into the form of the ex-
citation spectrum in both regions of the H-T phase dia-
gram.
In conclusion, we have measured the field-dependent
specific heat C(H) of UPts with Hllc at a variety of tem-
peratures below the zero-field superconducting transition
temperature. Although the sample shows smooth behav-
ior in C(T) at fixed H, we find distinct features in C(H)
at fixed T. These thermodynamic features occur at
roughly the same field values where neutron-scattering
studies reveal a qualitative change in the field evolution
of the AFM order parameter.
We are indebted to G. Aeppli both for generously sup-
plying the sample and for valuable advice. We thank D.
Bishop, E. Blount, R. Joynt, and C. Varma for helpful
discussions. The specific-heat measurements, made at
The University of Chicago, were supported by NSF
Grant No. DMR 88-16817. J. Yang acknowledges sup-
port from NSF Grant No. DMR 83-51992. B. Ellman
submits this paper in partial fulfillment of the require-
ments for the Ph. D. degree in the Physics Department of
The University of Chicago.
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Field-dependent specific heat and multiple superconducting phases in UPt_3

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