4. NEW ELECTRONIC PHASE TRANSITIONS UNDER HIGH PRESSURE
4.1 Pressure-induced crossover
In this section, as examples of pressure-induced crossover, we will show some data on electrical resistivity under pressure for three HF materials CeAl3, CeAl2, and CeInCu2. Brief discussions about these novel electronic states induced by high pressure will be introduced.
4.1.1 CeAl3
The temperature dependence of the electrical resistivity r of CeAl3 at various pressures up to 8 GPa and r of LaAl3 at ambient pressure are shown inFigure 76(Kagayama and Oomi, 1996). The electrical resistivity of LaAl3is similar to the ordinary nonmagnetic metal; it varies linearly
CeAl3
0 GPa
1.5 GPa
4.5 GPa 3 GPa
6 GPa 8 GPa
LaAl3
00 50 100 150
r (mWcm)
200 250
50 100 150
T (K)
200 250 300
FIGURE 76 The electrical resistivity,r, of CeAl3at high pressures as a function of temperature.rof LaAl3at ambient pressure is also shown for comparison (Kagayama and Oomi, 1996).
with temperature above 100 K without any anomalies. On the other hand, rof CeAl3at ambient pressure increases logarithmically with decreasing temperature until it reaches a maximum at 35 K and has a shoulder near 6 K. This behavior is due to the Kondo scattering on a thermally popu- lated level split by CEF (Cornut and Coqblin, 1972). With increasing pressure, the peak and the shoulder merge into one peak, which is shifted toward higher temperatures. Ther–Tat 8 GPa becomes similar to that of LaAl3. This result is interpreted as a pressure-induced crossover in the electronic state of CeAl3from a small-TKHF state to a large-TKIV state associated with an increase in the hybridization between the conduction band and 4f electrons.
In order to get the temperature-dependent 4f magnetic contribution rmag, ther(LaAl3) is assumed to be pressure-independent phonon part of CeAl3 and subtracted from the r(CeAl3) at various pressures, rmag¼ r(CeAl3)r(LaAl3).Figure 77illustrates thermagas a function of logT.
The maximum temperature Tmax in the rmag increases with increasing pressure. Since Tmax is roughly proportional to the TK, the pressure
0 10 20 40 60 80 100 rmag (mWcm)
120 140 160 180
CeAl3
0 GPa
3 GPa
6 GPa 8 GPa 1.5 GPa
4.5 GPa 200
100 T (K)
FIGURE 77 The magnetic part of the electrical resistivity,rmag, of CeAl3as a function of logTat various pressures (Kagayama and Oomi, 1996).
dependence of TK may be inferred. On the other hand, the logarithmic dependence ofrmagon temperature is observed in the wide range above Tmax. The negative slope becomes steeper at higher pressure reflecting the strong Kondo scattering with large enhancement ofTKat high pressure.
In order to examine the T2dependence in rmagat low temperature, rmagis plotted as a function ofT2inFigure 78up to 17 K forP1.5 GPa and to 80 K forP3 GPa. Above 0.8 GPa, theT2dependence is clearly observed in these temperature ranges as shown by straight lines. As pressure increases, the slope decreases and the temperature range showingT2dependence becomes wider.
The coefficientAof theT2term is shown inFigure 79as a function of pressure. The value ofAis reduced by three orders of magnitude com- pared to that at ambient pressure, which was reported previously to be 35mOcm/K2(Andres et al., 1975). The rapid decrease in the magnitude of Ais explained by the enhancement ofTKby applying pressure, which is
100 90 80 70 60 50 40 30 20
20
15
10
5
0 50 100
0 1000 2000 3000
rmag (mWcm)
T2 (K2)
4000 5000 6000 0.8 GPa
4.5 GPa 6 GPa
8 GPa 3 GPa
1.1 GPa
1.5 GPa
150 200
CeAl3
250
FIGURE 78 rmagof CeAl3as a function ofT2(Kagayama and Oomi, 1996).
consistent with the increase inTmaxbecause the coefficientAis inversely proportional to TK2. These results are qualitatively consistent with the pressure dependence of specific heat coefficientgas mentioned inSection 3.3.1. The value of g decreases significantly with increasing pressure.
Considering thatgis proportional toTK1, the large decrease ofgcorre- sponds to an enhancement ofTKby applying pressure.
4.1.2 CeAl2
Figure 80A shows the temperature-dependent electrical resistivity r(CeAl2) at various pressures and nonmagnetic LaAl2,r(LaAl2), at ambi- ent pressure for comparison. At ambient pressure, r(CeAl2) decreases with decreasing temperature and shows a peak around 5 K and a shoul- der around 60 K. The peak shifts to lower temperature and disappears above 2.5 GPa. On the other hand, the shoulder is enhanced at high pressure becoming a broad peak centered around 50 K at 3 GPa. The broad peak becomes less prominent by applying higher pressure. The magnetic part of the electrical resistivityrmagwas estimated by subtract- ing phonon contribution using ther(LaAl2),rmag¼r(CeAl2)r(LaAl2), where we assume that the phonon part ofr(CeAl2) is approximated by r(LaAl2).rmagcurves are shown inFigure 80B as a function of tempera- ture in the logarithmic scale.rmagat ambient pressure has two maxima due to combination of Kondo effect and CEF splitting, which were observed up to 2.4 GPa. These peaks become a single peak above 2.5 GPa through a shoulder in ther around 2.5 GPa. The temperatures of two maxima are defined asT1andT2. InFigure 80B,T1,T2, and the Ne`el temperature TN at ambient pressure are shown by arrows. rmag at T2
increases below 2.5 GPa with pressure, while it decreases above 2.5 GPa.
1.5
1.0
0.5
0.00 2 4
P (GPa) A (mWcm/K2)
6
CeAl3
8
FIGURE 79 The coefficientAof theT2term of CeAl3as a function of pressure (Kagayama and Oomi, 1996).
Figure 81 shows the pressure dependence of T1, T2, and TN. TN decreases with increasing pressure. Extrapolation of the data results in a critical pressure at around 3 GPa for a disappearance of the magnetism.
r(mWcm)rmag (mWcm)
1.5 GPa
0 GPa 4.5 GPa 3.0 GPa 2.5 GPa
2.4 GPa
0 GPa TN
T1
T2
m2
m1
1.5 GPa
4.5 GPa
6.5 GPa
8.0 GPa 2.5 GPa
3.0 GPa B
CeAl2
2.4 GPa
6.5 GPa
8.0 GPa
LaAl2
CeAl2
T (K) 00
40 80 120
0 40 80 120
100 200
1 10 100
T (K)
300 A
FIGURE 80 (A) Temperature dependence of the electrical resistivity of CeAl2under high pressure. (B) The temperature dependence ofrmag.T1,T2, andTNat ambient pressure were shown by arrows (Miyagawa et al., 2008).
T1 decreases with pressure and has a minimum value at 2.1 GPa. At higher pressures,T1begins to increase strongly indicating thatT1merges toT2. It is interesting to note thatT2decreases with pressure below 3 GPa but begins to increase above 3 GPa. Similar behavior forT1and T2 has been reported in other Ce compounds such as CeCu2Si2(Holmes et al., 2004) and CeCu2Ge2(Jaccard and Holmes, 2005).
Figure 82 shows the detailed behavior of r below 15 K. Electrical resistivity changes significantly as pressure increases:rat 2.1 GPa has a
T2
TN, T1,T2 (K)
P (GPa)
0 2 4 6
0 5 10 100 200
T1
TN
FIGURE 81 T1,T2, andTNof CeAl2as a function of pressure. The closed circles show the pressure dependence ofT1andT2. The open squares showTNas a function of pressure.
The dashed line is guide to the eye (Miyagawa et al., 2008).
CeAl2
2.1 GPa
T (K) 00
50 100
5
r(mWcm)
10 15
2.5 GPa 2.6 GPa
2.7 GPa
FIGURE 82 r(T) of CeAl2under high pressure down to 50 mK (Miyagawa et al., 2008).
broad peak around 4 K followed by a rapid decrease below that due to magnetic ordering, but as pressure increases, the peak disappears, and then r above 2.6 GPa shows a smooth increase with temperature. The electronic state near QCP is different from the normal FL behavior showingT2dependence in the electrical resistivity. Here, it was attempted to fit the temperature dependence of the electrical resistivity to the fol- lowing equation:
rmag¼r0þAnTn; (35) wherer0,An, andnare the residual resistivity, the coefficient, and expo- nent related to electronic state, respectively.
rmagbelow 2 K is shown inFigure 83. Fitting was carried out in the temperature range 50 mK < T <2 K. Solid curves inFigure 83show the result of fitting, in which these curves reproduce well the experimental data.
Figure 84shows the pressure dependence ofr0and n.r0has a peak around 2.7 GPa.nis 2 below 2 GPa but around 2.7 GPa, it is 1.3–1.4, which significantly deviates from the normal FL behavior (n¼2). Judging from these facts, it is concluded that CeAl2shows the QPT around 2.7 GPa. This indicates that the electronic state around 2.7 GPa is the so-called NFL state. Further,nbecomes 2 above 3 GPa (Miyagawa et al., 2008), indicating that the FL behavior is recovered.
Figure 85shows the pressure dependence of the electrical resistivity at several temperatures in the range 2.6< T <300 K, which was obtained from the results shown inFigure 80A. For all the isothermalr–Pcurves,
CeAl2
2.7 GPa 2.6 GPa (n= 1.3)
(n= 1.2) (n= 1.8)
(n= 2.1) 2.5 GPa
2.1 GPa
T (K) 00
20 40 60
1 2
rmag (mWcm)
FIGURE 83 Temperature dependence ofrmagof CeAl2below 2 K at high pressure. Solid curves show the results of fitting (Miyagawa et al., 2008).
rincreases with pressure followed by a maximum,rmax, atP¼Pmaxand Pmaxincreases with temperature. We also observed similar behavior for CeAl3 as shown in Figure 11 in Section 2.1.3.1. rmax is clear at low temperature (T < 35 K) but the maximum become broad at high tempera- ture (T>85 K). It has been reported that there is a crossover induced by pressure from weak Kondo (smallTK) to strong Kondo (largeTK) regime, which is largely different from the first-orderg–aphase transition in Ce metal (Barbara et al., 1986). In such a sense, the peaks observed in Figure 85 may correspond to the crossover in the electronic state of CeAl2.Pmax–T curve indicates that the Kondo temperatureTKincreases with increasing pressure. Pmax is plotted in Figure 86. Pmax increases gradually up to 2.6 GPa, and then rapidly above the pressure whereT2
also shows a similar change and then crosses room temperature at 5.5 GPa.
In summary, TK increases with pressure while TN is decreased by applying pressure and expected to disappear around 2.7 GPa. Since the coefficient of theT2term diverges around 2.7 GPa, the magnetic phase
P (GPa) 12.0
2
n
5 10 15 20
2.5 3.0
r0(mWcm)
FIGURE 84 r0andnof CeAl2as a function of pressure. The solid curves are drawn as guide to the eye (Miyagawa et al., 2008).
boundary exists at 2.7 GPa where NFL behavior is observed in r(T).
Above 3 GPa, the pressure effect on T2 changes and FL behavior is recovered.
CeAl2
290 K
85 K
35 K 18 K 8 K 2.6 K 0
0 40 80 120
2 4 6 8
P (GPa)
r(mWcm)
FIGURE 85 Pressure dependence of the electrical resistivity of CeAl2at several temperatures. The filled circles are obtained by isobaric temperature-dependent data at several pressures. The thick line shows isothermal pressure dependence of the electrical resistivity at 290 K (Miyagawa et al., 2008).
Pmax
T2
T1
300
200
100
00 2 4 6 8
P (GPa)
T (K)
CeAl2
FIGURE 86 T–Pphase diagram of CeAl2obtained from data discussed inSection 4.1.2 (Miyagawa et al., 2008). The filled circles show the pressurePmaxshowing the maximum ofr–Pcurves as shown inFigure 85. The open circles show the pressure dependence of T1andT2for comparison withPmax.
4.1.3 CeInCu2
The temperature dependence of electrical resistivity r of CeInCu2 at various pressures up to 8 GPa is shown inFigure 87. At ambient pressure, rincreases gradually with decreasing temperature, reaches a maximum around 27 K, and then decreases rapidly on further cooling. This behavior is similar to those of typical HF compounds (Brandt and Moshchalkov, 1984). The residual resistivityr0decreases rapidly with increasing pres- sure, which is different from the pressure dependence of r0for CeCu6
(Kagayama et al., 1991a). The temperature Tmax of the resistivity maxi- mum increases with increasing pressure and then the maximum in the r–Tcurve, which is a characteristic of CK system (CKS), is not observed above 4.5 GPa in the temperature range ofFigure 87. Since a large increase in Tmax corresponds to an increase in TK, as will be mentioned in the following section, the change in the overall behavior in ther–Tcurve in Figure 87implies a crossover from an HF at low pressures to an IV state at high pressures (Lavagna et al., 1983). Such crossover has also been observed in some cerium and uranium compounds (Aronson et al., 1989; Bellarbi et al., 1984; McElfresh et al., 1990; Ponchet et al., 1986;
Yomo et al., 1988).
0 GPa
2 GPa 3 GPa
4.5 GPa 6 GPa
8 GPa
CeInCu2
00 20 40 60 80 100 120 140
r(mΩcm)
50 100 150 T (K)
200 250 300
FIGURE 87 Electrical resistivityrof CeInCu2at high pressures as a function of temperature (Kagayama et al., 1992a).
In order to examine theT2dependence in ther, which is characteristic of a FL, r r0 is plotted as a function of T2 as shown in Figure 88.
rr0 has a T2 dependence at low temperatures. As the pressure increases, the temperature range in which r shows the T2 dependence becomes wider. The values of A and Tmax are shown in Figure 89 as functions of pressure. It is seen thatAdecreases rapidly with increasing pressure below about 3 GPa, and the rate of decrease becomes small at high pressures. On the contrary, Tmax increases rapidly with increasing pressure;Tmaxis 27 K at ambient pressure, but it increases to 300 K around 3.8 GPa.
It is well known thatTmaxis roughly proportional toTKand that the value ofAdepends onTKasA/TK2(Ponchet et al., 1986; Wire et al., 1984; Yoshimori and Kasai, 1983). Thusffiffiffiffi Tmaxis inversely proportional to pA
. To examine this relation, the value of 1= ffiffiffiffi pA
is plotted as a function of TmaxinFigure 90. The value of 1= ffiffiffiffi
pA
shows a linear dependence onTmax. The rapid decrease in A observed in Figure 89 arises from the large increase inTmaxorTKat high pressures, which is induced by the increase in the hybridization between 4f electrons and the conduction band. This indicates thatAandTmaxare dominated by a single energy scaleTK.
1.0 GPa 0.5 GPa
3 GPa
6 GPa 8 GPa 4.5 GPa
1.5 GPa
2.0 GPa
0 20
8 6 4 2 0
4
2
0 500 1000
T2 (K2)
1500 0
40
CeInCu2 r-r0(mΩcm)
FIGURE 88 r–r0versusT2of CeInCu2(A) below 2 GPa and 7 K and (B) above 1.5 GPa and below 40 K: the solid lines show theT2dependence (Kagayama et al., 1992a).
Next, Gru¨neisen parameterG forTmax is discussed. Extremely large values ofG, as high as 50, have been reported for HF compounds (Visser et al., 1990), while it is of the order of 1 for normal metals. TheGforTmax, G(Tmax) is defined as:
ln Tmax Tmaxð Þ0
¼ Gln V
V0 : (36)
0.8 B
A
0.6
A (mΩcm/K2)
CeInCu2
0.4
0.2
0.00 2 4
P (GPa) 0
0 200 400 600 800 1000 1200
1 2 3 4 5 6 7 8 9
P (GPa) Tmax (K)
6 8
CeInCu2
FIGURE 89 (A) The resistivity–maximum temperatureTmaxof CeInCu2, (B) the coeffi- cientAof theT2term as a function of pressure (Kagayama and Oomi, 1996).
CeInCu2 CeCu6
200 100
Tmax (K) 00
5 10
(mΩcm/K2)1 √A
1–2
FIGURE 90 1= ffiffiffi pA
plotted againstTmaxfor CeInCu2and CeCu6(Kagayama and Oomi, 1993b).
SinceTmax TK, this equation is the same as Eq.(3)inSection 2.Figure 91 shows a plot of ln[Tmax(P)/Tmax(0)] as a function of volume. The pressure coefficient of relative change ofTmax,@(lnTmax)/@Pis about 0.69 GPa1. G is determined to be 62.5 by substituting the observed value of k, 11103GPa1(Kagayama et al., 1990). InTable 1, we summarize the values of@(lnTmax)/@P,k, andG, for CeInCu2, CeCu6, CeA13, and CePd3. TheGvalues of three HF compounds are larger than those of IVS com- pounds such as CePd3,G¼9 (Oomi et al., 1990a). Large values ofGhave been observed for other HF compounds including uranium compounds (Visser et al., 1990). Taking into account these results, the largeGvalue seems to be one of the characteristics of HF compounds.
Figure 92 illustrates the relative change of the TK or the coefficient A/Tmax2of CeInCu2, CeAl3, and URu2Si2(Kagayama and Oomi, 1995, 1996; Kagayama et al., 1994b) as functions of volume. The origin of this plot is adjusted appropriately using theGvalue at ambient pressure, and the rectangle shows the range of the measurements ofTKandV, that is, 0 < P <8 GPa. The range of the horizontal axis of each rectangle depends on the magnitude of bulk modulus B0; it becomes wider with smaller value ofB0. The slope of this curve is the Gru¨neisen parameter at non- ambient pressure. It is seen that the magnitude of the slope becomes smaller as pressure increases. In other words, the volume-dependent Gru¨neisen parameter decreases with increasing pressure, which corre- sponds to a crossover induced by pressure. The magnitudes ofG(Tmax) at ambient pressure are summarized inTable 4. All observed points for the three HF materials fall on a ‘‘universal’’ curve, which is depicted using a solid curve inFigure 92. This fact suggests that the present treatment is a useful tool to analyze quantitatively the pressure dependent r curve.
CeInCu2
–In(V/V0) 0.00
In[Tmax(P)/Tmax(0)]
0 1 3
2 4
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
FIGURE 91 ln[Tmax(P)/Tmax(0)] of CeInCu2as a function of volume (Kagayama et al., 1994c).
Further, the applicability of this treatment should be examined for other HF and IV compounds.