4. NEW ELECTRONIC PHASE TRANSITIONS UNDER HIGH PRESSURE
4.2 Pressure-induced SC
Further, the applicability of this treatment should be examined for other HF and IV compounds.
Figure 93(Mori et al., 1999). The magnetic susceptibilitieswaandwcexhibit two anomalies atTN1andTN2. At low temperatures, the susceptibility increases slightly with decreasing temperature for both directions. This increase in susceptibility at low temperature may be due to a small amount of impurity.
Figure 94A and B shows the temperature dependence of the magnetic susceptibilitieswaandwcat various pressures. For all pressures,wcexhibits a sharp drop at the antiferromagnetic transition, which occurs just below the maximum in the susceptibility. The size of the anomaly due to the magnetic transition at TN1 decreases with increasing pressure. On the other hand, TN2 decreases linearly with increasing pressure and is no longer seen at high pressure inwa. The maximum ofwcthat indicates the Kondo temperature (Smith and Chu, 1967) increases with increasing pressure at a rate of dTmax/dP¼dTK/dP ¼0.046 K GPa1. The value of wc is about 4.5 times larger than wa at TN1¼36 K under ambient pressure. This anisotropy decreases by a factor of 2 at 1 GPa, suggesting that the magnetic anisotropy may decrease with increasing pressure.
Figure 95 shows the electrical resistivity r(T) for a- and c-axes as a function of temperature below 40 K under high pressures. At ambient pressure, the resistivity shows a sudden decrease near 35 K (¼TN1).
Moreover, the resistivity shows a small anomaly at 24 K (¼TN2). The anomalies atTN1 andTN2correspond to the magnetic phase transitions which were mentioned before. The anomaly nearTN1becomes less prom- inent with increasing pressure, corresponding to the pressure-induced decrease of the sublattice magnetization (Kawarazaki et al., 2000). TN1
decreases with increasing pressure and disappears above 1.0 GPa (Pc1).
TN2 also decreases with increasing pressure and disappears above 0.58 GPa (Pc2). At 0.4 GPa, on the other hand,r(T) shows a minimum
H= 1 T
H//a H//c TN2
TN1
00
–3c (⫻10 emu/mol) 5 21/c (⫻10mol/emu)
10 15 20
100 200 300 0
1 2 3 4 5 6
T (K)
P= 1 bar CeRh2Si2
FIGURE 93 Temperature dependence of the anisotropic magnetic susceptibility of CeRh2Si2at ambient pressure (Mori et al., 1999).
atTN214.5 K along thea- andc-axes. Further, as seen inFigure 96, the hysteresis is observed near TN2 along the a-axis, indicating that this transition is first order. Although no hysteresis is observed at ambient pressure, the results of quasielastic neutron scattering indicate that this transition is first order at ambient pressure (Graf et al., 1998; Severing et al., 1989).
The magnetic phase transition temperatures,TN1andTN2, were deter- mined by calculating the temperature derivatives of r(T), dr/dT. The pressure dependence of TN1 and TN2, plotted in Figure 97, is in good agreement with that determined by the measurements of thermal expan- sion (Figure 57; Honda et al., 1999). TN1 decreases gradually at low pressure and then abruptly near 1 GPa, which is the critical pressurePc1
where the transition disappears. Qualitatively, the phase boundary ofTN1
is interpreted on the basis of Doniach’s model for competing Kondo and RKKY interactions (Doniach, 1977). From the dHvA experiment under high pressure, the topology of the Fermi surface changes abruptly above 1.1 GPa (Araki et al., 2001). Since a new Fermi surface under high pressure
CeRh2Si2 CeRh2Si2
H//c-axis
H//a-axis A
B
0 GPa
0 GPa
c(10–3emu/mol)c(10–3emu/mol)
0.33 GPa 0.03 GPa 0.55 GPa 0.89 GPa 1.01 GPa
0.00 2.5 5.0 7.5 0 5 10 15 20
100 200
T (K)
300 0.47 GPa 0.86 GPa 1.00 GPa
FIGURE 94 Temperature dependence of the anisotropic magnetic susceptibility in CeRh2Si2at various pressures along (A)c- and (B)a-axes (Mori et al., 1999).
10 10 2 4 6 8 10 30 A
B
r(mΩcm)r(mΩcm)
10 20
T (K)
30 CeRh2Si2
i//c CeRh2Si2
i//a
TN2 TN2 TN1
TN1
TN1 TN1
0.86 GPa
0.86 GPa
0.59 GPa 0.40 GPa
2.3 GPa 0 GPa 0.40 GPa 0.59 GPa
2.3 GPa 0 GPa
40
0 10 20
T (K)
30 40
FIGURE 95 Temperature dependence of the electrical resistivity of CeRh2Si2along the a- andc-axes (Ohashi et al., 2003a).
TN2
i//a 0.40 GPa CeRh2Si2
i//c 10
2 3 4 5
5 10
T (K)
15 20
r(mΩcm)
FIGURE 96 The electrical resistivity of CeRh2Si2below 20 K at 0.4 GPa (Ohashi et al., 2003a).
is explained by the 4f-itinerant-band model, it is concluded that a change in the 4f electronic state from a 4f-localized Fermi surface to a 4f-itinerant Fermi surface occurs.
SC in CeRh2Si2was first discovered byMovshovich et al. (1996)for a polycrystalline sample in the pressure range abovePc2from 0.6 to 1.6 GPa.
The low-temperature resistivity for a high-quality single-crystal sample is shown inFigure 98(Settai et al., 2007). An indication of SC appears in the pressure region from 0.97 to 1.20 GPa; the zero resistivity is observed only atP¼1.05 and 1.06 GPa at 0.42 K. This implies that homogeneous bulk SC is realized in an extremely narrow pressure region aroundPc1.
The resistivityrfollows the FL formr(T)¼r0þAT2at low tempera- ture in a wide pressure range, wherer0is the temperature-independent residual resistivity andAis the constant.Figure 99shows theT2depen- dence ofr–r0below 2.3 GPa along a- and c-axes. TheT2dependence is observed even around Pc1 and Pc2. This is a characteristic feature of CeRh2Si2.
The values ofAfora- andc-axes are shown inFigure 100as a function of pressure. It is found thatA shows a maximum near 1.0 GPa for the current along thea-axis and near 0.86 GPa along thec-axis. This indicates that the valueAexhibits a peak at the critical pressurePc1between 0.86 and 1.0 GPa, pointing to strong spin fluctuations which corresponds to the pressure-induced QPT. It is noted that anisotropy for the value ofA changes drastically at Pc1. The inset in Figure 100 shows the pressure dependence of the ratioAa/Ac. A discontinuity is observed at the critical pressure Pc1, indicating that this transition changes the topology of the Fermi surface. On the other hand, no anomaly is observed nearPc2. Since this transition is first order, no fluctuations may exist and then the coeffi- cientAis not affected.
40
30
TN
(K) 20
10
00.0 0.5 1.0
0 3 2 1 0
–1 10 20
0.4 GPa
dr/dT(mΩcm/K)
T (K) TN2 TN1
30
P (GPa) TN2
Pc2 Pc1
TN1
1.5 2.0
FIGURE 97 Pressure dependence of the critical temperaturesTN1andTN2of CeRh2Si2. Broken lines are guide to the eye (Ohashi et al., 2003a).
Qualitatively, the pressure dependence ofAis consistent with that of Sommerfeld coefficient gof specific heat. As shown inFigures 101 and 102, g increases from 20 mJ/mol K2 at ambient pressure and passes through a broad maximum of 80 mJ/mol K2near 1.0 GPa. In the general case, the coefficientAofT2term in the resistivity and the linear specific heat coefficient g appear to have the relation A/g2 (Kadowaki and Woods, 1986; Miyake et al., 1989). For HF compounds, the value of A/g2¼1.0105mOcm/(mJ/mol K)2supports the viewpoint that the heavy mass is essentially due to the many-body dynamical effect between the lattice of local moments and the light conduction electrons. On the other hand, a small value A/g20.4 106 mO cm/(mJ/mol K)2 is observed in ordinary transition metals, which is due to a peculiar property of a single band.
Figure 103shows the value ofA/g2as a function of pressure. The value ofgis taken from the result of specific heat under pressure (Graf et al., 1997). The ratioA/g2is found to depend on pressure since the behavior of Ais different quantitatively from that ofg. The dashed line indicates the universal valueA/g20.4106mOcm/(mJ/mol K)2typical for ordi- nary transition metals (Miyake et al., 1989). At ambient pressure, we find
0.92 GPa
CeRh2Si2 J // [001]
1.28
1.16
1.01 1.06
1.05
0 0.5 1.0
0
0.2 0.4
Temperature (K)
r(mΩcm)
0.6
FIGURE 98 Low-temperature resistivity under pressure nearPc1of CeRh2Si2(Araki et al., 2003; Settai et al., 2007).
0.020
0.015
2.5
2.0
1.5
0.0 0.4 0.8 1.2
P (GPa)1.6 2.0 2.4
0.010
0.005 0.000
0.0 0.5 1.0 1.5
P (GPa) CeRh2Si2
a-axis c-axis A (mΩcm/K2)
2.0 2.5
FIGURE 100 Pressure dependence of the coefficient ofT2of CeRh2Si2along thea- and c-axes. Solid lines are guide to the eye. The inset shows the ratioAa/Acas a function of pressure (Ohashi et al., 2003a).
CeRh2Si2
CeRh2Si2 i//a
i//c A
B r–r0 (mΩcm)r–r0 (mΩcm)
0.86 GPa
0.86 GPa 1.2 GPa
1.2 GPa
2.3 GPa
2.3 GPa 0 GPa
0 GPa 120 100 80 60 T2 (K2)
T2 (K2) 40
20 0
120 100 80 60 40 20 0 0.0
0.0 0.5 1.0
1.0 0.8 0.6 0.4 0.2 1.5
FIGURE 99 T2dependenceofDr¼rr0of CeRh2Si2along thea- andc-axes (Ohashi et al., 2003a).
P (GPa)
0.0 400
200
0 0 Cm/T
10
1 10
T (K)
Cm/T (mJ/mol K2)
0 100 150 200
50
20 T
8
4
0
S
30 40
0.43 0.71 1.10 1.30 1.85
FIGURE 101 Magnetic specific heatCmCeRh2Si2divided by temperature as a function of temperature represented on a logarithmic scale at various pressures (Graf et al., 1997).
The lattice specific heat of CeRh2Si2was approximated by that of LaRhRuSi2(Calemczlk, 1990) and subtracted from the total specific heat to obtainCm. The inset is a plot ofCm/ TversusTfor CeRh2Si2at ambient pressure (open circles). The solid curve in the inset is the magnetic entropySm, calculated as the integral ofCm/T, and the dotted horizontal line corresponds toRln 2.
CeRh2Si2
0
0 20 40 60 80
0.5
0.5
1.0
1.0 P (GPa) g (mJ/mol K2)
P (GPa) TN(P)/TN(0)
0.0 0.5 1.0
1.5
1.5 2.0
FIGURE 102 Electronic specific heat coefficientgof CeRh2Si2as a function of pressure.
The inset is a plot ofTN(P) normalized to itsP¼0 value for two different samples.
In both cases, the dotted lines are guides to the eye (Graf et al., 1997).
A/g23.5106along thea-axis and 2.1106along thec-axis; these values are larger than that of transition metals by one order of magnitude.
This indicates that effect of magnetic scattering merges into theAand g value. At low pressures below 0.4 GPa, A/g2 decreases rapidly with increasing pressure, which comes from the vanishing of magnetic order.
Above 0.6 GPa, however,A/g2increases strongly with pressure below 0.9 GPa having a peak around 1 GPa. Moreover, it decreases again with pressure up to 1.7 GPa, at which pointA/g2because1.0106along thea-axis and 0.5 106along thec-axis, which are almost the same as that of transition metals. These data indicate that nonmagnetic scattering exists at the critical pressurePc1, and that a crossover from a HF state to an IV state occurs at high pressure.
Such behavior has been observed in the ferromagnetic U-compound UGe2(Oomi et al., 1998, 2000; Tateiwa et al., 2001). It shows a ferromag- netic ordering atTC¼52 K at ambient pressure. BothTCandTx, the latter is a characteristic transition temperature observed below TC, decrease with increasing pressure and become zero around 1.9 and 1.2 GPa, respec- tively. At low pressures below 1 GPa, the value ofA/g21.0105is near the empirical universal value for HF systems. Up to 1.1 GPa,A/g2 decreases to 1.0106, where the transition characterized by Txdisap- pears. Moreover,A/g2increases strongly again to 9106up to 1.3 GPa, where the value ofAshows a peak. These anomalies are related closely with the occurrence of SC. In other words, these anomalies are regarded as a precursor for the pressure-induced SC.
Next, the values of the residual resistivity r0 for a- and c-axes are shown in Figure 104 as a function of pressure. For both the a- and c- axes, r0 increases slightly as pressure increases and then decreases
00.0 1 2 3 4
0.5 1.0
P (GPa) c-axis
a-axis CeRh2Si2
A/g2 (10–6mΩcm2/(mJ/mol K)2)
1.5 2.0
FIGURE 103 Pressure dependence of the ratio ofA/g2of CeRh2Si2along thea- and c-axes. Broken lines are guide to the eye (Ohashi et al., 2003a).
exhibiting a minimum near 0.6 GPaPc2, where the magnetic phase transition disappears. The result indicates that the critical point nearPc2
affects the magnitude of r0. On the other hand, it is difficult to show whether an anomaly on r0exists near Pc1. In the case of ferromagnetic QCP, the anomaly of r0is expected to be observed at the critical point, while the less pronounced anomalies expected in the case of AF-QCP (Miyake and Narikiyo, 2002).
Figure 105shows the temperature dependence of the electrical resis- tivity over a wide temperature range at high pressures up to 8 GPa. No anomaly was detected in ther–Tabove 1.5 GPa, since magnetic ordering was suppressed completely. Instead, an inflection point atTmis observed in ther curve above 1.5 GPa.Tmshifts rapidly from 44 K at 1.5 GPa to higher temperature with increasing pressure.
T2dependence was observed at low temperature belowTm. The coeffi- cientAadecreases significantly with increasing pressure: the value ofAaat 8 GPa is smaller than the same at 1.5 GPa by two orders of magnitude.
Figure 106shows the pressure dependence of the value ofAaand 1/Tm2up to 8 GPa. It is found that the pressure versusAcurve is proportional to that for 1/Tm2, that is,Ais approximately proportional to 1/Tm2above Pc1. According to the theory ofYoshimori and Kasai (1983), the value ofAis proportional to 1/TK2. It has been confirmed that this relationA/1/Tm2 is valid in the HF materials (Kagayama et al., 1991a). These facts indicate thatTmis proportional toTK,Tm/TK(Yoshimori and Kasai, 1983).
TKis also estimated from the the temperature where magnetic suscep- tibility shows a peak. At ambient pressure,TKis approximately 35 K, and it increases with pressure (Mori et al., 1999). The slope of dTK/dPis calcu- lated to be 0.046 K GPa1from the result up to 1 GPa, while the large value is obtained, dTm/dP>15 K above 1.5 GPa in the present work. It is
3.0
2.5
2.0
1.5
1.00.0 0.5 1.0
P (GPa) r0 (mΩcm)
c-axis PC1
CeRh2Si2 PC2
a-axis
1.5 2.0 2.5
FIGURE 104 Pressure dependence of the residual resistivity ofr0of CeRh2Si2along the a- andc-axes. Solid lines are guide to the eye (Ohashi et al., 2003a).
considered that belowPc1, two kinds of interactions, RKKY interactions and the Kondo effect, are competing with each other, andTKis related not only to the Kondo effect but also to the RKKY interactions. Above Pc1, where the antiferromagnetic interactions disappear, TK is expected to become pressure sensitive as in a typical nonmagnetic HF compound.
TN1
Tm
Tm
1.5 GPa a-axis
CeRh2Si2
4.5 GPa
8.0 GPa 0 GPa
Tm
TN2
120
100
80
60
40
20
00 50 100 150
T (K)
r(mΩcm)
200 250 300
FIGURE 105 Temperature dependence of the electrical resistivity of CeRh2Si2along thea-axis up to 8 GPa (Ohashi et al., 2003a).
10–1
10–2
10–3
10–3
10–4
10–4
10–5
0 2 4 6
CeRh2Si2
1/Tm2 (K–2)
Aa (mΩcm/K2) a-axis
P (GPa)
8 10
FIGURE 106 Aaand 1/Tm2of CeRh2Si2as a function of pressure. Solid lines are guide to the eye to show that the curve ofAversusPis nearly parallel to 1/Tm2
versusPcurve (Ohashi et al., 2003a).
4.2.2 CePtSi2
In Ce-based heavy compounds, various interesting phenomena such as non-BCS SC and (NFL) behavior occur due to competition between the RKKY interactions and the Kondo effect. CePtSi2has the CeNiGe2-type orthorhombic layered structure (space groupCmcm), where the Ce and Pt–Si layers are stacked alternatively along the b-axis, with lattice con- stants a¼4.288, b¼16.718, and c¼4.238 A˚ (Geibel et al., 1992). The compound has the electronic specific heat coefficient gin the paramag- netic state of 600 mJ/mol K2(Geibel et al., 1990; Lee et al., 1990), and the characteristic Kondo temperatureTK of 3 K, indicating that CePtSi2is a typical HF compound. Below TN¼2 K, an AF transition occurs, as indicated by a downward slope in the electrical resistivity and a jump in the specific heat (Geibel et al., 1990, 1992; Lee et al., 1990; Nakano et al., 2009a; Oomi et al., 1999). TheTKand TNvalues indicate that the Kondo effect and RKKY interactions compete with each other in this compound.
From the high-pressure X-ray diffraction, the CeNiGe2-type structure is confirmed to be stable at least up to 6 GPa at room temperature. The lattice constants decrease continuously with increasing pressure. The pressure dependence of relative volumeV/V0is displayed inFigure 107.
The solid line is the result of least-squares fit to the Murnaghan’s equation (see Eq.(1)).B0andB00are evaluated to be 119 GPa and 3.0. These are in agreement with typical CK or HF compound such as CelnCu2
(B0¼90 GPa, and B00 ¼3.9). Absence of a significant anomaly in V/V0–Pcurve indicates that the crossover from the CK state or HF state (smallTK) to the IV state (largeTK) occurs gradually in CePtSi2.
Figure 108displays the temperature dependence of thermal expansion coefficientaat ambient pressure.aof CePtSi2is found to be the same as
1
0.98
0.96
0.94
0 2 4
P (GPa) V/V0
6 8
CePtSi2 RT
FIGURE 107 The pressure dependence of volume of CePtSi2at room temperature.
The solid line is least-squares fit to the first-order Murnaghan’s equation (Eq.(1)) (Oomi et al., 1994a).
that of LaPtSi2 above 60 K but shows a minimum around 22 K. The increase ina/Tat low temperature is well known as one of the character- istic behaviors of CK or HF compounds (Oomi et al., 1990a,c).ais gener- ally described at low temperature as described in Eq.(25). Thea/Tversus T2plot is shown inFigure 109. A large enhancement ina/Tis seen below 20 K. The value ofa/Tat 10 K is about 2.1107K2, which is much larger than that of LaPtSi2,0.05107K2, but is smaller than those of other HF materials such as CeAl3, 6 107K2(Kagayama and Oomi, 1993a). This indicates a large effective-mass (m*) enhancement or a high DOSs at the Fermi levelD(eF) due to the existence of 4f electrons; more- over,m*of CePtSi2may be smaller than that of CeA13. This result agrees
10 8 6 4 2
00 100 200
T (K) LaPtSi2
CePtSi2 a (10–6K–1)
300
FIGURE 108 Thermal expansion coefficientaof CePtSi2and LaPtSi2, as a function of temperature at ambient pressure (Oomi et al., 1993d).
CePtSi2
LaPtSi2 0
0 300
0.5 0.4 0.3 0.2 0.1 0
600 900
10 20
T (K) T2 (K2)
a/T (10–6 K–2)
30
FIGURE 109 a/TversusT2plot for CePtSi2and LaPtSi2(Oomi et al., 1993d).
with thegvalue of 600 mJ/mol K2, which is smaller than that of CeAl3, g¼1620 mJ/mol K2(Andres and Graebner, 1975).
Figure 110A shows the Dl/l of CePtSi2 below 10 K measured by a capacitance method (Nakano et al., 2009b).Dl/ldecreases monotonously with decreasing temperature down to 2.5 K. It increases rapidly below 2 K, due to antiferromagnetic ordering. Theain low-temperature region is shown inFigure 110B.aof 2106(K1) at 10 K is consistent with that measured by a strain gage method inFigure 108and is almost indepen- dent of the temperature down to 4 K.adecreases rapidly below 4 K and becomes negative below 2.4 K.abelow 1.2 K becomes6.5106(K1).
CePtSi2
10–5 A
B
0 20
–2 –1 0 1
T2 (K2)
40 60 80
–80 –6 –4 –2 0 2 4
a (10−6 Κ−1)
2 4 6
T (K)
8 10
aIT (10–6 K–2)
IIIdI
FIGURE 110 (A) Temperature dependence of the thermal expansion and (B) thermal expansion coefficientaof a polycrystalline sample of CePtSi2. Inset:a/Tas a function ofT(Nakano et al., 2009b).
Similar behaviors have been observed in other antiferromagnetic Ce-based HF compounds, such as CePd2Si2 (Dijk van et al., 2000), CeCu6(Tsujii et al., 2000), and Ce2RhIn8(Malinowski et al., 2003).
a/Tof CePtSi2at low temperatures is shown in the inset ofFigure 110B.
Below 10 K,a/Tincreases and shows a maximum atTmax¼3.6 K, which is almost consistent with TK rather than with TN. Thus, this maximum would appear to be due to the Kondo effect, indicating development of hybridization between the conduction band and the 4f orbital. Below 2.1 K,a/Trapidly decreases due to the antiferromagnetic ordering.
r of CePtSi2 polycrystalline sample under pressure up to 8 GPa is shown in Figure 111. At ambient pressure,r increases with decreasing temperature and shows maxima aroundT1¼5.4 andT2¼28.5 K. Above 2 GPa, these maxima merge into a single one which shifts to higher pressure with increasing pressure, indicating increase ofTK.
These two maxima are well known as characteristic features of Ce-based Kondo compounds, typical of interplay between the Kondo effect and CEF. At 8 GPa, FL like T2 dependence appears up to 50 K.
Similar behavior in ther–Tcurve at high pressure has been observed in other Ce compounds, which is interpreted as a pressure-induced cross- over from the CK state or HF state (smallTK) to the IV state (largeTK).
In the temperature range 2< T <10 K,rmagshows logTdependence as shownFigure 112(Oomi et al., 1998). The logTdependence seems to
CePtSi2
6 GPa 4 GPa 2 GPa
0 GPa
LaPtSi2 8 GPa
r(mΩcm)
Temperature (K)
0 50 100 150 200 250 3000
50
50 100 150 200
FIGURE 111 r–Tcurves of CePtSi2at various pressures up to 8 GPa.r–Tcurve of LaPtSi2
is also shown for comparison (Oomi et al., 1994a).
become wider and steeper as pressure increases up to 3 GPa. This is different from the normal FL behavior, suggesting that CePtSi2 is in NFL state between 1.5 and 4 GPa.
Figure 113 shows C/T under pressure up to 0.57 GPa. It shows a maximum aroundTN¼1.8 K due to the AFM order (Oomi et al., 1999).
This maximum shifts to lower temperature with a rate of dTN/ dP ¼1.5 K GPa1; thus, the AFM order is expected to disappear and QCP should occur around 1 GPa.
The details ofrhave been measured down to 50 mK under pressure by Nakano et al. (2009a), and these are displayed inFigure 114A. At ambient pressure,rincreases with decreasing temperature from 300 K and shows two maxima at 23 K (¼T2) and 7 K (¼T1). This is almost consistent with that shown inFigure 111.rshows a sharp decrease atTN¼1.8 K due to the AF transition followed by FL likeT2dependence below 1 K.
The pressure clearly changes the overall features inr. TN decreases with increasing pressure up to 0.79 GPa, as shown in the inset of
logT
1.5 GPa
rmag (mΩcm)
2 100 120 140 160 180
0.5 1 1.5
5 10 20
T (K) 2.0 GPa
3.0 GPa
4.0 GPa CePtSi2
FIGURE 112 Thermagas a function of temperature on the logarithmic scale. The solid lines indicate logTdependence (Oomi et al., 1999).