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Chapter 1 Overview of heavy fermion systems
Chapter 1
Overview of Heavy Fermion Systems
In the 1970s and early 1980s, there was an intense excitement about the mixed-valence (MV) systems [1-4]. It was well known that many rare earth and actinide compounds exhibited MV phenomena in the sense that the valence of the quasilocalized f electron was approximately 3.67 instead of 4. In fact, different experimental results would yield different fractional numbers. The valence fluctuations in such systems are highly correlated and have many remarkable properties. Some of them exhibit Fermi-liquid behavior with a large effective mass at low temperatures but have free moments at high temperatures. The interest in these materials became more widespread after quite a few cerium and uranium compounds were classified as heavy fermion systems [5-7]. The heavy fermion systems, which are also sometimes referred to as heavy electron systems, were originally assumed to be a loosely defined collection of intermetallic compounds containing lanthanide (mostly Ce, Yb) or actinide (mostly U, Np) elements. Recently, other compounds such as quasi two-dimensional CeCoIn 5 and "Skutterdites" such as PrOsaSb12 have been shown to exhibit such behavior. I will present an overview of the "exotic" properties of these systems, the only common feature of which is that they have large effective mass (50-1000 times greater than the mass of a free electron) below a coherence temperature T*. The effective mass is estimated through the electronic specific heat. In general, for temperatures much smaller than the Debye temperature and Fermi energy, the specific heat C of a metal can be expressed as C T
-- ~ + fiT 2,
(!.1)
where 7 = VmkfkB2m*/3h2, Vmis the molar volume, m* the effective mass of the electron, kF the Fermi vector, and T is the absolute temperature. Here, 7 is the electronic contribution and fl the contribution of the phonons to the specific heat. It may be noted that there is an additional spin-fluctuation term 6T31nT in the specific heat of UPt 3 and UA12. For normal metals like copper or aluminum, 7 is of the order 1 mJ/mol K 2 at low temperatures. For heavy fermion systems, 7 abruptly increases to a very large value below the coherence temperature T*. In fact, since 7 increases abruptly at low temperatures for many materials, an arbitrary but generally accepted definition of heavy fermions [5] is those systems which has 7 > 400 mJ/f atom mol K 2. In addition, 7 is generally normalized to a mole of f atoms so that there can be a comparison between systems with different structure. Some of the other properties of heavy fermions include: (i) an enhanced
2
Chapter 1. Overviewof Heavy Fermion Systems
Pauli spin susceptibility indicating a large effective mass; (ii) the Wilson ratio is approximately one; (iii) a huge T 2 term in the electrical resistivity; and (iv) highly temperaturedependent de Haas-van Alphen oscillations amplitudes at very low temperatures. CeA13 was the first heavy fermion system discovered by Andres et al. [8]. They measured the specific heat and electrical resistivity of CeA13 and found that below 0.2 K, there was enormous magnitudes of the linear specific heat term C = 7 T (7 = 1620 mJ/mol K 2) and the T 2 term in p = A T 2 (A = 35 ktf~ cm/K2). Earlier, CeA13 had been cited as an example of MV compound. The seminal results of Andres et al. [8] for variation of C and p with temperature are presented in Figures 1.1 and 1.2. The intense interest in heavy fermion systems started with the discovery of superconductivity in CeCu~Si 2 by Steglich et al. [9]. The big surprise was that, up to that time, magnetism and superconductivity were considered to be contradictory phenomena. However, in CeCu2Si 2, the 4f electrons which are responsible for local magnetic moments at higher temperatures are also responsible for superconductivity below the critical temperature Tc. After the discovery of high-Tc superconductivity in 1986 and the realization that both high-Tc superconductors and heavy fermions are not only highly correlated electron systems but also exhibit "nearness" to magnetism at the superconducting phase, the interest in these systems grew rapidly. In general, the traditional BCS theory of superconductivity does not apply to either of these two "exotic" systems. The experimental results of Steglich et al. [9] are presented in Figures 1.3 and 1.4. The main part of Figure 1.3 shows the resistivity and the inset shows the low-field ac susceptibility of CeCu2Si 2 as a function of temperature. Figure 1.4 shows in a C / T versus T plot, the specific heat jumps of two other CeCu2Si 2 samples. One can see that these jumps do not look very profound.
0.10
. . . .
'
I
-
w
I
l
I
o
_
j
Iz
(..)
0.05
Ca
-
~
0
\
~' = t . 6 2
J/mole Kz"
0.01
9 .... 0
1 100
l
1 200
. I ..
1 300
T (inK)
Figure 1.1. Specific heat of CeA13 at very low temperatures in zero field (O, A) and in 10 kOe ([2). Reproduced with the permission of the American Physical Society from Ref. [8].
Chapter 1. Overview of Heavy Fermion Systems
3
1.1
1.0 o cl
nk o.g
0.8 w
0.7
l 2
0
,_
1 4
I 6 ( K2 )
T2
1 8
IO
x 10"3
I
Figure 1.2.
Electrical resistivity of CeA13 below 100 mK, plotted against T 2. Reproduced with the permission of the American Physical Society from Ref. [8].
9
9
9
E
I
!
9
HI
I
(~ ,,m-, e-
v
.2 .
B (Tesla)
B
,4,-,
9 0 0.1
L
=
1
e0 o- 1
(p)
x -2
0.5
0.6
0.7
0.8
0.9 1 1.1 temperature (K)
! __jt,o!x, | 1.2
1.3
1.4
1.5
Figure 1.3. Resistivity (main part) and low-field ac susceptibility (inset) of CeCu2Si 2 as a function of temperature. Arrows give transition temperatures T~p) = 0.60 __. 0.03 K and T~X)= 0.54 _+ 0.03 K. Reproduced with the permission of the American Physical Society from Ref. [9].
4
Chapter 1. Overviewof Heavy Fermion Systems .
2.0-
-~ Q
#
.
=
.
-
.
_
_
'I
i
4
.I
__
9
I
,1
|'
_
' L
'
.
,
I--
,
v Q~ 0 E
; "....
,
, ~~
04
TIK)
Ceeu2Si 2
0.6
/
TJc)
LaCu2Si 2 ,,u
0.3
,
_
t
1
.
3
r oK)
10
Figure 1.4. Molar specific heat of CeCu2Si2 at B = 0 as a function of temperature on logarithmic scale. Arrow marks transition temperature Tff = 0.51 ___0.04 K. Inset shows in a C/T versus T plot, the specific heat jumps of two other CeCu2Si2 samples. Reproduced with the permission of the American Physical Society from Ref. [9].
The main part of Figure 1.4 shows, on a logarithmic scale, the molar specific heat of CeCu2Si 2 at B = 0 as a function of temperature. The inset in Figure 1.4 shows in a C/T plot the specific heat jumps of two other CeCu2Si 2 samples, which do not look very profound. However, the specific heat jumps below the coherence temperature T*, which is characteristic of heavy fermion systems, is elegantly displayed when one plots C/T versus T 2. Stewart [5] has plotted C/T versus T 2 of non-superconducting single crystals of CeCu2Si 2 and a piece of a superconducting single crystal of UBe13. These results are reproduced in Figure 1.5 in which the line through UBe13 serves as a guide to the eye.
Chapter 1. Overviewof Heavy Fermion Systems
1000
i
I
n
I
I
'"1
5
|
-I
1
_ t_
I - I
800
9 CeCu2Si 2 II UBe 13 600
400
200
n__,-~v
0
~
l
20
t
, I
40
L
1__
t__
60
80
I
100
I
..... 120
T 2 (K 2) Figure 1.5. Specific heat of non-superconducting single crystals of CeCu2Si 2 (O) and a piece of superconducting crystal (n) of UBel3. The line through the UBel3 serves as a guide to the eye. Reproduced with the permission of the American Physical Society from Ref. [5].
The heavy fermions have a wide variety of ground states such as superconductors UBel3 [ 10] with non-Fermi-liquid properties in their normal state, and UPt 3 [ 11] which orders antiferromagnetically below the Neel temperature (TN), exhibits a heavy Fermiliquid state well below TN, and has unconventional superconductivity with a multicomponent superconducting parameter [ 12]. At very low temperatures, some heavy fermions are antiferromagnets with weak moments (CeA12 [13], U2Zn17 [14]), narrow-gap semiconductors (CeNiSn [15], Ce3Bi4Pt3 [16]) with quasiparticles having large effective
6
Chapter 1. Overviewof Heavy Fermion Systems
masses, while a few are Fermi liquids with no long-range ordering (CeA13 [8], CeCu 6 [17]). Some other heavy fermions superconductors like CeColn 5 [18] are quasi twodimensional. At present, approximately 50 heavy fermion systems have been discovered and there is no uniformity in their properties. As we have noted, some heavy fermions are Fermi liquids with no ordering while some others are non-Fermi liquids. Both magnetic and superconducting quantum critical points [ 19] have been observed in some of these systems. The only common feature is the large effective mass and the fact that they are highly correlated electron systems. It has also been found that CeRu2Si 2 exhibits metamagnetism [20-22] which has a wide variety of technological applications. The intense interest is due to the many unsolved theoretical problems and a variety of possible technological applications. It is interesting to note that the magnitude of the nuclear relaxation rate [23] in UBel3 and of the ultrasonic attenuation [24] in UPt 3 in the normal state are of the same order as ordinary metals. The thermal conductivity measurements in CeCu2Si 2 [9], UBeI3 [24], and UPt 3 [24] yield results similar to ordinary metals. The local-moment relaxation rate of rare earth impurities in UBel3 [25] is approximately of the same order as in materials with normal effective mass. Hence, it is easy to conclude that the large effective mass of heavy fermions is not due to band-structure renormalization. The theory of these systems lags behind the experiment although several powerful techniques have been applied. These include Bethe ansatz method, 1/N expansion method, renormalization group technique, and exotic theories such as Dynamical Mean Field theory and Quantum Critical Point. There are several theories for heavy fermion superconductivity in which there is widespread interest because of some similarities with theory of high-Tc superconductivity. There are a large number of review articles and proceedings of international conferences on heavy fermions. It is impossible to discuss in detail the properties of these various heavy fermion systems in a monograph. In fact, it requires a series of volumes (may be titled as Hand Book of Heavy Fermion Physics) to discuss all the properties of these systems, one of the most intense areas of research in Condensed Matter Physics. I will summarize the relevant experimental results and provide extensive references for each system. Similarly, I will briefly summarize the various theories including the recently proposed exotic ones. It may be mentioned that extensive research is making the field grow very rapidly. This monograph is intended to introduce researchers to heavy fermion systems as well as to serve as a reference book for researchers already working in this area.
References [1] R.D. Parks (ed.), Valence Instabilities and Related Narrow-Band Phenomena (Plenum, New York, 1977). [2] J.M. Lawrence, ES. Riseborough, and R.D. Parks, Rep. Prog. Phys. 44, 1 (1981). [3] L.M. Falicov, W. Hanke, and M.B. Maple (eds.), Valence Fluctuation in Solids (North-Holland, Amsterdam, 1981). [4] E Wachter and H. Boppart (eds.), Valence Instabilities (North-Holland, Amsterdam, 1982). [5] G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984).
References
7
[6] C.M. Varma, Rev. Mod. Phys. 48, 219 (1976), Comm. Solid State Phys. 11, 211 (1985). [7] C.M. Varma, J. Magn. Magn. Mater. 47-48, 606 (1985). [8] K. Andres, J.E. Graebner, and H.R. Ott, Phys. Rev. Lett. 35, 1779 (1975). [9] F. Steglich, J. Aarts, C.D. Bredl, W. Lieke. D. Meschede, W. Franz, and J. Schafer, Phys. Rev. Lett. 43, 1892 (1979). [10] H.R. Ott, H. Rudigier, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 50, 1595 (1983). [11] G.R. Stewart, Z. Fisk, J.O. Willis, and J.L. Smith, Phys. Rev. Lett. 52, 679 (1984). [12] R.A. Fisher, S. Kim, B.F. Woodfield, N.E. Phillips, L. Taillefer, K. Hasselbach, J. Flouquet, A.L. Georgi, and J.L. Smith, Phys. Rev. Lett. 62, 1411 (1989). [13] B. Barbara, J. Boucherle, J. Buevoz, M. Rossignol, and J. Schweizer, Solid State Commun. 24, 481 (1977). [14] H.R. Ott, H. Rudigier, P. Delsing, and J. Fisk, Phys. Rev. Lett. 52, 1551 (1984). [15] T. Takabatake, F. Teshima, H. Fuji, S. Nishigori, T. Suzuki, T. Fujita, Y. Yamaguchi, and J. Sakuri, Phys. Rev. B 41, 9607 (1990). [16] M.F. Hundley, P.C. Canfield, K.D. Thompson, and Z. Fisk, Phys. Rev. B 42, 6842 (1990). [17] G.R. Stewart, Z. Fisk, and M.S. Wise, Phys. Rev. B 30, 482 (1984). [18] C. Martin, C.C. Agosta, S.W. Tozer, H.A. Radovan, E.C. Palm, T.P. Murphy, and J.L. Sarrao, Phys. Rev. B 71, 020503 (2005). [19] P. Haen, J. Floquet, F. Lapierre, P. Lejay, and G. Remenyi, J. Low Temp. Phys. 67, 391 (1987); L. Taileffer, J. Flouquet, and G.G. Lonzarich, Phys. B 169, 257 (1991). [20] L. Puech, J.M. Mignot, P. Lejay, P. Haen, and J. Flouquet, J. Low Temp. Phys. 70, 237 (1988). [21] J.M. Mignot, J. Flouquet, P. Haen, F. Lapierre, L. Puech, and A. Voiron, J. Magn. Magn. Mater. 76--77, 97 (1988). [22] D.E. MacLauglin, C. Tien, W.G. Clark, K. Glover, M.D. Lan, Z. Fisk, J.L. Smith, and H.R. Ott, Phys. Rev. Lett. 53, 1833 (1984). [23] D.J. Bishop, C.M. Varma, B. Batlogg, E. Bucher, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 53, 1009 (1984). [24] D. Jaccard, J. Flouquet, P. Lejay, and J.L. Tholence, J. Appl. Phys. 57, 2719 (1985). [25] F. Gandra, S. Schultz, S.B. Oseroff, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 55, 2719 (1985).
Overview of Heavy Fermion Systems
In the 1970s and early 1980s, there was an intense excitement about the mixed-valence (MV) systems [1-4]. It was well known that many rare earth and actinide compounds exhibited MV phenomena in the sense that the valence of the quasilocalized f electron was approximately 3.67 instead of 4. In fact, different experimental results would yield different fractional numbers. The valence fluctuations in such systems are highly correlated and have many remarkable properties. Some of them exhibit Fermi-liquid behavior with a large effective mass at low temperatures but have free moments at high temperatures. The interest in these materials became more widespread after quite a few cerium and uranium compounds were classified as heavy fermion systems [5-7]. The heavy fermion systems, which are also sometimes referred to as heavy electron systems, were originally assumed to be a loosely defined collection of intermetallic compounds containing lanthanide (mostly Ce, Yb) or actinide (mostly U, Np) elements. Recently, other compounds such as quasi two-dimensional CeCoIn 5 and "Skutterdites" such as PrOsaSb12 have been shown to exhibit such behavior. I will present an overview of the "exotic" properties of these systems, the only common feature of which is that they have large effective mass (50-1000 times greater than the mass of a free electron) below a coherence temperature T*. The effective mass is estimated through the electronic specific heat. In general, for temperatures much smaller than the Debye temperature and Fermi energy, the specific heat C of a metal can be expressed as C T
-- ~ + fiT 2,
(!.1)
where 7 = VmkfkB2m*/3h2, Vmis the molar volume, m* the effective mass of the electron, kF the Fermi vector, and T is the absolute temperature. Here, 7 is the electronic contribution and fl the contribution of the phonons to the specific heat. It may be noted that there is an additional spin-fluctuation term 6T31nT in the specific heat of UPt 3 and UA12. For normal metals like copper or aluminum, 7 is of the order 1 mJ/mol K 2 at low temperatures. For heavy fermion systems, 7 abruptly increases to a very large value below the coherence temperature T*. In fact, since 7 increases abruptly at low temperatures for many materials, an arbitrary but generally accepted definition of heavy fermions [5] is those systems which has 7 > 400 mJ/f atom mol K 2. In addition, 7 is generally normalized to a mole of f atoms so that there can be a comparison between systems with different structure. Some of the other properties of heavy fermions include: (i) an enhanced
2
Chapter 1. Overviewof Heavy Fermion Systems
Pauli spin susceptibility indicating a large effective mass; (ii) the Wilson ratio is approximately one; (iii) a huge T 2 term in the electrical resistivity; and (iv) highly temperaturedependent de Haas-van Alphen oscillations amplitudes at very low temperatures. CeA13 was the first heavy fermion system discovered by Andres et al. [8]. They measured the specific heat and electrical resistivity of CeA13 and found that below 0.2 K, there was enormous magnitudes of the linear specific heat term C = 7 T (7 = 1620 mJ/mol K 2) and the T 2 term in p = A T 2 (A = 35 ktf~ cm/K2). Earlier, CeA13 had been cited as an example of MV compound. The seminal results of Andres et al. [8] for variation of C and p with temperature are presented in Figures 1.1 and 1.2. The intense interest in heavy fermion systems started with the discovery of superconductivity in CeCu~Si 2 by Steglich et al. [9]. The big surprise was that, up to that time, magnetism and superconductivity were considered to be contradictory phenomena. However, in CeCu2Si 2, the 4f electrons which are responsible for local magnetic moments at higher temperatures are also responsible for superconductivity below the critical temperature Tc. After the discovery of high-Tc superconductivity in 1986 and the realization that both high-Tc superconductors and heavy fermions are not only highly correlated electron systems but also exhibit "nearness" to magnetism at the superconducting phase, the interest in these systems grew rapidly. In general, the traditional BCS theory of superconductivity does not apply to either of these two "exotic" systems. The experimental results of Steglich et al. [9] are presented in Figures 1.3 and 1.4. The main part of Figure 1.3 shows the resistivity and the inset shows the low-field ac susceptibility of CeCu2Si 2 as a function of temperature. Figure 1.4 shows in a C / T versus T plot, the specific heat jumps of two other CeCu2Si 2 samples. One can see that these jumps do not look very profound.
0.10
. . . .
'
I
-
w
I
l
I
o
_
j
Iz
(..)
0.05
Ca
-
~
0
\
~' = t . 6 2
J/mole Kz"
0.01
9 .... 0
1 100
l
1 200
. I ..
1 300
T (inK)
Figure 1.1. Specific heat of CeA13 at very low temperatures in zero field (O, A) and in 10 kOe ([2). Reproduced with the permission of the American Physical Society from Ref. [8].
Chapter 1. Overview of Heavy Fermion Systems
3
1.1
1.0 o cl
nk o.g
0.8 w
0.7
l 2
0
,_
1 4
I 6 ( K2 )
T2
1 8
IO
x 10"3
I
Figure 1.2.
Electrical resistivity of CeA13 below 100 mK, plotted against T 2. Reproduced with the permission of the American Physical Society from Ref. [8].
9
9
9
E
I
!
9
HI
I
(~ ,,m-, e-
v
.2 .
B (Tesla)
B
,4,-,
9 0 0.1
L
=
1
e0 o- 1
(p)
x -2
0.5
0.6
0.7
0.8
0.9 1 1.1 temperature (K)
! __jt,o!x, | 1.2
1.3
1.4
1.5
Figure 1.3. Resistivity (main part) and low-field ac susceptibility (inset) of CeCu2Si 2 as a function of temperature. Arrows give transition temperatures T~p) = 0.60 __. 0.03 K and T~X)= 0.54 _+ 0.03 K. Reproduced with the permission of the American Physical Society from Ref. [9].
4
Chapter 1. Overviewof Heavy Fermion Systems .
2.0-
-~ Q
#
.
=
.
-
.
_
_
'I
i
4
.I
__
9
I
,1
|'
_
' L
'
.
,
I--
,
v Q~ 0 E
; "....
,
, ~~
04
TIK)
Ceeu2Si 2
0.6
/
TJc)
LaCu2Si 2 ,,u
0.3
,
_
t
1
.
3
r oK)
10
Figure 1.4. Molar specific heat of CeCu2Si2 at B = 0 as a function of temperature on logarithmic scale. Arrow marks transition temperature Tff = 0.51 ___0.04 K. Inset shows in a C/T versus T plot, the specific heat jumps of two other CeCu2Si2 samples. Reproduced with the permission of the American Physical Society from Ref. [9].
The main part of Figure 1.4 shows, on a logarithmic scale, the molar specific heat of CeCu2Si 2 at B = 0 as a function of temperature. The inset in Figure 1.4 shows in a C/T plot the specific heat jumps of two other CeCu2Si 2 samples, which do not look very profound. However, the specific heat jumps below the coherence temperature T*, which is characteristic of heavy fermion systems, is elegantly displayed when one plots C/T versus T 2. Stewart [5] has plotted C/T versus T 2 of non-superconducting single crystals of CeCu2Si 2 and a piece of a superconducting single crystal of UBe13. These results are reproduced in Figure 1.5 in which the line through UBe13 serves as a guide to the eye.
Chapter 1. Overviewof Heavy Fermion Systems
1000
i
I
n
I
I
'"1
5
|
-I
1
_ t_
I - I
800
9 CeCu2Si 2 II UBe 13 600
400
200
n__,-~v
0
~
l
20
t
, I
40
L
1__
t__
60
80
I
100
I
..... 120
T 2 (K 2) Figure 1.5. Specific heat of non-superconducting single crystals of CeCu2Si 2 (O) and a piece of superconducting crystal (n) of UBel3. The line through the UBel3 serves as a guide to the eye. Reproduced with the permission of the American Physical Society from Ref. [5].
The heavy fermions have a wide variety of ground states such as superconductors UBel3 [ 10] with non-Fermi-liquid properties in their normal state, and UPt 3 [ 11] which orders antiferromagnetically below the Neel temperature (TN), exhibits a heavy Fermiliquid state well below TN, and has unconventional superconductivity with a multicomponent superconducting parameter [ 12]. At very low temperatures, some heavy fermions are antiferromagnets with weak moments (CeA12 [13], U2Zn17 [14]), narrow-gap semiconductors (CeNiSn [15], Ce3Bi4Pt3 [16]) with quasiparticles having large effective
6
Chapter 1. Overviewof Heavy Fermion Systems
masses, while a few are Fermi liquids with no long-range ordering (CeA13 [8], CeCu 6 [17]). Some other heavy fermions superconductors like CeColn 5 [18] are quasi twodimensional. At present, approximately 50 heavy fermion systems have been discovered and there is no uniformity in their properties. As we have noted, some heavy fermions are Fermi liquids with no ordering while some others are non-Fermi liquids. Both magnetic and superconducting quantum critical points [ 19] have been observed in some of these systems. The only common feature is the large effective mass and the fact that they are highly correlated electron systems. It has also been found that CeRu2Si 2 exhibits metamagnetism [20-22] which has a wide variety of technological applications. The intense interest is due to the many unsolved theoretical problems and a variety of possible technological applications. It is interesting to note that the magnitude of the nuclear relaxation rate [23] in UBel3 and of the ultrasonic attenuation [24] in UPt 3 in the normal state are of the same order as ordinary metals. The thermal conductivity measurements in CeCu2Si 2 [9], UBeI3 [24], and UPt 3 [24] yield results similar to ordinary metals. The local-moment relaxation rate of rare earth impurities in UBel3 [25] is approximately of the same order as in materials with normal effective mass. Hence, it is easy to conclude that the large effective mass of heavy fermions is not due to band-structure renormalization. The theory of these systems lags behind the experiment although several powerful techniques have been applied. These include Bethe ansatz method, 1/N expansion method, renormalization group technique, and exotic theories such as Dynamical Mean Field theory and Quantum Critical Point. There are several theories for heavy fermion superconductivity in which there is widespread interest because of some similarities with theory of high-Tc superconductivity. There are a large number of review articles and proceedings of international conferences on heavy fermions. It is impossible to discuss in detail the properties of these various heavy fermion systems in a monograph. In fact, it requires a series of volumes (may be titled as Hand Book of Heavy Fermion Physics) to discuss all the properties of these systems, one of the most intense areas of research in Condensed Matter Physics. I will summarize the relevant experimental results and provide extensive references for each system. Similarly, I will briefly summarize the various theories including the recently proposed exotic ones. It may be mentioned that extensive research is making the field grow very rapidly. This monograph is intended to introduce researchers to heavy fermion systems as well as to serve as a reference book for researchers already working in this area.
References [1] R.D. Parks (ed.), Valence Instabilities and Related Narrow-Band Phenomena (Plenum, New York, 1977). [2] J.M. Lawrence, ES. Riseborough, and R.D. Parks, Rep. Prog. Phys. 44, 1 (1981). [3] L.M. Falicov, W. Hanke, and M.B. Maple (eds.), Valence Fluctuation in Solids (North-Holland, Amsterdam, 1981). [4] E Wachter and H. Boppart (eds.), Valence Instabilities (North-Holland, Amsterdam, 1982). [5] G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984).
References
7
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