Magnetic susceptibility

CeCu ,(tore)

S. H. LIU

2. Review of key normal state properties

2.2. Magnetic susceptibility

The low-temperature part of the magnetic susceptibility, z(T), of CeA13 is shown in fig. 2 (Andres et al. 1975). Below 0.1 K it is clear that the susceptibility approaches a constant value, Z(0)-~36 × 1 0 - a e m u / ( m o l G ) , which is indicative of itinerant electron behavior, but its value is roughly 1000 times that of simple metals. The temperature-insensitive susceptibility, called the Pauli susceptibility, has the following expression for noninteracting band electrons (Blatt 1968):

Z(0) = 2U(,u),u~, (4)

where #B is the Bohr magneton. Thus the large value of Z(0) is also indicative of large effective mass and is entirely consistent with the large value of specific heat constant 7. With increasing temperature the function

z(T)

increases slightly and reaches a maximum at about 0.6K. Subsequently, it decreases with increasing temperature until for T >> 1 K it has the empirical expression

z(T) = N lz~ff/(T + to), (5)

where N is Avogadro's number, t0 _~ 30K is the paramagnetic Curie temperature, and #elf ~- 2-6/~B is the effective magnetic moment per Ce atom. This temperature dependence indicates local magnetic moments on Ce sites generated by one electron in the f orbital, which gives a theoretical value for #efr = 2.54#B (Stewart 1984). The positive value for to would indicate antiferromagnetic ordering around 30K, but there is no experimental evidence of that. All Ce-based materials have #elf around

0 . 0 3 7

i i i i I i i i ~ I

C e A l a

gx d, ~ ° ' ~ f~

E I I I

o H = 1 k O e A H = 3 k O e [] H = 5 k O e

E /x. r3

,,..,, [] d x

0 . 0 3 6 D

0 . 0 3 5 I I I I [ I I I I I I I I I

0 0 . 5 1 .O

T ( K )

m

o / k

1.5

Fig. 2. The low-temperature part of the magnetic susceptibility of CeA13 (Andres et al. 1975). The three sets of data were measured under different magnetic fields.

2.6/ZB, while U-based materials tend to cluster around 3.0#B. The theoretical value for #~ff of U 3+ in the 5f 3 configuration is 3.62#R. Most well studied systems have 69 ranging from 20 to 200 K. Some heavy-fermion materials, such as CeA12 (Barbara et al. 1979) and U2Znl 7 (Ott et al. 1984a) do develop antiferromagnetic order with N6el temperatures much below O while CeCu2Si 2 (Steglich et al. 1979), UBe13 (Ott et al.

1983) and UPt 3 (Stewart et al. 1984) become superconducting with critical tempera- tures of approximately 1 K, also significantly below their individual (9.

We now encounter the first serious challenge to the rules of separating core from itinerant electrons mentioned earlier. The high-temperature susceptibility data would lead us unequivocally into classifying the 4f electrons on the Ce atoms as localized, but if so, why is the susceptibility suppressed at low temperatures without antiferro- magnetic ordering? How are the heavy itinerant electrons, which generate the Pauli susceptibility, related to the localized f electrons, which generate the Curie-Weiss susceptibility? To complicate the matter further, we will discuss in detail later that recent neutron magnetic form factor measurements give evidence that the heavy fermions are f electrons themselves. In those materials which become superconducting at low temperatures, the superconducting electrons also have the spatial distribution of f electrons. This and other puzzling results, such as de Haas van Alphen measure- ments, will be the topics of later subsections.

The graph in fig. 3 is a plot of the specific heat 7 versus the low-temperature susceptibility Z(0) for a number of mixed-valence and heavy-fermion materials (Jones 1985). For noninteracting fermions, we see from eqs. (2) and (4) that the ratio of these two quantities contains only fundamental constants, and this is indicated by the solid line in the graph. Many-body interactions will modify the specific heat and the susceptibility, but in different ways. In particular, exchange interaction between pairs of fermions enhances Z(0) but not 7, and this would put the actual points to the right of the solid line. The deviation from the solid line, then, is a measure of

-.,/ioo0

E

2 ?

,.:-

IO0

I0 10 -4

i J i i 1 ~ 1 1 ] i i i i i i i i I i r i i i i f

B A. ,/ONES et oZ (/985)

4f 5f Z o

o • superconducting

El • magnetic

z, • not superconducling CeCuzS~2/'CeCu6

or magnetic 7 - uBel} ,dr=

//•UzZnl 7 UCdll 1 euPt3 NpBeR3/

/ Iu~I3 • zxYbCuAI

/ H ~ Nplr2

& U A I 2 NpOs? AIo_.~2 Ce

/ eU2PIC2

/ •Ulrz / eCeRu3Siz

S •U6Fe

/ YbAIz

/ ~' a-N~

, / o - ~ . •o~,.~ , ... L . . .

iO-3 iO-Z

X (0) ( e m u / m o l e - f - (:}tom)

Fig. 3. A plot of the specific heat 7 versus the low-temperature susceptibility z(O) for a Io- number of mixed-valence and heavy-fermion

materials (Jones 1985).

P H E N O M E N O L O G I C A L A P P R O A C H T O H E A V Y - F E R M I O N SYSTEMS 95 4.0

3.5

E 3.0 m,

2

2.5

2.0

I I

CeSn 3

I I I

o H II [100]

~. H II [110]

] I I I I

50 100 150 200 250

T (K)

Fig. 4. The magnetic susceptibility of 300 CeSn 3 as a function of T (Tsang et al.

1984).

exchange enhancement (see Lee et al. 1986). The graph shows that exchange enhancement exists, but not large enough so that the large susceptibility could be attributable to incipient ferromagnetic instability.

The magnetic susceptibility of CeSn 3 is shown in fig. 4 (Tsang et al. 1984). If we ignore the sharp rise below 50 K, the curve looks like that for CeA13 (fig. 2) except for changes of horizontal as well as vertical scales. The peak occurs at around 150 K, and the maximum value is around 2 x 103 emu/mol. The similarity in shape of the z(T) curves of these two materials suggests a scaling behavior (Lawrence and B6al-Monod 1981), which means that the same mechanism determines the suscepti- bility curves of both mixed-valence and heavy-fermion materials. On the other hand, the low-temperature tail in CeSn3, also seen in ~-Ce (Koskimaki and Gschneidner 1975) and CePd3 (Gardner et al. 1972) but not in heavy-fermion systems, seems to suggest a fundamental difference between these two classes of materials. This point will be developed further in later sections.

2.3. Resistivity and magnetoresistance

In normal metals we find three major components to the electrical resistivity p(T) (Blatt 1968):

p(T) = Pimp + Pe-e + Pe-ph, (6)

where Pimp is the temperature independent contribution due to scattering of charge carriers by impurities and other lattice imperfections, p,_e comes from mutual scattering of charge carriers and has a T z temperature dependence, and Pe-ph arises from scattering by phonons and has a T s temperature dependence at low temperatures and a linear T dependence at high temperatures. For metals which contain small amounts of magnetic impurities, such as Fe, Co, Ce or U, the impurity part, which is denoted by Pmag, has the approximate temperature dependence in the form

P'ma. = A + B ln(T/To), (7)

where A and B are two constants, for T _> T o, where T o is a characteristic temperature, and a constant value for T << T o. The physical meaning of this result is that, under suitable conditions, the scattering of mobile electrons from local magnetic moments undergoes a resonance at the Fermi level. The low-temperature resistivity samples the maximum scattering cross-section (Kondo 1969). With increasing temperature, more electrons are scattered off resonance. The resonance itself is produced collectively by the mobile electrons, and it broadens as the temperature increases. These two effects cause the resistivity to decrease with increasing temperature. The characteristic temperature T o, called the K o n d o temperature, is a measure of the width of the resonance. In dense magnetic systems, such as Fe or Gd, Pmag has a X-type cusp at the magnetic ordering temperatures and decreases with increasing magnetic ordering at lower temperatures (Blatt 1968). The K o n d o effect is usually unobservable in these systems.

With this background we now look at the electrical resistivity curve of CeA13, shown in fig. 5 (Ott et al. 1984b). Above 2 0 0 K the resistivity, about 180gf~cm, is roughly one order of magnitude higher than the typical p h o n o n contribution in simple and transition metals (Blatt 1968). This would indicate that there is a set of strong and incoherent scatterers in the material, similar to magnetic rare earth metals above their ordering temperatures (McEwen 1978). This picture is entirely consistent with the high-temperature susceptibility data, that the f electrons in Ce are localized and their magnetic moments are randomly oriented. Between 35 and 2 0 0 K the resistivity decreases with increasing temperature, and the temperature dependence resembles what one finds in metallic systems containing Ce impurities, with a K o n d o temperature T o ~- 100 K. This reinforces the notion that there are localized magnetic moments on Ce sites. Below 35 K, however, p decreases dramatically, indicating that the random magnetic scatterers are either disappearing or becoming aligned. The temperature of resistivity maximum is consistent with the Curie temperature 6), but

240

200

160

I

~z_ 120

80

40 ~--

0i

0

i _ e°%o°

• •

/

,

:

i I

Ce AI 3

"%,,..

" % " ' % ' ° , , , , , , , , • • • • •

i I ~ I J

o , e ,

- , l e -

_ ,ee e°°

0 .'9 oO,,.-

/

8 x l O 3 4

T 2 (K 2

i I I

100 200

T (K)

Fig. 5. T h e electrical resistivity of CeA13 as a function of T (Ott et al.

1984b). T h e inset s h o w s the T 2 300 d e p e n d e n c e of the l o w - t e m p e r a t u r e

resistivity.

P H E N O M E N O L O G I C A L A P P R O A C H TO H E A V Y - F E R M I O N SYSTEMS 97

the rounded resistivity maximum shows that no magnetic ordering takes place in this temperature range. At temperatures much below the maximum, i.e. T _< 1 K, the resistivity has a T 2 dependence (Remenyi et al. 1983, Andres et al. 1975), which is consistent with the picture that the charge carriers are mobile fermions which interact with each other, as deduced from low-temperature specific heat and magnetic susceptibility. On the other hand, the temperature of maximum resistivity is more than an order of magnitude higher than the temperatures for specific heat and susceptibility maxima. What is happening to the Ce moments between 1 and 35 K?

Other heavy-fermion materials may have somewhat different resistivity behavior at high temperatures (Stewart 1984). As shown in fig. 6, the Kondo anomaly is not at all apparent in UPt 3, while the compounds CeCu2Si 2 and UBe13 have a resistivity peak followed by a plateau at a higher temperature. Unlike the susceptibility curves, there is no semblance of scaling behavior among the resistivity curves.

In fig. 7 we show a set of resistivity curves of CexLa l_xPb3, for four different values of x to illustrate the effect of doping ( L i n e t al. 1987). The pure compound, x = 1, has a

p(T)

curve very similar to CeA13. Replacing some Ce by the nonmagnetic La results in much higher zero-temperature resistivity and a much lower temperature for the resistivity maximum. Larger amounts of doping (smaller x values) tend to remove the resistivity maximum altogether. The high-temperature part of the resistivity, when normalized to the Ce content, changes very little with doping. These data demonstrate further that at high temperatures the Ce sites scattering conduction electrons individually, but at low temperatures they act coherently such that any disruption of the Ce lattice causes a large increase in the resistivity.

In the case of magnetic metals or metals containing magnetic impurities, the resistivity is suppressed by an externally applied magnetic field, because the localized magnetic moments are better aligned and therefore more coherent (Van Peski- Tinbergen and Dekker 1963). As a result, a decrease of resistivity in a magnetic field is a reliable diagnostic tool for magnetic scattering. In particular, at temperatures where the moments are randomly oriented, the magnetoresistance depends on the

IOLA ~ I , I o.e'...o.O.'°

• o o o o o o ° ° -

~ 00.000"0 UPt 3

8~- ~ eeo oe

6 2 ~ • • u m m u m i m u • i m • @ AAA • A•

AAAk&AAAAA 4 '

. illlt i imi i I&AAAA &&

... ,HI I . . . li~

~ 4

-~" / U Idel3 t

2

0(~ ^{I ~ I , /}

I00 200 300

T E M P E R A T U R E (K)

Fig. 6. The resistivity curves for UPt3, CeCu2Si 2 and UBe13 (Stewart 1984).

8 ° L ~ I I A I I f ' I I I \

70 b~x-0-6 i (Cex Lal x) Pb3

6O

× 40

~ 3O e_-

2O 10

~ i I I I i [ I

0 0 5 10 30 90 150 210 270

T (K)

Fig. 7. The resistivity curves of CexLa 1 _xPb 3 for four different values of x to illustrate the effects of doping (Lee et al. 1986). The curves are normalized by the Ce concentration x. The temperature scale is expanded below 12 K, marked by the vertical dashed line, to show details of the low-temperature behavior.

field H a c c o r d i n g to:

p(T, H ) - p(T, 0) oc - H 2. (8)

In fig. 8 we s h o w the resistivity of U B e 13 at several t e m p e r a t u r e s p l o t t e d as f u n c t i o n s of the magnetic field (Remenyi et al. 1986). This substance has an estimated 7 value around l l00mJ/(molK2), a specific heat maximum around 2K. Its electrical

2 OL , , , , J

~ 150

- - 8 0 0 m ' ( ' - " ~ . ~ . L ~ ~ --

100 - - - 600 n 6oo n r - ' ~ ~ - ~ -

i zl Jl J

i -

2 4 6 8 10 12

H (T)

Fig. 8. The magnetoresistance of UBe13 for a n u m b e r of temperatures above and below the resistivity m a x i m u m . This is an abbreviated version of the results reported by Remenyi et al. (1986). The rapid drop of p(T,H) below 1 K at low fields is due to superconductivity.

P H E N O M E N O L O G I C A L A P P R O A C H TO H E A V Y - F E R M I O N SYSTEMS 99 25

2O

o_ 10

5

Lal. x Ce x Sn 3

I I

; ~ x = 0.8 --

x= 1.0 x = 0.6

• x = 0 , 4

. , /

100 200 300

T(K)

Fig. 9. The electrical resistivity curves o f L a 1 _~CexSn 3 (Maury et al. 1979). These curves are not normalized by the Ce concentration.

resistivity, shown in fig. 6, peaks at 2.35 K, and its Curie temperature is O = 70 K, which is close to the temperature of resistivity plateau• Above the resistivity peak, e.g. at 4.2K, the magnetoresistance is well described by eq. (8). Below the peak temperature, however, the field dependence is considerably more complex, and in the high field limit the resistivity seems to approach zero. Below 1 K the substance becomes superconducting, and the resistivity drops abruptly to zero at the upper critical field. These results demonstrate that the high-temperature resistivity is indeed due to magnetic scattering, but below the resistance maximum the cross section is very effectively quenched by the field•

The electrical resistivity of the mixed-valence system La t _xCexSn3 for four values of x are shown in fig. 9 (Maury et al. 1979). The resistivity of the pure CeSn 3 peaks at about 250 K, which indicates that the K o n d o temperature is higher than this value•

This temperature is of the same order of magnitude as that of the susceptibility maximum, roughly 150 K. The low-temperature part has the expected T 2 dependence (Stalinski et al. 1973). Thus, aside from a large difference in the temperature scale, the electrical resistivity of CeSn 3 behaves in a way similar to CeA13. Also shown in fig. 9 is that the effects of replacing Ce with La in CeSn 3 affects the resistivity in the same way as in CePb 3.

2.4. Neutron paramagnetic form factor

Under a uniform magnetic field H a paramagnetic material develops a macroscopic magnetization given by m = zH. The magnetization is not uniform on the microscopic scale, because it reflects the spatial distribution of the electrons which are polarized by the field. In a crystalline material the magnetic moment distribution re(r) is a periodic function of the lattice, and the neutron paramagnetic form factor measures its Fourier transform re(G), where G is a reciprocal lattice vector (Moon 1986, Stassis

1979, 1986). The data are collected over a discrete set of points in the reciprocal space. One defines

z(G)

by

re(G)

⁼

z(G)H,

(9)

whose G = 0 c o m p o n e n t is the static susceptibility Z- The shape of z(G) gives information a b o u t the orbital states of the electrons which respond to the magnetic field. F o r example, d and f electrons have distinct form factors, because the latter has a spatially much tighter wavefunction and a correspondingly b r o a d e r form factor. The s and p electrons in metals have orbital wavefunctions that spread almost uniformly throughout the solid, so their form factors are 6-functions in G. Since only the G # 0 components are actually measured by neutron scattering, one can only infer the s and p contributions to the susceptibility by the discrepancy between the static susceptibility and the value of z(G = 0) extrapolated from the data for nonzero G. In magnetically ordered materials where a distribution

re(r)

exists spontaneously, one can measure the form factor without the need for an external field. It has been determined in this way that in the 3d metals Fe and Ni the ferromagnetic m o m e n t s are mainly due to the d electrons, with small s and p contributions pointing in the opposite direction from the d m o m e n t s (Moon 1986).

Stassis and co-workers (Stassis et al. 1985, 1986) have measured the neutron para- magnetic form factors of three superconducting heavy-fermion systems, CeCu2Si2, U P t 3 and UBe13, both above and below their critical temperatures. The material CeCuzSi 2 has T c ~- 1 K. The form factor

z(G)

at 4.2 K is well fitted by the theoretical atomic form factor for Ce 3 + state, i.e. one electron in the 4f shell, as shown in fig.

10. The form factor remains the same at 300 K where the susceptibility is Curie-Weiss, and at 0.1 K where the material is superconducting. This indicates that the same set of electrons are polarized by the magnetic field at all three temperatures. The resistivity m a x i m u m of this material appears at about 2 0 K and the Curie temperature is

120

8O

×

40

FROM SUSCEPTIBILITY

~ ~ { Ce Cu2 Si2

E = 75 meV T=4.2K

~ { H = 50 kOe (//a-axis)

I i I i I I

0.2 0.4 0.6

sin 8/X

0.8

Fig. 10. The neutron paramagnetic form factor of CeCu/Si 2 (Stassis et al.

1985). The magnetic scattering ampli- tude F m is proportional to the form factor Z(6). The solid curve is the theoretical form factor for Ce 3 + state, which is the Fourier transform of the magnetic moment distribution for one 4f electron.

P H E N O M E N O L O G I C A L A P P R O A C H TO H E A V Y - F E R M I O N SYSTEMS 101

O = 170K, thus we can conclude from the data that the heavy mobile electrons at 4.2 K are the same 4f electrons which act like independent local m o m e n t s at 300 K.

Furthermore, they are also the electrons that become superconducting. The conclusion holds for the two U-based materials except that the form factors, s h o w n in figs. 11 and 12, are best fitted by that of the U 3 + ion. These experimental results drive h o m e the dual nature of the f electrons in heavy-fermion m a t e r i a l s - l o c a l i z e d at high temperatures and itinerant at low temperatures.

28

24

20

o 16

×

J '~ 12

4 - -

0 0

I I i I i I

U P t 3 H = 50 kOe T = 4 . 2 K - - 5 f U 3+

I I I J I

0.2 0.4 0.6

sin 0/;L

Fig. 11. The neutron paramagnetic form factor of UPt 3 (Stassis et al. 1986). The solid curve is the theoretical form factor for U 3 +, which is the Fourier transform of the magnetic moment distribution of three 5f electrons in Hund's rule coupled ground state.

800

600

%

x 400

200

t I I I I I

UBe~3 H = 50 k O e T = 4 . 5 K

I I I t E I

0.1 0.2 0.3 0.4 0.5 0.6 0.7

sin O / k

Fig. 12. The neutron paramagnetic form factor of UBe13 (Stassis et al. 1986). The solid curve is the theoretical form factor for U 3 +.

The form factors of all three materials remain unchanged in amplitude above and below the superconducting critical temperature, and the authors regarded this as a strong indication that the Cooper pairs are in the triplet spin configuration.

The same experiments have been performed on mixed-valence materials CeSn3 (Stassis et al. 1979a, b) and CePd3 (Stassis et al. 1982). The data for CeSn3 from 300 K down to 4 0 K are shown in fig. 13, and those at 4.2K are shown in fig. 14. For a

~ ' C2Sn3 '

6

i

0~ .I

13 12 5 '4 3 2 .2 .3 .4 .5 .6 .7

sin 0 X

Fig. 13. The neutron paramagnetic form factor of CeSn3 between 40 and 300 K (Stassis et al. 197%). The ratio between the magnetic scattering amplitude Fu(O)

and the nuclear scattering amplitude F N is a measure of the form factor x(G). The open and solid circles denote data points obtained for a 4 m m and a 2 m m thick crystal, respectively, with the magnetic field parallel to the [110] direction. The triangles are data obtained with the 2 m m thick crystal at 40 K with the field parallel to the [100] direction. The solid curves are the theoretical form factor for Ce 3+, which has one electron in the 4f orbital.

CeSn3 T ~ 4 . 2 K H = 42.5 k G(//! 110

"~ z

0 .I .2 .3 .4 .5 .6 .7

Sin 0 X

Fig. 14. The neutron paramagnetic form factor of CeSn3 at 4.2 K, obtained for a 4 m m (open circles) and a 2 m m (solid circles) thick crystal with the field parallel to the [110] direction (Stassis et al. 1979a). The triangles are calculated values a s s u m i n g that the induced m o m e n t consists of a 4f and a Ce 5d c o m p o n e n t of eg symmetry. The solid lines represent the theoretical 4f form factor.

P H E N O M E N O L O G I C A L A P P R O A C H TO H E A V Y - F E R M I O N SYSTEMS 103

%-

~3

( D

%

2

5 ~v I I L I I

t;, OoSn

4 •

m

@ @ • 6

I I I I I L

50 1 O0 150 200 250 300

T(K)

Fig. 15. Comparison of the magnetic susceptibility ofCeSn 3 reported by different workers by magnetization and neutron scattering measurements. The inverted triangles and squares represent data taken by Toxen (1979); the full circles were taken by Legvold (1979): the solid line was reported by Tsuchida and Wallace (1965); the dashed line reported by Malik et al. (1975), and the triangles and open circles are deduced from elastic neutron scattering measurements (Stassis et al. 1979b). The emergence of the 5d form factor coincides with the susceptibility tail.

wide temperature range, both above and well below the resistivity and susceptibility peaks, the material has the simple 4f form factor. The 4.2 K data show a significant anomaly, namely that a narrower component is added to the 4f form factor, and this component fits well with the spin form factor of a Ce 5d electron in the eg orbital.

As this added component grows rapidly with decreasing temperature, it causes the extrapolated z(G = O) to follow the susceptibility tail at low temperatures, as shown in fig. 15. The data for CePd 3 is similar, although the d part seems to be an even mixture ofeg and tzg orbitals. In both cases the d contribution has the same periodicity as the Ce sublattice, so it seems that the low-temperature susceptibility tail, which was widely believed to be an impurity effect, is actually intrinsic (Gschneidner 1985).

We will discuss the possible origin of this effect in section 3.

2.5. De Haas-van Alphen effect

In a strong magnetic field the motion of the itinerant electrons in a metal is modified from Bloch waves into quantized Landau orbits. With the field varying in time, various Landau orbits move in or out of the Fermi surface, causing the diamagnetic susceptibility to undergo periodic variations. The periodicity, measured in units of the magnetic field, determines directly the extremal cross-sectional area of the Fermi surface in the plane perpendicular to the field. If the Fermi surface is nonspherical, one can tilt the field relative to the crystalline axes to map out the cross sections in many directions, and thereby reconstruct the shape of the surface. In case the Fermi surface of a material has many pieces, one needs to Fourier-analyze the signal and