The lattice thermal conductivity and related properties of pure gallium single crystals at low temperatures

46 (1970) 507-530

Physica

THE

o North-Holland

LATTICE AND

OF PURE

THERMAL

RELATED

F. W. Natuurkundig

CONDUCTIVITY

PROPERTIES

GALLIUM

AT LOW

Co., Amsterdam

Publishing

SINGLE

CRYSTALS

TEMPERATURES

GORTER

and L.

Laboratorium,

J. NOORDERMEER van Amsterdam,

Universiteit

Received

Nederland

4 June 1969

Synopsis The

lattice

methods:

thermal

1) alloying

of a large

magnetic

temperature

below

unpractical

conductivity

of gallium

pure gallium field

to reduce

the critical

for gallium;

with the

electron

temperature.

only for 6 <

has been derived

small amounts

T <

higher,

5.3 x

10-4 Ts for the a axis

for the b and c axes, respectively. of k,,/k,,, states,

found to occur beyond The ideal thermal Debye

temperature

resistivities resistance possible

conductivities

Comparing

explanations

in the superconductive B.C.S.

theory

the to be

axis (a axis) a

method

IO-4 T2 and

x

Wid and Pid are compared

and with theory. to follow

8.5

reducing

is shown

yields values 6.0

x

10-4 Ts

the values

and the normal the difference

is

error.

of very pure single crystals is found

and

The magnetic

these results with

experimental resistivity

3) by

method

Using these values and data of Zavaritskii

the ratio of the lattice

are calculated.

impurity

of three

2) application

12 K and along one crystal

value of k, = 4.5 x lo-4 T2 W/k cm was derived. slightly

contribution,

The

by means

of impurities,

a power

of this behaviour

Rotation

diagrams

are compared law in H with

with

the variation

of the thermal

of the

and electrical

with earlier work. The magnetoa resistance-dependent

power;

are discussed.

1. Introduction. The study of the lattice thermal conductivity of pure metals is complicated by the fact, that the contribution of electrons to the observed conductivity is generally much larger. A method of suppressing this electronic conduction, which has been used in the past 1) is that of alloying the metal of interest with a varying amount of other metals. Impurities are chosen which mainly reduce the electronic conductivity and affect the lattice waves only to a minor extent. The thermal conductivity, k, can then be separated at the low temperature limit into an electronic term k, proportional to T, and a lattice term, kg proportional to Tz; the lattice conductivity of the pure metal can be estimated by extrapolation. By making some assumptions about ke, the separation can be made up to 80 K, see ref. 2. 507

508

F. W.

GORTER

AND

L. J. NOORDERMEER

An alternative methodapa) for reducing the electron thermal conductivity exists if the metal shows a strong magnetoresistance effect. If the thermal magnetoresistance does not saturate in high fields for a particular direction of crystal

axis and magnetic

field and if samples

used, k, can be varied over several magnetic field of the order of 10 kG.

orders

of very high purity

of magnitude

by applying

are a

It has been shown before that gallium shows very strong magnetoresistance effects both in its electricala) and its thermals) resistance. This may be partly inferred from its favourable position when comparing metallic elements in a Kohler diagrams), but also the high purity of galhum, as is commercially available at present, is very important. It has been shown79*) that in single crystals of gallium the electron mean free path may be as long as 1 cm at temperatures near 1 K, offering the possibility to reach very high values of ~~7 in moderately high fields, in which oc is the cyclotron frequency and 7 the electron relaxation time. In the present investigation the field dependence of the thermal conductivity of gallium has been measured in transverse fields up to 15 kG and values for the lattice conductivity along the main crystal axes have been derived at temperatures between 1.3 and 6 K. The impurity method, mentioned above, could not be applied effectively because of the low solubility of nearly all metals in gallium99 la). In one case, gallium with zinc as the impurity, kg could be separated with sufficient accuracy, from the measured k between 6 and 12 K. In this case, for an a axis sample, a comparison of the methods will be made in section 6. An important difference between the two methods is, that in the magnetic method only assumptions concerning the field dependence of the two terms are made, while in the impurity method some assumptions have to be made regarding the temperature dependence. A third possibility is to observe experimentally that the lattice conductivity exists for metals which become superconducting, the electronic contribution being strongly reduced below T,. Using this method in conjunction with the impurity method, Zavaritskiiii) derived values for the lattice conductivity of gallium between 0.1 and 0.5 T, (T, = 1.08 K). Clearly such values cannot be compared directly with data for temperatures above T,; since the electrons are the main scattering source for the phonons in the normal state, the lattice conductivity in the superconducting state (kgs) will increase relative to the normal state value (kgn). At sufficiently low temperatures the phonon mean free path will be determined by the sample size and a cubic temperature dependence of kg, results. The problem will be discussed in section 7, where the present data for kgn combined with kas from Zavaritskil will be compared with theory. Because of the possibility of realising very large values of l, the electron

PROPERTIES

OF PURE

GALLIUM

SINGLE

CRYSTALS

509

mean free path, gallium has been studied extensively at low temperatures. A relation between both the electrical and the thermal resistivity and t has been derived from measurements of the size effect 7983is), the electron relaxation time T has been calculated from cyclotron resonance experimentsis) and one has even succeeded in directly measuring the Fermi velocity by a heat-pulse methodid). Data on the temperature-dependent part of the electrical resistivity, the ideal resistivity pid and its thermal counterpart Wid as derived from our measurements in zero field up to 30 K, will be compared to those of Rosenberg is), Powell 16~17), Olsen-Bar and Powellis) and Reich la). 2. Ex$eriment. The measurements were made with a double Wheatstone bridge, in which the two thermometers are connected in parallel arms of the bridge. If the thermometers have very similar resistance VS. T curves, the setting of the bridge consisting of these arms will not be affected by TO PUMP

4%

Fig. 1. The low-temperature part of the apparatus. 1) the sample, 2) carbon thermometer, 3) thermal anchor, 4) thermal heat connection with the cooling bath, 5) copper ring for closing the vacuum can; H1 and Hz heaters.

510

thermal

F. W.

fluctuations,

GORTER

AND

which influence

L.

J. NOORDERMEER

both thermometers

to the same de-

gree. Thus temperature differences can be measured to a higher precision with such thermometers. The requirement of similarity and equal magnetoresistance is fulfilled by De Vroomen typeso) carbon-film thermometers made according to the prescription of Star et al. sl). These thermometers can be matched within 5% over the whole range of 1 to 40 K, in which region they are used in practice. The magnetoresistance of the assembly of thermometer and constantan leads shows a negative maximum at 1o/o with a tendency to change sign at the higher fields. For different temperatures and for other specimens the curves can be fitted within 20% to a single function of H.R. The thermometers were calibrated between 1 and 4.2 K and at liquidnitrogen temperatures. By plotting the ratio of the resistance and that of a standard thermometer, of which the resistance was measured in a separate calibration using gas thermometry, the large temperature gap can be interpolated, the ratio being only slightly temperature dependent. This procedure introduces an error in T of only about 9% above 10 K. The resulting curve was fitted to a power-series expansion of log R in log T, see ref. 21. Fig. 1 shows the low-temperature apparatus. The samples (1) were grown in plexiglass moulds from 99.9999% starting material obtained from various sources. On two arms thermometers were attached by melting their copper body into the gallium. The leads were anchored to a thermal station at With (3) and led through an octal-type feed-through to the helium-bath. heater Hi the temperature of the sample can be varied by adjusting a heat current through the copper thermal resistance (4). The thermal gradient is made by means of heater Hz. This setup is less suitable for accurate measurements in case of a high thermal conductivity at temperatures near that of the cooling reservoir, which is not very important since the present investigation is mainly centred on conductivities in high fields. The remaining arms of the sample were used to attach potential leads, enabling us to measure the resistance within the limits imposed upon by the voltmeter. The german silver section makes it possible to solder the vacuum can at (5), while the bottom is cooled in order to keep the sample below its melting point (29°C). A magnetic field of up to 15 kG was used which could be rotated in the plane perpendicular to the sample axis. Isothermal series of points at 800 G intervals of the magnetic field were taken by adjusting the heat current, thus keeping temperature difference and mean temperature constant. The orientation of the single crystal samples was chosen along the main crystal axes. This was achieved by aligning the seeds by means of Laue back reflection photographs. The misorientation was generally smaller than 2 degrees. Our room temperature resistivities are equal to those reported by other authors179 7,8).

PROPERTIES

OF PURE

GALLIUM

SINGLE

511

CRYSTALS

As a measure of the purity of metals one generally quotes the residual resistance ratio (RRR). In our case the electrical resistivity at low temperatures cannot be measured easily; we therefore derived the RRR from lowtemperature thermal resistance data, thus assuming the Wiedemann-Franz law to be valid near T = 0. It is also possible to derive values for the electron mean free path directly from the thermal resistivity W x T at T = 0, using the results and Yaqubrs)

for the thermal

size effect,

reported

by Boughton

for the a and b axes. TABLE I Sample data

sample a49 a56 I a56 II Zav. a-3P b34 b113 b56 I Zav. b-3P c45 c56 I c56 II Zav. c-3P

cross section form (mm) 0 0 q o 0 0

0 o 0

0 Cl o

kolT (W/K2 cm)

Wid/T2 (lo-* cm/WK)

RRR (103)

lo (mm)

source

2 2 2 1.6

39 140 62.5 29.5

4.25 3.14 3.6 4

27.5 100 44 21

0.35 1.25 0.56 0.26

JM AS Z AS -

2.6 0.42 2 1.6

77 33 250 72

2.3 1.7 2 1.8

25 10 80 23

0.16 0.07 0.50 0.14

JM JM AS -

8.6 11.3 32 14

0.11 0.14 0.40 0.18

JM AS AS -

2 2 2 1.6

3.80 5.00 14.1 6.25

16.2 16.2 11.2 15.5

Samples are denoted by their crystal axis followed by the geometry-factor and the number of the series; Zav. taken from Zavaritskii, ref. 11. The residual resistance ratio RRR and the impurity mean free path are derived from ko/T, see text. JM: Johnson and Matthey; AS: Alusuisse (Z: 20 zones passed through).

As may be seen from table I, which collects sample data, the RRR for gallium is not a good measure for the electron mean free path (in contradiction to ref. 9). Zone refining is seen to be succesful for sample a56r; its zero resistance has not been corrected for size effects. Thermal conductivities as high as 350 watt/Kcm have been measured in sample b56. Even higher values (850 watt/Kcm) have been reported by Boughton and Yaqubls). 3. Experimental results. In figs. 2 to 4 we show representative data of the thermal conductivity as a function of temperature for the three crystal axes. In zero field the impurity scattering is seen to dominate below about 2 K, resulting in a T dependence. The maximum is reached at 2 K except for

F. W.

512

GORTER

AND

L.

J. NOORDERMEER

[wott/cmK] IO3

-’ a

k

axis

100

,0-4t, ] 0.1

,

,

1

,

,

10

,

,

100

300

[r

-1

Fig. 2. The thermal

conductivity

H // c axis on logarithmic Powell’s ductivity

work

(dashed)

scales. at high

in the superconducting

3D measurements by Zavaritskii GaZn - a sample

of a single Full lines: temperatures.

crystal

of gallium

this work, Dotted

(kgS) and normal

(k,,)

with

connected lines: state.

Q’ // a axis and

with

a thin line to

kgs and kgn: lattice Dash-dot

lines:

con-

3P and

on pure (3P) and impure (3D) gallium. of Ga with about 0.02% Zn impurity.

the c axis crystals (fig. 4) where it is found between 3 and 4 K. At higher temperatures electron phonon scattering takes over, Wid = Kidi being proportional to T2. Above 80 K, k becomes roughly constant; values have been taken from Powell et ~1.16). In the region between 1 and 4 K two curves taken from Zavaritskiilr) are also shown. From data on the impure series denoted 3D, Zavaritskii estimated the lattice conductivity kgs between 0.1 and 0.5 K, which is shown

PROPERTIES

OF PURE

GALLIUM

SINGLE

CRYSTALS

513

as a dotted line. In both the 3D and the more pure 3P series a rapid decrease below Tc (1.08 K) has been found. The lattice conductivity reaches a cubic T dependence near or below 0.1 K. The dotted lines above 1.2 K refer to the lattice conductivity as derived by the magnetic method (lower

T range) and for the a axis also by the impurity method 20 K, fig. 4). The total conductivity of the sample Ga(Zn), for the latter, is seen to be more than a decade higher.

(between 6 and which was used

Fig. 5 shows results for the electrical resistivity; data from various authors have been collected. Above the melting point (302 K) gallium behaves as a

13-

I2-

1c

/

.’

,/.Y

/’

H-0

3P,’ /

)t

l(

I

I

I’

-:\

I

3D/’

i

!’

-I’

! ;

i i

i

lo-

L____

/’

i -1_ IO

\

/.’

i

! k

,

I

,o

ia

/

i H=MkG

!

!

_* 1

/

i

; i i

lo-

i

j

/’ .-., *.‘~&

p*

16

-,

Fig. 3. The thermal conductivity

of a gallium single crystal with 0 // b axis, H // c axis. For legend see fig. 2.

514

F. W. GORTER

AND L. J. NOORDERMEER

mttlcm Kj 1

I

c

J

I

I.1

Fig. 4. The thermal conductivity

I

I

1

I 10

I

oxis

I 100

I 300.7

K

--1

of a gallium single crystal with (? // c axis, H /I a axis. For legend see fig. 2.

free-electron metalaa~sa) but on solidification a large anisotropy becomes apparent. The resistivity value for the c axis is seen to be anomalous in the sense that it is higher in the solid than in the liquid state, as observed in bismuth24). From the melting point down to 83 K, the measured values can be fitted to a Bloch-Grfineisen equation p~/pe = -0.17 + 1. I7 T/025), with values of 19of 208, 181, 160 K 16) for the a, b and c axes, respectively. Around 80 K the temperature dependence changes to a T2.5 law as shown by our data in the liquid Na region. Extrapolating down to 20 K, we find a good fit to the results of Olsen-Bar and Powelll*). For the c axis we obtained similar results in the liquid Hz region. Between 20 K and 12 K the temperature dependence changes gradually from T3.9 to a limiting power T4.5, as

PROPERTIES

OF

PURE

GALLIUM

SINGLE

515

CRYSTALS

[Rcml

-E Liq.

‘“6 lCiS-

l&

lt+P

1 1Ci*-

log-

-10 10 t

12’

L ,s 0.1

Fig. 5. The electrical lines. N.N.:

Dashed

lines denote

Newbower,

field of about

-T

resistivity

c axes, and in the liquid

I

I

1

I

1.0 of gallium

(1iq)state.

14 kG are shown are connected

C.Y.

single crystals

300

resistivity

Cochran,

of samples

along

the a, b and

are connected

pa of samples

Yaqub,

[K]

oriented

Full lines from experiment

the residual

Neighbor;

100

”

see text.

from Curves

various

with thin sources:

in a magnetic

a561 (H // c axis) and ~561 (H // a axis) and

to zero field curves

with a dashed

line.

may be seen from fig. 5. In the range from 1 to 4 K various results for the bulk resistivity have been both deduced from size-effect measurements by Yaqub and Cochran 7) and measured directly in a thick crystal: Newbower and Neighborss), and Reich’s). Describing all data available for the c axis around 4 K in terms proportional to TO, T2, and T5 we get the results collected in table II. In fig. 5, two curves are shown for H = 14 kG. In both cases a minimum

516

F. W.

GORTER

AND

L. J. NOOKDERMEER

TABLE II The low-temperature source

TO. 1011 CIcm

Olsen-Bar a.o. ~561, this work Reich Newbower Yaqub

a.o.

a.o.

electrical T2.1011

resistance

Rem

for the c axis

Ts. 1013 Qcm

remarks

550

3.6

4-10

490

3.4

lo-15

37.2

1.36

1.7

12.7

1.05

1.16

1.0

1.87

<8

K, T4.5 K, T4.5

relative to B (4.2) orT1.9 + pT4.5

is found near 20 K. The u-axis sample a561 reaches a limiting resistance of about 1.4 x 1O-4 !_&m below 2 K. The c-axis sample c56r has a mean free path 9 times shorter (table I). The curves reflect the behaviour of I rather than that of the anisotropy. The magnetoresistance in figs 2-5 depends on the field strength as well as on the crystal orientation. If a rotation diagram (fig. 6) is taken in which the current flows along one of the axes and the transverse field is rotated, large variations in both electrical and thermal magnetoresistance are found. Between these cases qualitatively no differences are found. Since we did not measure the two simultaneously, slight differences in alignment were possible and therefore we did not calculate Lorenz parameters (cf. Alers27)). There are only slight differences between the rotation diagrams reported by Reed and Marcus (fig. 6a, left) and the present ones. We note that the data are more easily obtained in the thermal experiment than in the electrical one, since inductive effects are avoided in the first case. Besides the rotation diagrams all field measurements, including the derivation of the lattice conductivity were carried out with the field along the axis of highest magnetoresistance. In the case of the b-axis crystal of fig. 6 the field was

transverse

magnetoresistance

Ib

=

0 Eiamp Hz

14

kG

”

Aw, i -90

-60

-30

0

30

60

90

cL

a

c

Fig. 6. Rotation diagrams for the electrical (left, a) and thermal (right, b) magnetoresistance of gallium single crystals oriented along the b axis. The transverse field is rotated in the ac plane.

PROPERTIES

OF PURE

GALLIUM

SINGLE

517

CRYSTALS

c axis

\

6.376K

\

5.25rlK

\1800

K 1.76w

--I

Ke\\Id>,.39 5K I

I

\I,

I

1 3 10 30 I -cI Fig. 7. Thermal conductivity as a function of magnetic field (c axis). The dashed thin line shows qualitatively the electronic term k,, the constant k, can be found by subtraction; see text.

along

the c axis, as in the case of the a-axis crystal;

for the c-axis crystals

the field was parallel to the b axis. 4. The lattice conductivity by the magnetic method. An estimate of the lattice thermal conductivity, K,, can be made by assuming that the observed values are the sum of a field-independent lattice conductivity and a monotonously decreasing electronic term. The electrical resistivity p(H) increases, in our range of magnetic field and current, approximately quadratically with the field. It is reasonable, therefore, that k’(H) in the form (W,(H) - We)-1 also follows a power law with a power vz near to -2, and with this assumption ki is obtained by simply extrapolating the straight-line sections in fig. 7 to the highest fields and reading off the differences. More accurate values are obtained by fitting the curves by adjusting n and kg. A value for n can be found from the field derivative of k(H) at high fields, where Wa can be neglected in the thermal resistance.

518

F. W.

GORTER

AND

L. J. NOORDERMEER

The results are shown in fig. 12, together with kg obtained from other experiments. Some scatter in the data may be expected from oscillatory effectsss) which are not seen in the figure because of the relatively large field steps; in our purer samples the relative amplitude is very small. The values for n as derived from the electrical resistivity data are somewhat higher than the corresponding ones from the thermal data. Typical values for the c axis are n = 1.978 in the electrical and PZ= 1.95 in the thermal case, both taken in the residual resistance region. This difference makes separation methods in which p(H) is used directly to determine the field dependence of W,(H) somewhat dangerous. Ignoring this difficulty the apparent Lorenz parameter L(H) = LO + k,p(H)/T gives kg values not very different from those presented in fig. 12, due to the fact, that the second term dominates. The high-field behaviour of galvanomagnetic effects has been predicted by Lifshitz et al. 29) and reviewed by Fawcett6) in terms of the topology of the Fermi surface. This theory does not differentiate between the thermal and electrical resistance and the generally shorter mean free path in the thermal case would lead to smaller values of ~0~7 only. A saturation of the magnetoresistance will occur if the number of holes is not compensated by an equal number of electrons. In case the magnetoresistance would saturate (in spite of the fact, that this does not occur in the electrical case) different results will be obtained if a corbino geometry is usedso). An additional test on the reliability of the values for the lattice conductivity presented has been made by studying the thermal magnetoresistance of a gallium corbino disk also. This consists of a circular disk in which the current flows radially and the magnetic field is perpendicular to the disk (and the current). In this geometry no Hall voltages can build up and the magnetoresistance behaviour is changed. Although in an anisotropic metal like gallium the isotherms (or potential lines) are not circular and the geometry factor is not easily obtained, it is interesting to compare the results. The power n is found to be somewhat smaller than in the rod-type samples, but the lattice term is clearly present, its magnitude being equal within 10% to values from fig. 12, taking an average over the directions in the plane of the disk and using a calculated geometry factor. Furthermore the analysing procedure has been applied to a case where field and heat current direction were such (a, b) that the electrical resistivity saturates (though not completely in 15 kG). The results could be described by means of n = 1.3 and as one should expect for this case of rather low magnetoresistance no field-independent contribution was observable. As mentioned before, n, which characterises the field dependence of the electrical and thermal magnetoresistance, is not equal in the two cases. Furthermore it is not independent of temperature. Similar deviations have

PROPERTIES

OF PURE

GALLIUM SINGLE

CRYSTALS

519

O.Ol_ ----+T Fig. 8. The power n of the magnetoresistance plotted against temperature (a) and against the zero field resistance W(0) T (b). The scales are 2--n on the left-hand side and n on the right-hand side. Note the straight line in (b).

been noted in gallium by Yaqub and Cochran7) and by Reed and Marcusa). The latter ascribe the effect to either disruption of open orbits or to a change of the compensation by impurities. Fawcette) mentions a number of experimental conditions that may be responsible for such differences. One of these is the use of finite contact areas, which were present in our experiments. However, the results are not essentially different from ref. 4, where point contacts were used. We have studied the dependence of n on temperature. Fig. 8 shows the VS. temperature (8a) and for results for the sample ~5611 m a plot of 2-n the thermal case also as a function of the zero-field resistivity W(O) T (8b). The results for the other crystal axes are similar. From fig. 8b it may be seen that 2-n can be described by a constant times W(0) T suggesting that the deviation from a quadratic field dependence is proportional to the electron mean free path. This rather surprising result indicates that a lower n is essentially a mean free path effect, rather than an impurity effect only. From this discussion it is clear that a Kohler-plot cannot be a temperature-independent representation of the magnetoresistance outside the residual resistance region. Fig. 9 shows such a plot, in which the thermal magnetoresistance is shown as a function of Ht. The transformation from the variable H/W(O) T to HI was made in order to take account of the

520

F. W. GORTER

1

AND L. J. NOORDERMEER

/’ 1

I -

ti’

I Hi

I

102

I

I

1

103 [Gaussc

Fig. 9. A Kohler-type plot of the thermal magnetoresistance of gallium single crystals. The thick lines represent the electronic terms in the residual resistance region. The line (a) has been shifted a decade for clarity. Y.C. denotes electrical resistivity measurements of Yaqub and Cochran. The ordinate HI is used to allow this comparison, Mean free paths i have been calculated with ref. 12. The curvature of the lines is due to the lattice conductivity.

anisotropy of W(0) TI!. The values of W(0) Tl which were used are: 8.94, 1.98 and 28.7 x IO-4 for the a, 6 and c axis respectively (from ref. 12). The straight lines a, b and c are derived from k, in the residual resistance region. In order to show the effects of the lattice conductivity and the temperature, curves derived from k(H) are shown for some selected cases. The u-axis curves, shifted over a decade for clarity, exhibit the above mentioned decrease of n and also a gradual shift as a function of T. The same is found for the c-axis crystals. Comparing samples of different purity, such as ~561 and ~5611 at about the same temperature (T m 3.5 K), only the influence of the lattice conduction differs widely. For comparison we have plotted the dashed line, marked Y.C., taken from Yaqub and Cochran’) who measured the electrical magnetoresistance of a thin sample (I = d, the diameter) oriented along the c axis.

PROPERTIES

OF PURE GALLIUM SINGLE CRYSTALS

521

The features described above are not in contradiction to the Kohler rule, because the latter has only been derived for elastic scattering, while the discrepancies occur at temperatures, where inelastic scattering dominates. 5. The impurity

method. The high purity of commercially

available

gal-

lium, which proved to be so useful for the magnetic method, is partly due to the low solubility of nearly all other metals in gallium, but the same property makes an application of the impurity method very difficult. In earlier worksvia) si1ver and aluminium were reported to dissolve up to a few percent. Aluminium has the same valency as gallium and thus it is unsuitable; for silver the resistance increase was found to be very small. Other metals which were tried are Zn, Cd, Ge, Tl and Cu. Of these, only Zn proved to have a large enough resistance increase to be useful, the RRR being about 50. Zn, Ag and Al have an atomic volume only slightly smaller than that of gallium which accounts for their larger solubility. Even in this case the total thermal conductivity was a decade higher than the lattice conductivity (fig. 2). From the measured k(T), we subtracted the quantity k,JT) obtained from

k,’ = W, = /3T-1 + aT2. Here B is purity dependent and is calculated from the Wiedemann-Franz law: B = LO/PO, taking for the Lorenz number its normal value Lo = = 2.45 x 10-s Vs/Ks and using the measured value of the residual resistance pa. The coefficient 01of the electron-phonon scattering term is not very sensitive to the purity (which is the same as saying, that Matthiessen’s rule is roughly obeyed) and therefore taken from purer samples. As we shall show, there is an increase of 01with decreasing purity in the temperature region where both terms contribute to the thermal resistance. Furthermore, (Yis temperature dependent which altogether makes W, uncertain at high temperatures, where the second term dominates. The lattice conductivity is found from k = kg + k,. In fig. 12 the result is shown. Above 12 K an improbably steep rise may indicate that we have indeed used a wrong W,. Below 6 K the scatter becomes too large although the general trend is a T2 dependence. Comparing the results from the two methods one may conclude that the lattice conductivity is nearly quadratic in T. For the b axis the number of points (isothermal field curves) is not large enough to decide whether a slightly steeper dependence may be present as in silversi). The two u-axis curves both have a T2 dependence, but the lattice conductivity of the impure sample is displaced towards lower values. The magnitudes expressed in kg/T2 (watt cm units) are as follows: Magnetic method: a axis 5.3 x 10-a; b axis 8.5 x 10-4; c axis 6.0 x 10-4;

522

impurity purities remains

F. W’. GORTER AND L. J. NOORDERMEER

method:

a axis 4.5 x

10-J.

is to lower the coefficient, unaffected.

Klemensl) has discussed lated the resulting thermal

Evidently whereas

the influence

the temperature

various scattering mechanisms conductivity. He showed that

of the imdependence

and has calcudislocations do

not change the temperature dependence but only lower the coefficient, as we have found. By assuming that dislocations are pinned to impurities, our results are reasonably explained. For the c axis a decrease of the slope is observed at the high temperature side, indicating the onset of some other scattering mechanism, as for instance point defect or Umklapp scattering. In very impure samples (which can as yet not be made) K, is expected to become proportional to T. This should occurss) when the electron mean free path becomes shorter than a phonon wave-length or ql < 1 in which q is the phonon wavenumber. Using q&m = 1.6 kT/hv, and the sound velocity

vs = 4.2 x 105 cm/s (a axis) 33) and pel from ref. 8 it follows that 15 x 10-6 T/PO,which is equivalent to K, = 1.6 x 10-s watt/cmK as the lower limit for which a TZ dependence can be expected for K,. In Ga(Zn) we reached k, = 6.5 x IO-2 T watt/cmK, indicating that even a ten times lower conductivity would not affect the temperature dependence of kg and would give a better accuracy. It was found that impurities hinder the growth of single crystals along the c axis, thus eliminating a possibility of reducing the k, relative to Kg.

qi=

6. Discussion. Rotation diagrams, in which the magnetic field is rotated in a plane perpendicular to the current, have been used in the past to study the topology of the Fermi surface‘i). In the course of this investigation we have taken rotation diagrams under virtually the same experimental conditions, though our gallium was more pure while our directional accuracy may have been smaller than in the work of Reed and Marcus 4). In figs. 6 and 10a some of the results are shown. The curves consist of a number of minima superimposed upon a sins 4 background. This is shown in fig. lob, in which sins 4 is used for the ordinate. A small minimum now becomes visible (arrow) at about & = 30”. In the case of larger minima, like those in fig. 6, subtracting the background will result in minima at slightly different angles than would be found by taking these angles at the minima in the original plot. In table III we have collected these angles. In this work we subtracted the background and have taken an average over a number of diagrams (different samples, temperatures and both electrical and thermal). From ref. 4 we have taken angles referring to absolute minima and some special crystal directions, thought by Reed and Marcus to represent these angles. Summing up the differences one notes, that for the b axis (field in UC

PROPERTIES arbitrary

OF PURE

GALLIUM

SINGLE

CRYSTALS

523

units

AWH

T I

c

I

120

90

60

0

30

30

(o-

0

1

-sin2@

I

*

’

0.;

10

Fig. 10. A rotation diagram of a crystal 4 // a and H in the (bc) plane, plotted against the angle 4 between field direction and b axis (a) and against sin2 +b (b). The dashed line shows the influence of the lattice conductivity. The two curves have been fit at low 4 values to show this. TABLE III Angles at which minima occur in the rotation diagrams “ideal direction” angle with b this work Reed and Marcus

b 0 0 0

ideal direction angle with a this work Reed and Marcus

a 0 0 0

ideal direction angle with a this work Datars 34) Reed and Marcus

a 0 0 0

(031) 30 30 30? (106) 15.8 20 15

(105) 18.7

9; 90 90 (102) 40.2 37 37

(101) 59.5 57.5 72.5 62 (250) 68.1 70.7 70 67

(201)‘ 73.5 74.5 -

c 90 90 90

(130) 71.5

72

plane, fig. 6) a splitting of the (101) minimum occurs which can also be seen in the work of Reed and Marcus, though less clearly. The shoulder at 15 degrees has become a minimum at 20”, while a small minimum near 74.5” was found. For the a-axis crystals we found no new minima and for the c-axis crystals the double minimum near 70” has become a single deep one at 70.7”. Since the comparison with the ideal angles seems less justified in view of

524

F. W.

GORTER

AND

L. J. NOORDERMEER

these results, we did not try to correct the proposed present results.

Fermi surface with the

In the preceding sections Discussion of some zero field properties. we have introduced values of Wid and pia, the lattice conductivity, k,, and some values for the Debye temperature 0~ derived from pid. These properties are interrelated since the ideal resistivities are due to electron-phonon scattering

while

the lattice

conductivity

is mainly

limited

by phonon-

electron scattering. As far as these properties are influenced by the phonon spectrum a temperature-dependent en can be used to describe differences between the real and the Debye spectrum. In most metals the en(T) derived directly from the heat capacity and the en(T) derived from fitting the pid curve to the Bloch-Grtineisen function, are nearly equal, although in the latter case the interactions of transverse and longitudinal phonons with electrons do not necessarily contribute equally to the resistance. Other 0’s derived for instance from the expansion coefficient or the elastic properties only fit at high or very low temperatures, while the one derived from the melting point 8, = 125 K is anomalous. The heat capacity has been measured by Adams and co-workersas) down to 16 K. After correcting their C, data with a term C, - C, = ACiT following Gopals6) (A -3R = 19 x 10-S) we find a constant value of en = = 265 K from 300 K down to about 100 K decreasing to 8n = 170 K at 15 degrees K. Below 4 K both Seidel and Keesom 37) and Phillips 3s) find terms with a temperature dependence steeper than T3, showing that the apparent en decreases with increasing temperature. Their values at 0 K are 317 and 324.7 K, respectively; 19n(4 K) = 296 K for the former. From these data we expect a minimum in en(T) between 10 and 17 K while Rosenberg 15) quotes (Wolcott, unpublished) a decrease to en(O)/ en(T,i,) = 1.78. A similar behaviour of 0 is generally found for the hexagonal metals (Cd and Zn) though the minimum is less pronounced in those cases. If this behaviour is also present in en, a drastic change in the temperature dependence of pid at T min should be expected. As we have shown before, such a change occurs near 17 K but the power of T never exceeds 4.5, which may indicate that small-angle processes are effective - or that en continuously increases with temperature up to 100 K. At higher temperatures en is anisotropic; using ~(90 K)/p(273 K) both Blomsg) and OlsenBar and Powelli*) find values near 230, 220 and 210 K, whereas using p(T)/p(90) the latter find values en = 132, 135 and 131 K, at T = 6 degrees increasing to 165, 163 and 155 K at 20 degrees, as compared to en(6) = = 260 and 8n(20) = 190 K. Reichrs) using four different methods to evaluate en obtains 0n = 220, 220 and 205 K above 50 degrees. The ideal thermal resistance Wid, also shows a change in the temperature dependence. This was noted by RosenbergIs) as a sudden change in the

PROPERTIES

OF

PURE

[cm/wottK3j

I

I

5.10-3

GALLIUM

SINGLE

CRYSTALS

I

I

I

525

//’

1O-4-

1.0

Fig.

11. The

and

c directions.

sample

ideal

(b) from

to the ordinate

I/I 2.0

resistivity The

-T coefficient

I

I

I

10

20

50

Wia/T2

for a number

for the u and b axes

values

Rosenberg at T =

5.0

is slightly

below

of samples

4 K are mean

out of the b axis. The curve

1 K. The ratio at the minimum

WI

en(T)

in the a, b values.

The

0~~ is normalised

is shown

as a horizontal

line.

slope of a plot of WT VS. T3, which is generally used to separate the impurity resistance WoT from the total. If, however, not the slope, but the quantity Wid/T2 itself is considered, the change is smaller and spreads out in T. Fig. 11 shows the results. It is immediately clear that Wid/T2 depends on the purity of the sample; lower values correspond to purer gallium. Roughly, the values for our purest and the less pure samples differ by a factor 0.6 which is the ratio of the theoretical predictions in the limits Wid > WO and Wid < WO. The full expression is Wid/T’ = 76D,18-2(T)

Nf’3W(~)

f~.

(1)

As in the case of the electrical resistance W(co) represents the high temperature resistance, N, is the number of free electrons, 03 is a constant which is 1 in the ideal case and B(T) again contains information on the phonon spectrum and may be called 0~. The factor p may be 0.61 or 1 as mentioned above. In eq. (l), only the thermal resistance due to “vertical” scattering processesl) is taken into account. A contribution to Wid proportional to T4 arises from “horizontal” processes, which are similar to those producing

F. W. GORTER

526

the ideal electrical

resistivity.

AND L. J. NOORDERMEER

While

this term is generally

negligible,

the

high pid indicates that it must be taken into account. Subtracting the quantity WE/T2 = pid/LOT’ from the data presented in fig. 11, it was found that the residue no longer shows the typical temperature dependence of O,“(T), which is also shown in fig. 11. Using values of Pid from Olsen-Bar and PowellI*) to correct the data of Rosenbergis) regarding the same crystals we find that WiV,/T2 (V = vertical) is a slowly decreasing function of T. The ratios between the values calculated from (1) with B(T) = OD(T), Ds, are of the order of 10, when N, is taken equal to 3, and increase with T. In eq. (1) the calculation of the electron-phonon interaction parameter C, is circumvented by inserting W(co) in the equation. The assumption is made that C, is independent of temperature. An alternative method is to use experimental values of kg at the same temperature, which contains C, in a different manner. The product (kg/T2)(Wid/T2) will be independent of

T,

iL

18

0.1

II

I

-T

1.0

I

I

10

rK1

0

Fig. 12. The lattice conductivity in the superconducting state (kgS) taken from Zavaritskii, and in the normal state kgn for the three axes. The curve (a) at the higher temperatures was obtained with the impurity method.

PROPERTIES

OF PURE

C when the parameters (Bloch scheme)

GALLIUM

SINGLE

C, and es are simply

or Ct = CZ (Makinson

CRYSTALS

interrelated,

527

i.e. if Ct = 0

scheme).

One obtains @p/T’) ( wid/T2)

= 490 &fio-4N;4’3

(4

in which Dg = 1 for the Makinson scheme and Dg = 0.23 or smaller for cases in which the Bloch assumption is made. In the theory, kg/T2 is independent of 8 and Na, so the anisotropy seen in fig. 5-1 is not accounted for. Wid/T2 has an anisotropy different from that of k,/T2, making Dg anisotropic. Since the experimental values of W$/T2 are much smaller than calculated with 0 = 8n from (1) and the experimental values of (kg/T2)(Wz/T2) are higher than those calculated with eq. (2), it is reasonable to assume that N, is much reduced. This is confirmed by the size effect data of Boughtonrs) from which we derive N, = 0.26, 2.5 and 0.05 for the a, b and c axis, respectively. 7. The lattice conductivity of gallium below T,. The measurements in the present investigation did not extend to below the critical temperature. However, the values for kgn (n = normal state) derived for pure and impure gallium allow more conclusions as to the behaviour of kg, in the superconducting state, because the theory gives the ratio k,,/k,, and not the absolute magnitude. Trying to compare experiment and theory we have taken values from Zavaritskiirr) who measured kes(T) for two different purities, denoted the 3P and the 3D series in figs. 2 to 4. The lattice conductivity was given for the c-3D crystal (where it showed up most clearly) by extrapolating a vs. T,/T. It was also found that the lattice concurve of L(T)/&(Tc) ductivity for the b axis was smaller by a factor 1.5-2 than for the other two directions. This differs from the present results, for which kgn is largest along the b axis. In fig. 12 we have collected the c-3D values taken from Zavaritskii’s fig. 1 and our measurements. The BCS theory of superconductivity has been adapted to the lattice thermal conductivity by Bardeen, Rickaysen and Tewordt *a) and the effect of point defects in the impure sample has been calculated by Klemens and Tewordtdr). The influence of point defects is compared with phononelectron scattering through a parameter To which appears as (T/To)3. Even in the most impure samples we find To = 250 K, so that the metal may be called pure as regards mass difference scattering. When plotted as kgs/kgn vs. T/T, (note the difference with k,,) a universal curve for each To occurs. In fig. 13 we have shown two such curves denoted K.T. for To = co (CL= 0) and To = 25 K (011). Our experimental values, calculated with kg, for the pure metal, are also shown and are seen to differ

528

F. W.

GORTER

AND

L.

J. NOORDERMEER

appreciably. The discrepancy may be attributed to several causes. I) The conductivity at very low temperatures was calculated with K, = 1/3C, .v .I = = 0.46T3i and taking i = d, the diameter of Zavaritskii’s sample. However, Gregory42) finds an indication that the scattering at the walls may not be diffuse so that i may be larger than d. The effect of subtracting a smaller d-l from the total ii&, is to increase the K,,. 2) The results on the lattice conductivity K,, from the magnetic and impurity method indicate the presence of dislocations. Since the frequency dependence of the relaxation time, the results for dislocation scattering and scattering by electrons are the same, the analysis of Hulm43) may be followed, which puts K;i = = W, + IV, and K&r = W,/h + IV,, where the lattice thermal resistivity We due to phonon-electron scattering is reduced in the superconducting state by a function h, and W, is the resistivity due to other (here dislocation-) scattering mechanisms. In the limit h 3 We; this means that Kgs/Kgnbecomes about equal to We/W,, showing that small extra resistances W, may have a large effect at low enough temperatures. However, the temperature dependence would be different from that in fig. 13. Departures may also occur from the anisotropy of the energy gap d(T) which leads to a value 2Aa/KT, = 3.3244) instead of the B.C.S.-value of 3.5. Summarising, the results show qualitatively the expected behaviour, but no definite conclusions can be made about theory or experiment.

!I 0.1

-

0.3

TlTc

Fig. 13. The increase of the lattice conductivity kg, below T, plotted as vs. T/Tc for c-axis crystals. The kg, of pure gallium was used. The theoretical curve of Klemens and Tewordt (K.T.) for metals of comparable purity (0~1)is also shown together with that for a completely pure metals (CX= 0).

PROPERTIES

OF PURE

GALLIUM

SINGLE

CRYSTALS

529

8. Concluding remarks. One method of analysis of the lattice conductivity, the method of Backlund45), also used by Powell et al. for galliumrs). This method is said to produce the lattice conductivity above 100 K, where the Borelius approximation (section 3, ref. 25) p(T)/p(B) = -0.17 + 1.17 T/B can be used. They calculate k, from p(T) = l.l7p(B) T/B. In view of the fact, that the resulting lattice term is just 17% of the total conductivity for two of the axes, while the method was not given any argument but its reasonable results, we do not incorporate it in our results. For a limited group of metals the magnetic method seems to be a promising way of finding the kg of a pure metal, while the result for a sample with O.Olo/o Zn impurity shows, that the impurity method must be used with much care. In the course of this article values of i, the electron mean free path, the Fermi velocity ZIF and the relaxation time T have been used, which had been measured by different methods. It is gratifying that the classical formula I = z&T is well obeyed: for a direction lying in the bc plane we find .&,, = 2.3 T3 ems), 7-1 = 2.4 x lo7 s-l is) and this vcaic = 5.5 x 107 cm/s should be compared with 6.0 f 4 x 107 and 5.5 & 8 x 107 for two directions as found by Von Gutfeld et al. 14). Finally we wish to remark that the possibility of using pure gallium for a magnetoresistance heat switch, which was exemplified in a preceding articles), is confirmed by these experiments. The limits are set by the lattice conductivity and the size effect, but a switch ratio of 104 even at 10 K seems possible if high enough fields can be achieved. Acknowledgements. It is a pleasure to acknowledge stimulating discussions with Professor Dr. A. R. Miedema, who supervised this research, and Professor Dr. G. de Vries. We are much obliged to the staff of the Natuurkundig Laboratorium of the University of Amsterdam, in particular to Mr. J. F. M. Klijn and Mr. J. M. L. Engels. We are grateful to Mr. P. Bloembergen for adapting his computer program for this work. This work is part of a research program der Materie” (F.O.M.).

of the “Stichting

Fundamenteel

Onderzoek

REFERENCES 1) Klemens, P. G., Solid State Phys. 7 (1958) 83. 2) Griineisen, E. and Adenstedt, H., Ann. Physik [5] 31 (1938) 714. 3) 4)

Sondheimer, E. H. and Wilson, A. H., Proc. Roy. Sot. A 190 (1947) 435. Reed, W. A. and Marcus, J. A., Phys. Rev. 126 (1962) 1298.

5) 6)

Gorter, F. W. and Miedema, A. R., Cryogenics 8 (1968) 2, 86. Fawcett, E., Advances in Phys. 13 (1964) 139.

7)

Yaqub,

M. and Cochran, J. F., Phys. Rev. 137 4A (1965) 1182.

PROPERTIES

530 8)

Cochran,

9)

Weisberg,

J. and Yaqub,

10)

Von

Zavaritskii,

Frederking,

13)

Moore,

14)

Von

T. W.,

15)

Rosenberg,

Rev.

H. M., Phil. Trans.

Powell,

R. W., Woodman,

Powell,

R. W.,

18)

Olsen-Bar,

19)

Reich,

20)

Physics, Moscow, 1966. De Vroomen, A. R., Thesis,

21)

Star,

Proc.

M., Van

(Ser II) 2 (1969)

Roy.

998. 108; Proc.

Int.

18 (1967)

855.

138.

581. Rev.

Sot.

Letters

R. W.,

Proc.

Sci. 258 (1964)

Dam,

14

(1963) 432.

A 209 (1951) 525.

Leiden,

Roy.

Sot.

3298, 2814;

A 209 (1951) 542.

Proc.

Xth

Int. Conf.

Low Temp.

1958.

J. E. and Van

Baarle,

C., J. sci. Instrum

(J. Phys.

E.)

257.

Busch,

G., Phys.

kondens.

Materie

24)

Mott,

25)

Publications (New York, 1936). Borelius, G., Ann. Physik [5] 8 (193 1) 261.

R. P. and Smirnov,

N. F. and Jones,

Newbower,

20 (1968)

9 (1964)

441.

Yurchak,

Alers,

Letters

Materie

33 (1960)

1069.

Sot. 247 (1955)

22)

26)

Acta

A. H., Phys.

23)

27)

Phys.

Rev.

18 (1966) jr.,

2174.

124 (1961) 36.

37, 6 (1960)

Phys.

CRYSTALS

M. J. and Tye, R. P., Brit. J. appl. Phys.

Roy.

M. and Powell,

R., CR Acad.

Rev.

kondens.

Letters

17)

W.

M.,

R. J. and Nethercot,

16)

140 A (1965)

- JETP

1968; Phys.

Phys.

SINGLE

R., Helv.

Physics

J. and Yaqub, Zurich,

Gutfeld,

Rev.

R. M., Phys.

T. and Reinmann,

R.

MFPS,

GALLIUM

M., Phys.

N. V., Soviet

Boughton, Conf.

PURE

L. R. and Josephs,

11) 12)

OF

H., The theory

R. S. and Neighbor,

P. B., Phys.

1 (1963)

B. P., Soviet

Rev.

J. and Marcus,

78. Physics

J. E., Phys.

101 (1956)

Rev.

of Metals

Letters

and Alloys,

18 (1967)

28)

Yahia, Lifshitz,

30)

875. Barron,

31)

Van Baarle,

32) 33)

32 (1966) 1700. Pippard, A. B., Phil. Mag. 46 (1955) 1104. Munarin, J. A. and Eckstein, Y., Phys. Rev.

Letters

34)

Cook,

Int.

35)

Andrews, 1968. Adams jr., G. B., Johnston,

36)

4784. Gopal,

E. S. R., Specific

37)

Seidel,

G. and Keesom,

38) 39)

Phillips, N. E., Phys. Rev. 134 (1964) A 385. Blom, J. W., Magnetoresistance for crystals

40) 41)

Hague, 1950). Bardeen, J., Rickaysen, G. and Tewordt, L., Phys. Rev. 113 (1959) Klemens, P. G. and Tewordt, L., Rev. mod. Phys. 36 (1964) 118.

C., Roest,

113 (1959)

W.

D. K. C., Physica

R.,

Proc.

XIth

Physics

24 (1958)

Heat

at Low

Low

Rev.

of gallium,

Gregory,

Hulm, J. K., Proc. Roy. Sot. A 203 (1950) 74. Gregory, W. D., Sheahen, T. P. and Cochran, J. F., Phys.

45)

Backlund,

Solids

Physics,

Phys. Books

St.

74 (1952) (London).

1083.

42)

Chem.

20 (1958)

1426.

J. them.

43) 44)

N. G., J. Phys.

Letters

F. W., Physica

Temp.

Heywood

112, 4 (1958)

8 (1959)

S102.

19 (1967)

Conf.

Temperatures, Rev.

- JETP

M. K. and Gorter,

H. L. and Kerr, E. C., Amer.

P. H., Phys.

538.

137.

M. I., Soviet

G. J., Mrs. Roest-Young,

J. R. and Datars,

W. D., Phys.

Rev.

M. I. and Kaganov,

T. H. K. and McDonald,

1065. Dover

41.

29)

I. M., Azbel’,

J. A., Phys.

- Solid State 10, 5 (1968)

of properties

Martinus

Nyhof

(The

982.

53.

20 (1961)

1.

Rev.

150 (1966)

3 15.