Uniaxial strain dependence of the Fermi surface of

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The other shears y,, y,, y,,, and yyZ can be similarly defined. .... Laplace transform notation, the output of the ..... We wish to express our gratitude to Dr. N. A..
Uniaxial strain dependence of the Fermi surface of tungsten"

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D. J . S T A N L E Y J . ,M. ~ PERZ. M . J . G . L E E , A N D R. G R I E S S E N ~ Dcporrrrierrt c!f'P/rysic.s,Urri~~c~~sity ~f'Torotrlo.Torotrto, Orit., C(ortrt/oM5S l A 7

Received June 14. 1976" The cleriv;~tivesof five representative cross sectional areas of the Fevmi sul-face of tungsten with respect to ~iniaxialstressand stwin along the c ~ ~ axes. b e are determined experimentally from simultaneous measurements of quantum oscill:rtions in magnetostriction and torque, ancl also in sound velocity and torque. The 1.esu1tsare resolved into two components. the dependence on volume-conserving tetl-agonal shear. and the dependence on isotropic dilation. The tetlxgonal sheardepender~ceis f o ~ ~ ntodbe in generally good agreement with the resultsof KKR calc~~lations based on a fit to the unstrained Fermi SUI-face. The isotropic dilation dependence agl-eeswell with hydl-ostatic press~11-e measurements: from these results, the volume derivatives of the scattering phase shifts are deduced. This work is the first step towards a point-by-point determination of the distortion of the Fermi s ~ ~ r k of ~ ctungsten e in a lattice s~ibjectedto an arbitrary homogeneous straln. Les dirivees des surfaces de cinq sections representatives de la surface de Fermi du tungstene par rappol-1 des efforts et diformations uniaxiaux le long des axes du cube sont determinees experimentalernent B partir de rnesures simultnnees d'oscillations quantiq~~es dans la magnetostriction et ~ L couple. I de mime que de la vitesse du son et du couple. Les rCsulti~tssont resolus en deux composantes exprimant respectivement I'effet du cisaillement tetragonal conservant le volume et celui de la dilatation isotrope. Dans le premiei-cas,on trouve en gCn61-alun bon accord avec les resultats des calculs KKR bases sur Lln ajustement B la surface de Fermi non deformke. Pour I'influence de la dilation isotrope, il y a (In bon accord avec les mesures par pression hydrostatique. A partir de ces rCsultats on d@duitles dkrivCes volumiques des dephasages de diffusion. Ce travail constitue une premikre &tape vers la determination point par point de la distorsion de la su~fncede Fermi du tungstene dans un reseau sournis i Line deformation homogkne arbitl-airs. [Traduit par le journalj Can. J . Phys.. 55. 344( 1977)

-. -

Introduction Within the last two decades, the shapes of the Fermi surfaces of all except a few technically intractable inetallic elenlents have been determined experimentally by measuring the frequencies of quant~lnl oscillations in high magnetic fields in such properties as magnetization (the de Haas-van Alphen (dHvA) effect), resistivity, magnetostriction, and the velocity and attenuation of ~~ltrasonic waves. The a m ~ l i t~idesof these oscillations contain a wealth of additional information; for example, cyclotron --ltfasses can be detmmined from their temperature dependence, impurity scattering rates from their field variation, and spin splitting from the relative alnplit~tdesof harmonics. The a m p l i t ~ ~ dofe oscillatory magnetostriction 'Research supported in part by National Research Council of Canada. 'Present address: Materials Physics Division, AERE Harwell, Didcot, Oxfordshire, England OX11 ORA. 3Present address: Natuurkundig Laboratorium der Vrije Universiteit, Amsterdam, The Netherlands. "Revision received September 9, 1976.

is proportional to the derivative of the relevant Ferini surface cross sectional area with respect to uniaxial stress cri, and the ainplit~lde of oscillations in the elastic constant cij is proportional to the products of the derivatives with respect to strains E~ and E ~ The . absolute amplit ~ ~ dof e sthese oscillations depend on such factors as the spin splitting and thermal broadening of the conduction electron energy levels, and scattering by impurities and dislocations. These factors can be best accounted for by measuring q ~ l a n t ~oscillations ~m in the magnetostriction or elastic constant, and simultaneously one other dHvA-type oscillation. Then the ratio of the amplitudes determines the stress or strain derivative. The most ~ l s e f measurements ~~l prove to be magnetostriction in co~nbinationwith magnetization or torque (Griessen and Sorbello 1972, 1974), and of elastic constants, or sound velocity, with inagiletization or torque (Testardi and Condon 1970). The combination of magnetostriction and elastic constant oscillations (Stanley et al. 1976) proves to be less convenient because stress and strain derivatives are mixed. In

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STANLEY ET AL.

345

principle, any one of these combinations is sufficient to determine both stress and strain derivatives, because stress and strain are simply related by the elastic constants; in practice, whenever the experimental uncertainty is large, the different combinations are complementary. An arbitrary strain in a cubic metal is conveniently resolved into seven components : three tetragonal shears, three angular shears, and an isotropic dilation. These three types of strain are depicted in Fig. 1. A tetragonal shear along the z direction (y,) is defined as the volumeconserving combination of conventional dilational strains : [I I

&

x

Ez

= &

Y

=-'

2Y:

= YZ

An angular shear in the xy plane (y,,) is defined as the volume-conserving rotation of the x and y axes, each toward the other in the xy plane, so that the angle between the axes decreases by y,, radians. The other shears y,, y , , y,,, and yyZ can be similarly defined. An isotropic dilation composed of three equal dilational strains along the cube axes

is usually represented by the fractional change in the volume of the unit cell (AR/R). The seven strain components represent only six independent quantities, since the sum of the tetragonal shears is unity. Because infinitesimal volume-conserving shears bring about no change t o first order in either the Fermi energy or the phase shifts of a cubic metal (Gray and Gray 1976; Griessen et al. 1976), it is useful to resolve the experimental data into derivatives of the cross sectional area of the Fermi surface with respect t o tetragonal shear, angular shear, and isotropic dilation. - Neglecting possible' Strain-induced anisotropy of the crystal potential, the effect of volumeconserving shears on the Fermi surface can be calculated rather directly by an extension of the KKR method of band structure calculation. A siinilar calculation of volume derivatives involves the volume derivatives of the scattering phase shifts, which are not known a priori. These can be determined, however, by making a least-squares fit to the experimentally-determined volume derivatives of the areas. By combining these results, the response of each

FIG. 1. Distortions (--- dashed lines) of a unit cube solid lines) by (0)tetragonal shear y,, (b) angular shear -/,., and (c) isotropic dilation A!J/!J. (--

point on the Fermi surface to a general infinitesimal strain can be calculated. An interpretation of strain response data in molybdenum, based on this approach, has recently been carried out by Griessen et al. (1976). Tungsten is typical of the cubic transition metals, without complicating characteristics

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346

CAN. J. PHYS. VOL. 55. 1977

such as ferromagnetism; it is chemically and metallurgically stable, and good single crystals of high purity are readily obtainable. Although the density of states in tungsten is dominated by the d bands, the Fermi surface has regions of widely varying symmetry character. Moreover, because spin-orbit effects are much stronger relative t o crystal field effects in tungsten than in molybdenum, a relativistic spherical muffintin potential model is expected to give a better description of the shear dependence of the Fermi surface. In this paper, we present the results of simultaneous measurements of quantum oscillations in magnetostriction and torque, and in sound velocity and torque, corresponding to five principal orbits on the three sheets of the Fermi surface of tungsten. The tetragonal shear dependence of the orbital cross sections is deduced, and compared with the predictions of K K R calculations based on the phase shift Fermi surface model of Ketterson et al. (1975). The validity of the spherical potential model is discussed in the light of the results. The dilation dependence of the scattering phase shifts is determined from the volume derivatives of the Fermi surface cross-sections, and the point-bypoint strain dependence of the Fermi surface is discussed. A discussion of angular shear effects in tungsten is presented elsewhere (Lee et al. 1976).

Experimental The experimental techniqiles used here for measi~ring simultaneously oscillations in magnetostriction and torque are identical to those described by Griessen et al. (1976) for molybdenum, so only a brief outline is given here. Magnetostriction oscillations were detected by measuring capacitance changes in a conventional three-terminal capacitance cell (Fig. 2) using -~za"Variable-rdii6-iransformerbridge as described by White (1961). Torque oscillations were measured by mounting the sample cell on beryllium-copper crossed springs and detecting torque-induced rotations with another threeterminal capacitor and bridge. Typical noise levels were lo-'' cm for the length measurements and l o - ' dyn cm for the torque measurement. The strongest oscillations of magnetostriction and torque in tungsten both had amplitudes approximately 1000 times greater than noise.

FIG. 2. (a) Three-terminal capacitance cell. The capacitance between the sample S and guarded electrode E is measured; the bulk of the sample holder A, B, and C is the grounded guard electrode. A shim D is inserted to adjust the gap between S and E. The sample is glued to the base F ; both F and C are made of tungsten to minimize stresses on the sample produced by thermal contraction. The rest of the sample holder is Be-Cu. I is electrically insulating epoxy holding E in A and F in C. (b) For sound velocity measurements the top of the capacitance cell is removed and a transducer T is bonded to the exposed face of the sample. Radio frequency voltage is applied to an electrode R plated on the top of the quartz transducer; the sample is grounded through G to provide the R F ground connection to the back plating of the transducer.

T o measure oscillations in an elastic constant, the top of the magnetostriction cell was removed and a piezoelectric quartz transducer was bonded to the exposed face of the sample; pulse modulated ultrasonic waves were generated a t this face and propagated normal to it,

S T A N L E Y E T AL. B L A N K I N G PULSE

,

COUNTER

C

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FREQUENCY SYNTHESIZER

STEP f

- UP

TRANSFORMER

+

PULSED RF OSCILLATOR

30 M H z

RF

+++

AMPLIFIER

A N D DETECTOR

SWITCH

\

/

TRANSDUCER SYNCHRONIZING \

SAMPLE

PULSE PULSE

BOX

INTERNAL R E F E R E N C E

- CAR

P H A S E - SENSITIVE

DETECTOR

O S C I L L A T O R OUTPUT

I

I

OUTPUT

FIG.3. Block diagram of self-tracking pulse-superposition apparatus for measuring oscillations in transit time r.

reflecting from the parallel opposite face at F (Fig. 2) and returning to the first face after a transit time T . Oscillations in this transit time, which are related directly to oscillations in elastic constants and sound velocity, were measured by a self-tracking pulse superposition technique based on that of Testardi and Condon (1970). Figure 3 shows a block diagram of our experimental arrangement. The frequency synthesizer (Adret Electronique Model 201) produced a sinusoidal signal of approxin~ately1 V amplitude at a frequency f (typically 100 kHz), which could be modulated by application of an external voltage. This signal was stepped up using a transformer of 100: 1 ratio and then clipped to provide a well-defined triggering pulse for the - - radi8 frequency o s c ~ l ~ a ~The o r . latter has been described by Au (1974); it was designed in our laboratory, and built entirely with solid state components for high stability. The oscillator could deliver up to 35 W (peak) to a 50 R load at 50% duty cycle, and could be gated precisely with very little overshoot or ringing. The R F voltage at 30 MHz was applied to a piezoelectric quartz transducer, whose resonant frequency was matched to that of the R F oscillator, to generate an ultrasonic wave in the sample. The reflected ultrasonic echo in turn

generated an electrical signal at the transducer which was amplified and detected (LEL model IMM-2 I F amplifier; gain, with detector, is 105 dB). An R F switch (Saunders model DS11A) was used to isolate the amplifier while the ~ u l s e d oscillator was turned on. The amplitude of the detected echo was measured with a box-car integrator (designed in our laboratory), gated by a delayed pulse sufficiently wide to allow for small variations in the transit time. Under conditions of constant transit time T , the output of the box-car integrator was maximized when the pulse period was f =p ~ where p = 2, 3, 4, ...; in practice, p = 3 or 4 proved most useful. The system could be made to track changes in the transit time by modulating the pulse repetition frequency, typically at 200 Hz. The resulting amplitude modulation of the echoes was amplified by a phase sensitive detector (Ithaco 391A). The PSD output was combined with the output of the reference oscillator at 200 Hz to drive the frequency synthesizer. With proper choice of phasing, the system locks to a maximum response corre= p ~ and , changes in the PSD sponding to f output are a measure of changes in f or T . The operation of the system, for changes that

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-'

-'

,

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348

CAN. J . PHYS. VOL. 5 5 , 1977

are slow in comparison with the modulation frequency, can be represented by the first order feedback loop illustrated in Fig. 4. For convenience all the electronic gains and transfer functions and the curvature of the response curve (echo amplitude vs. synthesizer frequency) at the resonant condition f = p~ have been lumped into the open loop gain A. Using Laplace transform notation, the output of the phase sensitive detector is determined by

(1976), changes in an elastic constant c i j can be related to the observed changes in the frequency f by

provided that the fractional changes are small. The noise level of the system corresponded to fractional changes in frequency of 2 parts in lo7, whereas the stronger oscillations in tungsten had amplitudes typically 1000 times as large. Measurements were made on a cylindrical single crystal sample of tungsten of length 6 mm and diameter 5 mm, cut from an ingot of Solving, and noting that A >> 1 (typically 99.999% purity, with residual resistivity ratio A = 1000), P ~ ~ ~ =/ 5500. P ~ The , ~ cylinder axis was nominally the [I001 crystal direction; the ends were A Ud spark-planed to within 112" of (100) crystal Uo = 1 A 1 + (sTl(1 A)) faces, and were flat and parallel to each other to within _+ 1 pm (i.e. within k0.006 of a sound wavelength at 30 MHz). This sample size and shape proved suitable for both magnetostriction Two features of [4] are worth noting. First, the and sound velocity measurements, as well as effective time constant is reduced to TIA; for the accompanying torque measurement. The this allows one to have the PSD output time sample was mounted on a tungsten base in the constant T typically at 1 s Lo reduce output cell described above. noise, while measuring rapid variations in the Quantum oscillations were recorded in contransit time. Secondly, the steady-state response tinuous magnetic field sweeps up to 108 kOe is independent of A, so that the output will provided by a Nb3Sn superconducting magnet. track transit time changes even when A changes The data were recorded digitally, and sub(as it normally does, since A depends on attenua- sequently Fourier analyzed as a function of tion, and ultrasonic attenuation oscillations reciprocal field to give the amplitudes and phases accompany velocity oscillations); this will be of the various frequency components. In general, true so long as A >> 1. onlv the fundamental dHvA oscillations were The system was calibrated by disconnecting used in the analysis, because their amplitudes, the PSD input, and measuring on the counter and hence their signal-to-noise ratios, were the change in pulse frequency that corresponded significantly higher than those of the harmonics. to a given change in the PSD output, which was Where the harmonic amplitudes were sufficonveniently brought about by adjustment of the ciently large to permit reliable analysis, the internal offset. As shown by Stanley et al. results were found to be consistent with those obtained from the fundamental oscillations. All measurements were made in a cold finger containing helium exchange gas at reduced pressure inside an insert dewar containing 4He pumped to 1.1 K. The sample could be rotated about an axis perpendicular to the magnetic field. It was mounted so that the rotation axis was either [loo], the axis of the cylindrical FIG.4. Block diagram of the feedback loop representing sample, or [OIO], perpendicular to that axis. the self-tracking apparatus. The open loop gain is A ; The latter configuration was more flexible: in the output time constant of the phase sensitive detector one orientation (sample axis parallel to the is T. The PSD output is u,. The input u, is proportional to the instantaneous departure of the resonant fre- field), the derivative of an area with respect to quency f, = (pr)-I from the free-running synthesizer stress or strain parallel to the field could be frequency. determined; by ;otating the sample by 90" the

-'

+

+

+

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STANL.EY E T AL.

derivative of an equivalent area with respect to stress or strain perpendicular to the field could be measured. The former configuration was used to improve the resolution of the perpendicular strain derivative for one of the ellipsoidal crosssections. As the torque oscillations vanish in symmetry directions, measurements were made on both sides of these directions and the stress or strain derivatives along the symmetry directions were obtained by interpolation.

349

1954) modifies longitudinal velocity oscillations when the propagation direction is not parallel to the magnetic field. They give expressions valid for propagation parallel and perpendicular to field; we give here a generalization of their result for propagation at an arbitrary angle cx to the field direction:

Each Fourier component of the elastic constant oscillations leads the corresponding torque component by n/2; the ambiguity in the sign of (a In Ao/asl) + sin cx can be resolved only by reference to other measurements or calculations, except in special cases where a ln Ao/asl = -sin cx, which causes the elastic constant oscillations to vanish. The crystal structure of tungsten is bodycentred cubic. The Fermi surface consists of an electron jack centred at the zone centre r, hole octahedra centred at H, and small ellipsoidal Mere h is Planck's constant and e is the electronic hole pockets centred at N. Girvan et al. (1968) charge; cgs em units are used. By taking deriva- have made an extensive study of the Fermi tives of Fosc with respect to magnetic field H, surface geometry using the dMvA effect, and stress oij, and strains s i and sj, the oscillatory their notation is used in labeling orbits here. contributions to the magnetostriction 61/1 and We have investigated the stress and strain elastic constant cij, respectively, can be deduced. derivatives of 5 types of orbits, and have found Details are given by Gold (1968), Griessen and them to be well-characterized by the values Sorbello (1972, 1974), and Testardi and Condon with H along the cube axes. In Table 1 we list (1970); these have been summarized recently the measured derivatives of area with respect by Stanley et al. (1976), and we quote only the to stress along the three cube axes, for the prinfinal results relevant to the present experiments. cipal extremal orbits illustrated in Fig. 5. The tetragonal shear derivatives were calThe stress derivative is determined from the ratio of the amplitude of the nth Fourier com- culated from the stress derivatives, by means ponent of the magnetostriction oscillations, of the relationship: (61/1),,, to that of the corresponding torque z,,, measured about a given axis

Analysis Explicit expressions for the quantum oscillations in the thermodynamic properties of a metal associated with a single extremal area A, of the Fermi surface can be obtained directly from the oscillatory part of the free energy density Fosc in a magnetic field H, which was first derived by Lifshitz and Kosevich (1955) :

where 0 is the measured in a plane Perpendicular to that axis; the angular derivative can be determined by measuring the variation of the dHvA frequency. The relative phase of the two oscillations (either 0 or n) determines the sign of the stress derivatives. Oscillations in the velocity of longitudinal ultrasonic waves propagating along [I001 correspond to oscillations in the elastic constant c,,. Testardi and Condon (1970) have pointed out that the Alpher-Rubin effect (Alpher and Rubin

and using the elastic constants measured by Featherston and Neighbours (1963). They were also calculated from the strain derivatives, by means of the relationship: alnA - alnA (112)(t31nA [lo] -- -a ~ i asj

a1nA) ask

These results, together with earlier results obtained from simultaneous measurements of elastic constants and magnetostriction (Stanley

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CAN. J. PHYS. VOL. 55. 1977

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TABLE 1. Experimental stress and strain derivatives of extrernal cross sectional areas on the Ferrni surface of tungsten

Orbit

Area (a.u.)

o , neck orbit 1[001]

0.0164

n, ball orbit 1[001]

0.0583

v, octahedron orbit 1[001]

0.3836

p,, ellipsoid at (l,l,O)n/n

0.0158

i

aInAa 85, (10-"bar-')

aInAb

aInAc

orbit 1[001] p2, ellipsoid at (l,l,O)n/a

0.0228

orbit 1[100] "Magnetostriction and torque experiment, his work. b E l a ~ tconstant i~ and torque experiment, this work. CElasticconstant and magnetostriction experiment (Stanley e l ol. 1976).

et al. 1976) are presented in Table 2. There is good agreement among the results of the three types of experiments. In calculating the best experimental estimate of the tetragonal shear derivatives, the result of each experiment was weighted in inverse proportion t o its probable error. The tetragonal shear derivatives for these 5 orbits were calculated by an extension of the relativistic K K R method of band structure calculation (Korringa 1947 ; Kohn and Rostoker 1954; Onodera and Okazaki 1966; Segall and H a m 1968). In the presence of an arbitrary strain E, the shape of the Fermi surface can be determined by locating the wave vectors k that satisfy the implicit equation

- I

_

where h is the smallest eigenvalue of the K K R matrix, and where in general the Ferini energy parameter EF and the phase shifts q l , j are '"f"u"iictions o f 'stiairi. The strain is accompanied by a normal displacement of the Fermi surface which is given by (Griessen et al. 1976)

vector in the plane of the orbit. The quantities ahl la^) can be evaluated by differentiating the elements of the K K R matrix analytically, and using the Hellman-Feynman theorem. A fill1 account of the method will be published elsewhere. The 5 relativistic phase shifts of the spherical muffin-tin potential in tungsten were taken from the parameters of the best fit t o the Fermi surface area data by Ketterson et al. (1975). In order to achieve fill1 agreement with the area data, these authors found it necessary t o take into account departures from spherical symmetry within the muffin-tin sphere, and their phase shifts are therefore labelled by the irreducible representations of the cubic crystal field. We estimated q2,512by making an appropriately-weighted average of q r , + , 2 , s 1 2and qr8+ , 2 , s 1 2 . Following Ketterson et al. (1975), we took the equilibrium lattice parameter to be a, = 5.9810 a.u., and the Fermi energy paraineter t o be EF = 0.850 Ry. This leads to the following set of spherical phase shifts (radians):

and the strain derivative of an orbital area can be evaluated by integrating this quantity around the orbit. The result is:

where k, is the projection of the Fermi wave

The phase shift derivatives of orbital areas of the model Ferini surface are given in Table 3. Because the various extremal orbits on the Ferrni surface of tungsten sample regions of direrent

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STANLEY ET AL.

FIG.5. Orbits on the Fermi surface of tungsten studied in this work; p , , R, 0 , and v correspond to H along [OOI], while pZ occurs with H along [IOO]. TABLE2. Tetragonal shear dependence of areas of some extremal orbits on the Fernii surface of tungsten

-

0,

Orbit

Area (a.u.)

i

neck orbit _L[001]

0.0164

x, y

z

R, ball orbit _L[001]

0.0583

x, y Z

v, octahedron orbit _L[001] 0.3836

x, y

p,, ellipsoid at (1,1,0)~/ci orbit _L[001]

0.0158

x , JI

p,, ellipsoid at (I,l,O)~lo

0.0228

z

orbit _L[100]

z s y z

b

Av. exp.

Calc.

0 . S5(35) - 1 .70(70)

0.27 -0.55 2.05 -4.10

1.0(7) -2.0(14)

0.80(35) - 1.6(7)

1.65(15) -3.30(30) 0.75(15) - 1.50(3)

1.65(30) -3.30(60) 1.0(2) -2.0(4)

2.05(25) -4.10(50)

1.75(15) -3.50(30)

-9.1(5) 18.2(10)

-9.75(130) 19.50(260)

-9.2(7) 18.4(15)

-9.25(50) 18.50(100)

-7.2(7) - 7.2(7) 14.4(15)

- 10.10(230) - 6.35(230) 16.45(320)

OMagnetostriction and torqlre, this work. *Elastic constant and torque, this work.