Anisotropy in the Kondo Effect Occurring in Zinc Manganese Single Crystals

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ANISOTROPY IN THE KONDO EFFECT OCCURRING IN

ZINC MANGANESE SINGLE CRYSTALS

Murray J. Press

M.Sc. Degree

The resistivity of single crystals of pure zinc and

zinc manganese (64 ppm) were measured below 7°K along the

principal axes of the axially symmetric crystals to determine whether any anisotropy exists in the strength of the

exchange scattering occurring in an anisotropie dilute mag-

netic alloy exhibiting the Kondo effect.

In pure zinc it was found that the resistivity parallel to the c-axis was greater than that perpendicular to the

c-axis by 4%, i.e., p"/p~ = 1.04. In the zinc manganese single crystals a further small

anisotropy was found which is interpreted as being due to the

spin scattering perpendicular to the c-axis being larger than

that parallel to the c-axis by 8%, i.e., J~/Jn = 1.08 where

J is the direct or s-d exchange integral. For the ZnMn

system, J has an average value of -0.23 ev. Within the experimental error, the product of the density of states and

the exchange constant is isotropie which implies that the

ternperature of the onset of the ~pin compensated state (i.e., Kondo temperature) is isotropie.

Department of Physics Eaton Electronics Laboratory McGill University, Montreal ANISOTROPY IN THE KONDO EFFECT OCCURRING IN ZnMn SINGLE CRYSTALS

PRESS • ANISOTROPY IN THE KONDO EFFECT OCCURRING IN ZINC MANGANESE SINGLE CRYSTALS

by Murray J. Press

A thesis submitted to the Facu1ty of Graduate Studies and Research in partial fu1fi11ment of

the requirementsfor the d~gree of Master of Science.

Eaton Electronics Research Laboratory DeparL~ent of Physics McGill University Montreal May, 1969

_T p __ ~o \ 1"::\ v .. __. , Q'7() j TABLE OF CONTENTS

ACKNOWLEDGEMENTS i ABSTRACT ii CHAPTER

l INTRODUCTION 1

II EXPERIMENTAL TECHNIQUES 7

II.l Survey of Experimental Techniques 7

II.2 Preparation and Characteristics of SLUG 12 II.3 Electronic Circuit 16

II.4 Cryostat 27

II.5 Sample Preparation 33

III EXPERIMENTAL RES~LTS AND ANALYSIS 39

III.l Experimental Results 39 III.2 Analysis of Data and Discussion 44 III.3 Further Discussion 49 IV CONCLUSIONS 51 APPENDIX

JOSEPHSON TUNNELLING 53 BIBLIOGRAPHY 62 i

ACKNOWLEDGEMENTS

The author wishes to express his deepest gratitude to Professors F.T. Hedgcock and W.B. Muir for their encouragement and help during the course of his research.

Thanks are also due to Messrs. B.E. Paton and P.L. Li for valuable discussions on the theoretical aspects of the research problem and to Mr. T. Petermann for his assistance in producing the samples. Finally, the author is grateful to the National Research Council for the grants which supported this research project and for the National Research Council Scholarship held during the course of the research. ii

ABSTRACT

The resistivity of single crystals of pure zinc and zinc manganese (64 ppm) were measured below 7°K along the principal axes of the axially syrnmetric crystals to de termine whether any anisotropy exists in the strength of the exchange scattering occurring in an anisotropie di lute mag netic alloy exhibiting the Kondo effect.

In pure zinc it was found that the resistivity parallel to the c-axis was greater than that perpendicular to the c-axis by 4%, i.e., p"/p~ = 1.04. In the zinc manganese single crystals a further small anisotropy was found which is interpreted as being due to the spin scattering perp~ndicular te the c-axis being larger than that paraI leI to the c-axis by 8%, i.e., J~/J" = 1.08 where J is the direct or s-d exchange integral. For the ZnMn system, J has an average value of -0.23 ev. within the experimental error, the product of the density of states and the exchange constant is isotropie which implies that the temperature of the onset of the spin compensated state (i.e.,

Kondo temperature) is isotropie. 1

CHAPTER l

INTRODUCTIûN

At low temperatures, the resisitivity of a pure metal and most alloys varies according to an equation of the form

(I.l)

where T is the tempe rature in degrees Kelvin; Po is the residual resistivity due to impurities, lattice defects, S etc.; and PL = AT is the lattice resistivity due to inter actions of the conduction electrons with phonons. A typical experimental curve would be the one in figure la.

However, for certain dilute (0.001%-1%) alloys (Cu,

Ag, Au, Mg, Zn with Cr, Mn, Pe, Re, Os impurities), a tem perature variation as in figure lb is obsèrved, i.e., the resistance decreases from high temperatures, levels off, and then starts to increase again forming a so-called resistance minimum. Careful measurements have shown that below the temperature of the resistance minimum, T , the resistivity M1N is proportional to -lnT. Magnetic measurements on these alloys show that the transition metal impurities aIl possess a localized magnetic moment.

Numerous theorists attempted to explain the observed behaviour assuming an exchange interaction between the spin of the conduction electrons, ;, and the spin of the electrons 2

" ...

RESISTlVITY O~ A METAL WI TH NORMAl. .IMPURI TI ES

FIG la

·RESISTIVITY OF f!. MET~L WITH MAGNETIC IMPURITI.ES

FIG lb 3

~ localized on the impurity ions, S. The harniltopian con-

~ ~ taining the spin dependence was taken to be Hs- d =-2JS.s where the constant J, called the direct or s-d exchange integral, is a measure of the strength of the exchange interaction. Finally in L964, Kondo, calculating the effect on the scattering probability of the conduction electrons to the second Born approximation found that the

Pauli principle had to be taken into account when considering intermediate electron states in the scattering process.

This introduced the fermi surface and the fermi function into the scattering probability and even the simplest case of a spherical enerqy surface gives rise to a singularity in the resistivity which involves InT.

This term when combined with the lattice resistivity

(ATS ) gives rise to a resistance minimum provided the s-d exchange integral J is negative, i.e., the conduction and localized electrons are coupled anti-ferromagnetically.

The full expression predicted by Kondo for the electrical resistivity including the spin scattering term is

(I.2)

where PA is the normal or Coulomb resistivity per unit im purity; PM is the temperature independent part of the re sistivity due to the magnetic or spin scattering of the conduction electrons per unit impurity (second order terrn in the perturbation theory); Z and Ef are the conduct.ion electron to atom ratio and the ferrni energy of the host metalj and c is the concentration of impurity ions.

Equation I.2 displays the observed temperature dependence of the resistivity and also shows the experimentally observed variation of TM1N with concentration, narnely

1/5 TM1N Œ c •

Kondo's expression, equation I.2, was derived assuming a free electron model, i.e., the fermi surface is spherical.

The next problem is to deterrnine whether any modification is needed fOT non-free electron-like behaviour, i.e., when the ferrni surface is anisotropic. A non-spherical fermi surface could conceivably lead to an anisotropy in the value of the ex change constant J implying that the spin scattering is stronger along one crystalline axis than along another.

That this is possible can be seen by exarnining the defining

equation for J, narnely

~ where ~LOC(r) is the wavefunction for a d-electron localized

on the impurity site; ~CON(r') is an s-wave conduction band electron wavefunction; and v(r,r') is the potential between

the localized d-electrons and the conduction electrons.

It will be sorne effective potential containing the effects

of neighbouring atoms and the screening electron cloud as 5

weIl as the direct Coulomb interaction between the electrons

being considered. In a non-cubic crystal structure, the distance between atoms along the basic axes will be different. Thus the

periodicity of the potential and hence the periodicity of

the conduction electron wavefunctions, ~CON' will differ in different directions. Added to this is the fact that the screening cloud of electrons around the impurity will

be polarized by the non-spherical crystal potential making V(r,r')~ ~ anisotropic (E. Daniel, 1963). Thus it is physically

reasonable for J to be anisotrop5.c in an anisotropic crystal environment. In fact, Miwa and Nagaoka (1966) have derived

general expressions for the values of J in an anisotropic environment. They find that for an axially symmetric

crystal the J tensor can be characterized by two principle

values ~~ and J".

The same crystal potential that determines J also determines the band structure and hence the density of states

at the fermi energy, N(O). Thus it would not be surprising if there existed sorne close connection between the two. To determine experimentally whether any su ch anisotropy does exist, measurements were made on single crystals of pure zinc and single crystals of dilute zinc manganese (64 pprn). Pure zinc forms a close-packed hexaqonal structure and is known to have an anisotropic fermi surface.

The difficulty with measuring the resistivity of single crystals at low temperatures is their very low resis tances (10-10_10 -6 ohms). Thus in order to make measurements 6

on them a device must be built that is capable of measuring very small voltages. A circuit involving a superconducting sensing element capable of detecting 10-14 volts will be described in the chapter on experimental techniques. 7

CHAPTER II EXPERIMENTAL TECHNIQUES

II.1. Survey of Experimental Techniques

A technique will be described for the detection of very small voltages; 10 -10 volts -10 -15 volts. These voltages are detected by sensing small currents through low resistance circuits. By employing a suitable feedback circuit, resistances of the order of 10-6 ohms to 10-10 ohms can be measured to an accuracy of a few tenths of a percent. The apparatus is similar to that described by McWane et al, (1966). The heart of the device is a Josephson Junction, whose operation was first predicted theoretically by B.D. Josephson in 1962. This device consists of a superconducting-insulator-superconductinq junction where the insulating layer is the oxide layer on one of the superconductors. Josephson was able to show that tunnelling of superconducting pairs of electrons, called the junetion current, across an oxide barrier was possible up to a maximum current, called the critical current, le' after which a voltage across the barrier appeared. The maximum super-current which can be passed is a funetion of the two superconductors' density of eleetron states, the thickness and area of the oxide layer, and the magnetic flux trapped both in the oxide layer and in the region of the flux penetration in the two

superconductors. 8

For a given junction, the density of electron states and the physical dimensions of the barrier are constant. However, the quantity of magnetic flux trapped in the oxide layer and in the penetration depths of the superconductors can be increased (or decreased) by driving a second current, called the galvanometer current, through one of the superconductors. Theoretical calculations (see Appendix) show that the critical current is a roughly periodic function of the trapped magnetic flux and therefore of the galvanometer current (fig. 2a). For a junction current exceeding this critical current, a junction voltage is observed which is proportion al to the excess current, defined as

Iexcess = Ijunction - Icritical·

Since l ·t. l is a function of the trapped magnetic cr~ ~ca flux, for a constant I. t. , the excess current and hence Junc ~on the observed junction voltage are also roughly periodic functions of the galvanometer current (fig. 2b). A Josephson device with these properties is called a SLUG for "Superconducting Low-Impedance Undulating Galvano- meter" • Figure 3 shows a circuit in which a SLUG is used to measure small resistances. Any voltage differences between the points A and B will cause a current to flow between them through the SLUG and thus will modify the galvanometer current. This in turn will cause a change in the junction voltage provided the galvanometer current is chosen so that 9

1 t I, UMCTlOH ; 1 ~IIIlCI"./'! 1 1 1 1

C RITICAL

CURRENT

( MA)

F~G 2.

IUNCTION ·II.CIII

GAL V A NOM ETE le U IlE N T (MA)

FIG 211 VAIIATION WITH MAGNETIC fUI.D Of ., CIITICAL CUIUNT It) JUNCTION VOLTAGE 10

f-V-7

-- ... 12 ~' -""\ tl 1 , B A 1 " .. _--~" 5 LUG

RI R2

SLUG S·UBCIRCUIT,.

FIG 3 Il

~ junction voltage # 0 ~ galvanometer current such as point "a" in figure 2b. By connecting the junction voltage to the imput of a suitable feedback servo-mechanism which controls the current 1 through the resistor R a null condition between 2 2 the points A and B can be achieved to within 10-15 to 10-12 volts depending on the size of the resistors RI and R , 2 the characteristics of the SLUG, and the gain of the feedback network.

By driving a current Il through the resistor RI' the potential of A changes with respect to that at B, causing the servo-system to change the current 1 until an equili 2 brium condition is achieved at which time the points A and B are again at equal potential. By measuring the change in the current 1 produced in 2 response to current change Il' the ratio of the resistances of the two resistors can be determined'and is given by

In this manner resistors between the values of 10-10 ohms and 10-6 ohms can be measured. The lower value is set by the sensitivity and noise level of the circuit while the upper value .is determined from time constant considerations of the circuit. The second restriction arises because as the resistance of the circuit containing the SLUG 12 and the resistors RI and R increases, the time constant 2 L/R, decreases. The SLUG has an inductance of approx imate1y 10-8 henries. Thus resistances of the order of

10-9 ohms result in a time constant of a few seconds while a resistance of 10-6 ohms results in a few millisecond time constant. However, in order for the overall circuit to be stable, the effective time constant of the feedback circuit (-.milliseconds) must be shorter than that of the SLUG subcircuit. This is to prevent the circuit from os cillating at high gains and also to prevent jumping an oscillation in the junction voltage versus galvanometer current characteristics in figure 2b. The requirement that the time constant in the SLUG circuit is greater th an a few mi1liseconds thus places an upper limit of 10-6 ohms on the resistance which can be measured using this circuit. II.2. Preparation and Characteristics of the SLUG

The SLUG is prepared in essentially the same manner as described by Clarke (1965), McWane et al (1966), and Shapiro (1967). A piece of thin (- .003") niobium wire is scraped clean to remove the insulating enamel and any previously formed oxide. A current is passed through the niobium wire from a capacitance discharge unit in order to create an oxide layer on the niobium wire a few Angstroms thick. A bead of m01ten 50-50 tin-lead solder with excess zinc chloride flux is dropped around the niobium wire and two copper wires and a110wed to freeze. Three copper wires are connected to the niobium to be used as current and voltage connections as in 13

C E

( V ) B

'~N tl.,fI"N

P P C

TO TO Ra R L co,. ",a TUI.NO OXIDE lAYER

1~(------~5mm.------~)~

A SLUG SHOWING CONSTRUCTION

FIG 4 14

figure 4. The connections to the niobium wire are made by pressing copper capillary tubing around the niobium and soldering the copper leads to these. Connections marked P in figure 4 are of this type. To make very low resistance connections (less than 10-9 ohms) to the SLUG via the niobium wire (for use in the gal vanometer circuit) the niobium is first scraped vigourously with a razor blade and then molten solder is dropped onto the wire. This forrns a Josephson Junction with a very high critical current (greater than fifty milliamperes) and a resistance which i5 very low, probably zero. Connections marked C in figure 4 are of this type. Tinned copper wires, connected to ~ and RR are then soldered to the solder bead making a superconducting path between the two resistors (see figure 7) through the SLU~. To observe the junction characteristics, a current is passed between the two superconductors via connections A and

C (fig. 4). No junction voltage is observed at D and E until

3 current exceeding the critical current is flowing. Once this critical current is exceeded a junction voltage is seen which is a function of the junction current, Ijn' flowing, (fig. 5) and of the temperature. This junction voltage appearing when the junction current exceeds the critical current can be varied by changing the galvanometer current, I H, flowing through the niobium wire. A typical characteristic curve is given in fig. Ga. The SLUG's characteristics are extremely tempe rature de pendent. As the temperature is lowered, the critical current 15

.lUNCTION

VOLTAGI y. y) 10

°0~--L---~------~------~3~~----~4~------~5------

JUNCTION CUIIINT ("' •• ),1, ..

fiG 5 .

.lUNCTION

YOLTAGI

ycY)

GALYANOMETII CUIIENT, 1. C"'~

50 fIG·6_

40 .lUNCTION

YOLTAGE 30

y.Y) 20

i l 1 2 3 4 ,i GA LYANOMITEI CUIIENT. 1. C.... ) 1 1 1 1 1 1 FIG 6 .. le CHARACTERISTICS OF ,A SLUG 1 1 i 1 1 16

increases. The initial slope of the junction voltage versus junction current, for a junction current exceeding the critical current, becomes greater, as shown in figure 5. The oscillations in the junction voltage as the galvanometer current is varied increase in amplitude, as in figure 6b. This results in a much larger effective junction "resistance" defined as the change in the junction voltage produced by unit change in the galvanometer current. Typical values range from 0.01 to l ohm.

This increase in the sharpness of the junction characteristics is due directly to the increase in sharpness of the fermi surface of the two superconductors as the tempe rature is lowered.

II.3. Electronic Circuit

In the circuit in figure 7, the SLUG is used to achieve a null in the voltage across the left hand and riqht hand resistors, ~ and RR respectively. When this condition is ohtained one has

i.e., the ratio of the two resistors is inversely proportional to the ratio of the currents flowïng through these resistors.

In order to have the required stability from the SLUG so that it can be used as a reference amplifier, both the junction and galvanometer currents as weIl as the temperature 17

H P KUTHUY DIFFiRENTlA .. NA N OVOLT M ET ER AMPL 1 FI E R 1 T fDVM4

1.2!.~.. !0.20 K"f'-....!~J1. HP6106A ~ 0-100 VOLTS.....-r-JUNCTION CURRENT

H,. 6106~_ ~ OALVANVOMETER CURR'NT 0-100 VOLI~

~ J.;t NP6217A D r ...... J 10~ V DO-----.-I~ <1 0·40 VO LTS H

V 10.GOA. ~ ~ 1 K.A.110KA M <1> ~ ~LP_ 4;/ ...... ;' REVUSING SWITCH

1....

C P P C

s 0

~SUPE.CONDUCTINO CONNECTIONS

SLUG ELECTRON le e IReUIT

FIG 7 18

of the SLUG must be kept constant.

To achieve a constant current, a constant voltage source is put across a large temperature-independent resistor. The voltage sources used for the junction and galvanometer currents are Hewlett packard 6l06A Constant Voltage supplies with a stability and noise level of around 0.001% or ±lOOuv. The resistors used are General Radio type 500 series with an accuracy of 0.025% and a temperature coefficient of 0.002% per degree centigrade. To further stabilize the temperature of the resistors, they are irnmersed in an oil bath. In order to have the power supplies operatinq at their maximum stab ility an effort was made to always use them ab ove 10 volts

(in order that the 100uv fluctuations be negligible). To accomplish this, for the junction current there is a choice of using a 20KQ, 10KQ, 5KQ, 2KQ or lKQ resistor chosen by a rotary switch, while for the galvanometer current, the re sistors available are 100KQ, 50KQ, and 20KQ.

The leads from the power supplies to the SLUG, made of #40 copper wire, have a resistance of about 20 ohms at room temperature and about 10 ohms at helium temperatures. Any resistance changes of the leads and of the SLUG, once tem perature equilibrium is achieved, are then negligible compared to the larger General Radio resistor in series with it and thus the currents could be kept constant to at least 0.01%. The junction voltage of the SLUG is measured with a Keithley Nanovoltmeter Model 148 with a maximum sensitivity of 10-8 volts full scale, noise level of one nanovolt, band 19

width of around 0.7 cycles per second, and an output of 0 to ±l volt proportional to the scale reading. Tt also con tains a zero suppress so that small changes in a large vol tage can be observed.

The circuit in operation employs feedback from the nanovoltmeter output to the circuit imput, i.e., across ~ in figure 7, which creates the null condition between the resistors RR and ~. To use the circuit first the junction characteristics are determined and a convenient operating point is chosen along a long steep part of the junction voltage versus qal vanometer current curve, such as point "a" in figure 6b. The voltage is then suppressed on the nanovoltmeter sc ale and the feedback loop is closed in the direction so as to obtain negative feedback. The zero suppress is readjusted so that the nanovoltmeter is accurately zeroed, i.e., no feedback current flows into the right hand resistor. The actual curr ent is found by measuring the voltage across a 10.00 ohm standard General Radio resistor with an accuracy of 0.025%, using a Hewlett Packard D.C. Null Meter, Model 419A with an accuracy of 1% or a Hewlett Packard 2401C Digital Voltmeter

(DVM) with an accuracy of 0.001%. The left hand current is similarly measured. The junction and galvanometer currents are determined by measuring the voltage across 10 ohm re sistors with the Digital Voltmeter. The particular voltage and hence current being measured is chosen by means of a

Centralab 6 position rotary switch. 20

Current is then slowly increased through the le ft hand resistor, ~, by passing a voltage from a Hewlett Packard 62l7A power supply (0-40 volts) through a large (300Q-lOKQ) constant resistor connected in series with~. The resulting current changes the voltage across ~ compared to that across

~ causing a current to flow in parallel with the galvano meter current through the SLUG. This changes the junction voltage resulting in a deflection on the nanovoltmeter.

Because the output of the nanovoltrneter is fed across RR' a current flows through it proportional to the scale deflection of the nanovoltmeter. Equilibriurn is reached when the voltage difference between the le ft hand and right hand re -14 -12 sistors (10 -10 volts) is such that the scale deflection of the nanovoltmeter causes exactly the right amount of current to flow through ~ to maintain the null condition. Because of the internaI construction of the nanovoltmeter, namely the fact that one of the imput leads is directly connected to one of the output leads, it is impossible in most cases to achieve negative feedback using the nanovoltmeter alone. An equally serious problem is that of the 10Q lead resistance between the nanovoltmeter and the cryostat. A feedback current of 1 milliampere flowing in the leads will cause a deflection of 10 millivolts making it impossible to zero suppress the nanovoltmeter. To overcome these problems, an isolator between the output of the nanovoltmeter and the right hand resistor must be used to effectively float the output of the nanovoltmeter. This is accomplished using a 21

Hewlett Packard Sanborn differential amplifier model 8875 with an imput impedance of 20 megohms and floating imput and output. Using the differential amplifier, it is possible to supply up to 100 milliamperes in the feedback circuit. A standard reference resistor for use as the feedback resistor, i.e., for RR' was made using brass with a resistance ratio of about 0.5. First the resistivity for the brass was measured usin~ a long thin sample. Then a resistor was made with the following resistance characteristics:

8 R 9.0 x 10- ohms ± 5%; 300 = 8 R77 = 6.15 x 10- ohms ± 5%; -8 R4 • 2 = 6.0 x 10 ohms ± 5%.

Thus at liquid helium temperatures, i.e. from 1.3°K to 7°K, the resistance of the brass resistor is temperature-independ 8 ent and has a value of 6.0 x 10- ohms ± 5%. The sensitivity of the circuit can be determined from the characteristics of the SLUG, the sensitivity of the measuring instrument, i.e., the nanovoltmeter, and the SLUG circuit resistance (consisting of the SLUG, left and right hand resistors, and the connections in this loop). From the junction voltage versus galvanometer current characteristics, one can obtain the effective "resistance" of the SLUG at the operating point [defined as av J NiaI H 1opera t' 1ng p01n, t ( p01n't

"a" in f1g., 6b)J, of the order of 0.1 ohms. Then from the sensitivity of the nanovoltmeter on the scale used one can determine the minimum current that can be detected, for example, if one can detect 10 x 10-9 volts on the nanovoltmeter this 22

means a current sensitivity of

l = V/R

10-8 = volts 10-1 ohms

= 10-7 amperes.

Then from the circuit resistance egua1 to the sum of the resistances of the standard resistor, RR (6 x 10-8 ohms), and the resistor beinq measured, ~, one can determine the voltage sensitivity. If the circuit resistance is 10-7 ohms, then

10-7 amps x 10-7 ohms ~V.m1n = = 10-14 volts.

As the resistance being measured decreases, the effective gain of the SLUG increases. This is because the SLUG is sensing a minimum current determined by the nanovoltmeter and the SLUG characteristics through a smaller resistance.

Because of the noise 1eve1, the minimum resistance which can be measured is about 10-10 ohms to an accuracy of about a

percent. The upper 1imit of the resistance which can be measured is determined by time constant considerations. The circuit consists of two main subcircuits: 1) SLUG circuit; 2) Nanovoltmeter circuit.

1) The SLUG circuit is composed of the SLUG, 1eft and right hand resistors and the connectors between these components. 23

The SLUG circuit has an inductance of 10-8 henries (estimated working backwards by knowing the resistance and measuring the time constant). Thus the time constant of the SLUG cir cuit (= L/R) is of the order of seconds for a SLUG circuit resistance of 10-8 ohms and decreases to mi11iseconds for a circuit resistance of 10-5 ohms. 2) The nanovo1tmeter has a time constant on the more sensitive sca1es of the order of a few seconds. This is reduced, however, by the overal1 gain of the circuit. Thus working with an overa11 loop gain of one thousand resu1ts in an effective time constant of the order of mi11iseconds. In order for the overa11 circuit to be stable, the nanovo1tmeter must be able to compensate for changes in the SLUG circuit faster than the SLUG circuit can drift. This requires that the effective time constant of the nanovo1trneter c~rcuit be shorter than that of the SLUG circuit. As pointed out, the time constant of the SLUG circuit is inversely oro portional to the SLUG circuit resistance. This places an upper limit of approximately 10-6 ohms on the resistor which can be measured using a SLUG as the sensing element. Re sistances greater than 10-6 ohms can be measured using the nanovo1tmeter as the primary sensing element. This is done by rep1acing the SLUG direct1y with the nanovo1tmeter as in figure 8. Using this circuit resistances from 10-6 ohms to 10+3 ohms can be measured to a high degree of accuracy.

As mentioned previously, the minimum measurable resistance is determined by the noise 1evel. The main sources of noise 24 e

.. '.217A

0-40 VOLTS

H , 10.04. N::~~~~~iTER DI~~:~r,1::L

REVElS'Na SW'TC ..

R•

ELECTRONIC CIRCUIT FOR LARGER RESISTORS

FIG' 25

are induced voltages from stray electrical and rnagnetic fields; fluctuations in the SLU~ ternperature; amplifier noise; and Johnson noise. To reduce electrical and magnetic noise aIl important wires, such as nanovoltmeter leads were twisted in pairs, and aIl circuit components placed inside grounded metal boxes wherever possible. The low voltage part of the circuit, in cluding the SLUG, left and right hand resistors and connecting wires are placed inside a lead can which since it is super conducting in liquid heliurn provides good magnetic as weIl as electrical shielding. AlI the electrical grounds of the circuit are connected together at one of the power supplies for the SLUG. This configuration produced the minimum amount of noise pickup.

The second main source of noise is the tempe rature changes of the SLUG due to temperature fluctuations in the helium bath. The SLUG is used at 4.2°K or at 1.3°K by purnping on liquid heliurn until an equilibrium vapor pressure and hence temperature is reached. Experiments showed that this temperature could be maintained constant to within a few millidegrees. The fluctuations are caused by changes in the heat imput in the system, such as raising the sample tem peratures and radiation from warrner parts of the system.

Amplifier noise is only a srnall fraction of the total noise as the nanovoltmeter has an inherent noise level of less than a nanovolt.

Johnson noise is due mainly to the lead resistance of the Figure 9

Eguipment Set-up showing: (a) cryostat (b) Keithley nanovoltmeter 148

(d) digital voltmeter HP2401C (e) voltmeter HP419A

(f) junction current power supply HP6106A

(g) galvanometer current power supply HP6106A

(h) power supply supplying current through ~, HP6217A

(j) temperature regulatinq system

(k) temperature measuring circuit including Pluke

galvanometer 841A (1) vacuum gauge meter, NRC type 701

(m) current measuring box

(n) Tectronix 516A oscilloscope '.:-:., ...... '_.~~-...... _. m ---)-~ i- 1· . t :;:=....g::~::::Il

--r-ifJ" .- .1 '-- 1, , ~, 7 27

nanovoltmeter as the low ternperature part of the circuit

has a very low resistance and is at a very low temperature.

This lead resistance of around 10 ohms results in a Johnson

noise level of about two or three nanovolts and is again

small compared to the two main contributions to noise, namely,

electrical and magnetic noise, and temperature fluctuations

of the SLUG.

II.4. Cryostat

As previously pointed out, the junction voltage versus

current characteristics of the SLUG are very tempe rature

dependent. Thus in order to be able to obtain a constant

reference point from the SLUG, its temperature must be main

tained constant. This implies placing the SLUG directly in

the helium bath.

In order to be able to do a resistance versus L~lmperature

run on a sample the temperature of the sample must be varied.

However, to use the SLUG at its most sensitive point and to

avoid recalibrating it for each tempe rature , the SLUG must

be kept nt a constant temperature, either 1.3°K or 4.2°K,

while the temperature of the samples is varied. This is

accomplished by placing the samples on a copper plate inside

an evacuated can. The two resistances to be compared are placed

in close thermal contact with the copper plate but are elec

trically insulated from it by a thin layer of paper and varnish.

Thermal contact is maintained by leaving a residual exchange

pressure of five microns of helium gas.

The temperature is measured with an Allen-Bradley 39 ohm, 28

1/8 watt carbon resistor, in thermal contact with the copper plate. This thermometer was previously calibrated against a calibrated Texas Instruments Inc. Model 340, Germanium

Thermometer, SeriaI # 984, Type 104A. Also mounted on the

copper plate is an Allen-Bradley 27 ohm, 1/8 watt resistor

used as the sensing element in a temperature regulating de vice which supplies the current to a copper-nickel heater with a resistance of 200 ohms.

The tempe rature is measured using a three terminal arrangement "to the Allen-Bradley 39 ohm resistor, and similarly

to the germanium resistor, in a Wheatstone Bridge arrangement

as shown in figure 10. This three terminal arrangement effectively cancels the lead resistance.

The resistance of the 39 ohm resistor went approximately as follows:

40 ohms at 300 o K,

51 ohms at 77°K,

323 ohms at 4.2°K,

5 kohms at 1.3°K.

The germanium thermometer had the following resistance

changes:

8 ohms at 300 o K,

3.5 ohms at 77°K, 2.6 Kohms at 4.2°K, 2 Mohms at 1.3°K. 'f

1.5 VOLTS

~~..... FLUKE GALVANOMETE R .

RllAD

R T)

TEMPERATURE MEASURII\'G BRIDGE ·e FIG 10 30

The copper plate is attached to a piece of thin-walled stainless steel tubing which is then attached to a brass flange as in figure Il. This minimized the amount of heat conduction between the sample plate and the helium bath. A small copper can is fitted over the copper plate to provide a radiation shield. A larger can made of brass fitted with a stainless steel flange is used to enclose the vacuum space. The vacuum se al is made with an indium O-rinq fitted into a 0.030" groove in the stainless steel flange. When the can is pressed against the brass flange by means of eight equally spaced boIts, the indium O-ring is squashed against the brass flange forming a vacuum seal that is superfluid helium leak tight over many tempe rature cycles. As stated previously it is necessary to place the SLUG directly in the helium bath while the rest of the circuit is inside a vacuum can. To accomplish this it is necessary to take several copper and niobium leads from the inside vacuum space of the can to the outside helium bath. This is done using an epoxy electrical feedthrough as described by

A.C. Anderson (1968). This consists of a thin copper cyl in der which is soldered into the brass flange (fig. Il), a bakelitedisk to hold the epoxy and to space the wires, and

Stycast 2850GT as the sealant. To protect the SLUG which is exposed outside the can from the turbulence when liquid helium is first added, an upper can consisting of a stainless steel cylinder with an Figure Il

Cryostat showing:

(a) standard resistor

(h) sample resistor

( c) SLUG

(d) epoxy seal (e) radiation shield

(f) inner can

(g) (juter can (h) upper can (j) heater

(k) vacuum line .~.- 32

r-- y.. VALVE

THERMOCOUPLE J \/ VACUUM 1 - X .i °OAUOI , -" 1 HE LlUM IXC HANOI \ J 0 AI )( . 0 1 ~ .~ 'IIALVI -

• RADIATI ON TaA. 1 ..... J

. " 0 -ounR CAN

...." - INNIR CAN '"

10 .UM.

COLD TRA. J

VACUUM SYSTEM

FIG 12

1 33

aluminum top with large perforations is placed around the

SLUG and the exposed wires (fig. Il). The large perforations (_ 5 mm.) allow the helium liquid and vapour free access so that the temperature may be stablized yet they are small enough to break the rough initial flow of helium during the helium transfer. Around the whole set of cans is placed a layer of thin lead foil. This serves as a magnetic and electric shield for the SLUG and the rest of the low temperature part of the circuit. The cryostat arrangement that has been described here suffices to reduce the noise in the SLUG circuit to a few percent of the total voltage measured. The noise is a result of fluctuations i~ the temperature of the helium bath due to heat conducted to it from the heater inside the vacuum can by the residual helium exchange gas. To further reduce this noise, two thermally isolated helium baths should be used, one maintained at atmospheric pressure and used strictly for the SLUG while the second is thermally connected to the samples and used to vary their temperature. With this design a noise level not greater than 10 -14 volts would be obtained over the whole tempe rature range measured.

II.5. Sample Preparation

The pure zinc and zinc ~anganese single crystals were grown by the Bridgeman technique starting with Cominco 99.999% pure zinc.

To make the zinc manganese dilution first a master alloy 34

was made with 0.5% manganese (Cominco 99.99%). The mixture was agitated for twenty hours at 700°C and then quenched in cold water. After etching in concentrated HC1 acid it was annealed in an argon atmosphere at 400°C for six days to further homogenize the alloy. Pure zinc was added to a piece of the 0.5% manganese master to obtain a 0.01% manganese concentration which was then p1aced inside a pyrex ampoule with an argon atmosphere, heated to 450°C and agitated to mix the solution. To grow the crystals the ampoule containing the solution

(either the pure zinc or the zinc manganese a11oy) was lowered vertically through a two zone furnace as shown in figure 13 (Bridgeman technique). The rate of des cent was 50 mm. per hour for the pure zinc and 7 mm. per hour for the zinc man ganese. The slower rate for the ZnMn was a compromise be tween growing large single crysta1s and minimizing the con centration gradient. Single crysta1s grown in this manner had a cross-section of 0.5 inches and a length of 3 to 7 inches. To check the concentration gradient in the ZnMn alloy, thin slices were cleaved off each end of the central portion of the ingot produced. These were rol1ed, etched in RCI acid, annea1ed for 24 hours in a vacuum at 40QoC and then quenched in cold water.

The resistance ratio, defined as R4.2/(R273 - R4 • 2 ) was measured for each end piece. Then using the result derived experimentally by Hedgcock and Rizzuto (1967) relating the resistance ratio with the concentration of manganese for 35

ROTAT ING CLOCKWOR K

1',.,00 MM./HOUA -TUNG5TEN WIRE

--..,....0·

I-+~- PYR"E X A M POU LE

-+"""""'+-ii-MO LT E N 50 L UTiON

+-+~-5 INGLE CRYSTAL

---12'

-HEATI NG ELEMENT

---24'· BRIDGEMAN FURNACE

4000CCr-~~~~-~0Y--'~~~1~2~--- TEMPERATURE PROFILE OF FURNACE

FIG 13 36

polycrystalline samples having undergone the same heat treatment, namely, R Conc. of Mn in atomic percent = (0.334) 4.2 R273 -R4 . 2 the concentration gradient was measured. The centre piece of the alloy ingot was then etched, annealed at 400°C in a vacuum for a week and the concentration gradient again determined. This process was repeated until no concentration gradient was observed. It required about a month for this to be accomplished. The final concentration of manganese was estimated to be 64 parts per million using the above formula. To prepare crystals for resistivity measurements, the single crystal ingots were cleaved a10ng two basal planes at a convenient distance apart (_ .400"). They were then etched in HC1 acid to remove any surface impurities and the basal planes were covered with a thick layer of solder

(- .020") for future electrical connections. visua1 examin- ation of the crystals after this process showed that the solder had not penetrated in~o the crystal but had remained on the surfaces. Crystals of the desired cross-section were then cut using a fine wire spark cutter (see fiqure 14 for detai1s of cutting the crystal). This procedure produced crystals suitable for measuring the resistivity para11e1 to the c-axis. To make crystals suitable to measure the resistivity perpendicular to the c-axis, a slice was cut perpendicular to the basal plane, soldered on the ends and then cut with a spark cutter, as in figure 14. 37

1oE------11:

~ ....,...,..~-...-. ~ ...... ______--- ...... __ .;.:::,~_':::a...... " ------.-----~--~~.--,:.- ... ~ ..."I_-.------_._---_--=:.:..~ ,1...... ":. "'-, ...... w Q.... 0 '" u .' '. 0' K ...... 1 1 ~' 0< 1 ... '" 1 1 • >< 1 • • >-'" 1 CI:: 0< 1 J • u J ,• 1 J 1 • u 1 , \T ~. J ~ 1 1 ,• z 1 1 , ... 1 , ... ~ J J \, ::::) 1 .. u - 1 \ ...,. ~ J \ . J 1 , u.. II. 1 , 0 1 ) '\ '\ Q J 1 '\ 1 1 ,. 0 ------.. ------.... --... ::c - ..:;-;;. ~: -.~: 1 1 J . '" . .~ ....--.' ... t- '.', '-k ...... w ~ ..... I ...... :i .' .... , ~ '~ 38

The pure zinc single crystals produced in this manner exhibited a high degree of purity and freedom from lattice defects as evidenced by a resistance ratio of the order of

2 x 10-5 •

The ZnMn single crystals exhibited the expected resistance ratio to correspond to the previously measured manganese concentration. To make connections to the crystals in order to attach them to the SLUG circuit, # 30 copper wires, covered with a

layer of solder (superconducting critical temperature of about 7.5°K), were then soldered to the solder caps on the ends of the crystals. As determined by measuring the resis-

tance of a very thin slice of pure zinc single crystal prepared with this type of soldered connection, it was found that the resistance of this connection was not greater than 10-11 ohms. 39

CHAPTER III

EXPERIMENTAL RESULTS AND ANALYSIS

111.1. Experimental Results - Pure Zinc

Measurements made on the single crystals of pure zinc are shown in figure 15 where ·it can be seen that the resis tivi ty in the temperature range measm:ed differed in the two directions by approximately 4%. The resistivity with the current paraI leI to the c-axis is greater than that with the current perpendicular to the c-axis, i.e.,

p (Zn)" / p (Zn).1. = 1. 04 ± • 02 . (111.1)

The possible error shown in p" /P.1. of .02 is due to the un certainty in the geometrical factor (length of the crystal along direction of current flow/cross-sectional area) for the crystals measured. Within this accuracy, the result agrees with that estimated by B.N. Alexandrov and I.G. D'Yakov (1963) of about 1.03 in the tempe rature range measured.

The results for the resistivity perpendicular to the c axis were then plotted on a logarithmic plot as in figure 16 to determine the temperature dependence of the resistivity.

The resistivity approached a T5 law at higher temperatures

(above about 5°K) while at lower temperatures it follows a 2 T or a T3 law. The difficulty in determining the exact relationship at low tempe rature resulted firstly from the fact that peT = OaK) had to be estimated by extrapolation and secondly, because of the small variation of the resis tivity with temperature below 4°K, the numbers peT) - p{O) .0

u Z N

~ U . Z t;;: ... ! 0 It'I N - 1 >- .... ~" •! > .... '" • '"w • jX ~ • ~•

.\,

ii ,1 i, ! .,

R(O): .98x10·9./2. l •• PURE ZINC • -LC • • •• • •• • P(T)-P(O) . ••• • • Cl-CM- • • • • • •• • • •• • • • • • • • •

•

• •

10'12~""" _____--:a- ___-:, __ --::--_-=-_-x_-,;:--::-- __ 1 2 3 4 5 8 TEMPERATUR E (0 K) FIG 16 : 42 had a large uncertainty. Assurning that one can write

5 peT) = p(O) + AT + B~ it was possible to estimate that the coefficient of the phonon scattering term, A, has a value of 1 x 10 -14 ohm-cm./{OK).5 In their paper on the resistivity of polycrystalline samples of several metals, Garland and Bowers (1968) find that in sorne cases n = 2, due to electron-electron scattering. As- suming that it is the same process in zinc, i.e., n = 2, it is possible to estimate that B has the value of about 1 x 10-12 ohm-cm./{OK)2. The values of A and B for zinc compare qualitatively with those measured by Garland and Bowers for aluminum which has a comparable Debye ternperature (9 {Zn) D = o 3l0 K, 9D (Al) = 428°K) and is of the same order of purity. Zinc Manganese

The experimental results for the resistivity of the Zn~n single crystals containing 64 ppm of manganese are presented in figure 17. The experimental scatter in the points is about 1%. To analize the data, the points from 1.3°K to 5.5°K were assumed to obey an equation of the form

peT) = p(lOK) + (dp(T)/alnT)lnT predicted theoretically by Kondo. Fitting the parameters p(lOK) and (dp{T)/alnT) by the method of least squares (see for example Worthing and Geffner 43

U u -1 -- • ,.... --• •-• • • • l .:: · c If- • :E .1 c • N •• ;- l"'- 0 .1 • • >- .... c ,r/' 1- ~ c •'. > -C> N -1- ... !a . / V') ./-. w ~ ./ •~ ./ ~ • • j: ;'. ••• / • • • •

:2 y cn.------J- = Co ~)(

1 •! 44

1943), the values obtained are:

8 p(lOKj~ = (10.20 ± 2%) x 10- ohm-cm., 8 p(lOK) Il = (l0.80 ± 2%) x 10- ohm-cm., -8 (III.2) (op/olnT)~ = (-0.744 ± 4%) x 10 ohm-cm., 8 (op/oInT)" = (-0.658 ± 4%) x 10- ohm-cm.

These values agree closely with the sarne values measured for polycrystalline sarnples by Hedgcock and Rizzuto (1967) for the same concentration. The errors listed corne from two sources. The first error is a probable error, defined as the arnount away from the mean value that contains 50% of the experimental points. The probable error in p(lOK) was about 0.1% while in (op/oInT) it was about 2%. The second source of error is that which arises from the measurement of the geometrical factors of the crystals and is of the order of 2%. Cornbining these two errors gives the estimated errors listed in equation 1II.2. The measurements were aIl made against a standard brass resistor, RR' which has a numerical value of 6.0 x 10 -8 ohms

± 5%. The errors listed above include only possible relative errors since quantities su ch as (p(lOK) Il and (op/oInT)" /(op/olnT)~ are independent of the absolute values of ~. III.2. Analysis and Discussion The equation derived by Kondo (1964) for the resistivity of a di lute alloy containing magnetic impurities is 45

(111.3) where PL a T5 is the lattice resistivitYi c is the concen tration of impurity atomsi PA is the normal or Coulomb scattering per unit impuritYi Z and Ef are the electron to atom ratio and the fermi energy of the host metal which are assUl,:ed to be unchanged by the addition of a small quanti ty of impuritYi and PM is the temperature independent contri bution to the spin or magnetic scattering per unit impurity.

Kondo's expression for PM is

(111.4)

where m is the conduction electron maSSi S is the spin of the impurity ion: and N/V is the number of atoms per unit volume. Equations 111.3 and 111.4 were derived by Kondo assuming a free electron model, i.e., the fermi surface is perfectly spherical. The question then arises about the effect that a non-spherical fermi surface will have on these equations.

Nagaoka (1965) derives an equation essentially similar to Kondo's but allows for the effects of non-free electron- like behaviour by allowing the effective mass of the conduction electrons to be a function of direction in momentum space [(l/m*)= (1;n2) (a2E/ak2 ) lE ]. The expression he obtains for f the spin scattering contribution to the resistivity is

cP = Km*J2[1 + N(O)Jln(T/.77D)] (111.5) M 46 where K is a constant involving things like the fermi energy, atomic volume, impurity spin, etc., which will be constant for any particular solvent and solute~ m* is the effective mass of the conduction electrons which is taken as a function of direction of electron motion; N(O) is the density of states/atom/spin at the fermi energy~ and D is the width of the conduction band and is usually taken to be of the order of the fermi energy. Thus using Nagaoka's expression, equation III.5, the total resistivity can be written as

2 (III.6) p(T) = PL + cPA + Km*J [A + N(O)JlnT] where A = l - N(O)Jln.77D is a constant. Since the experimental data indicates that in a certain tempe rature range the resistivity varies as the logarithm of temperature, it appears that the temperature variaticn of PL and PA are negligible in magnitude compared to that due to the spin scattering in this temperature range. Thus differentiating equation III.6 with respect to InT, one finds

ap(T) = Km*J2[N(O)J]. (IIL7) alnT

Forming the ratio of (ap/alnT) parallel and perpendicular

to the c-axis and assuming a nearly free parabolic band for the density of states, i.e., N(O) a (m*)3/2, one obtains

(ap(T)/alnT).L m*.L\ (J~)2 [N(O)J]~ (IIL8a) ( ap (T) 7~nnT) " = ( m* ,,-) J,;- [N ( 0 ) J J "

= (::~15/2(~~)3 (III.8b) ."1 47

Therefore

1/3 J~ = [(ap/nnT)~ (nt*"tj (III.9a) J:" (ôp/ôlnT)" m*.1.

= [ (1.13 ± .08) (1.04 ± .02)5/2]1/3 (III.9b) = 1.08 ± .04

where use has been made of equations 111.1 and 111.2. The

assumption has also been made that m*(Zn)~/m*(Zn)" =

m*(ZnMn)~/m*(ZnMn)II since the addition of a small quantity

of impurity atoms will not affect the fermi surface and

hence the appropriate effective mass for the alloy will be

the same as for the host metal.

Thus a small anisotropy is observed in the s-d exchange

constant J and hence in the strength of the spin scattering

paraI leI and perpendicular to the c-axis.

From equation III.8a and using expression III.9a for

J~/J" one finds 1/3 [N(O)J] Il = (ôp/ôlnT) .l r ( * / * )2/3 [N(O)J]~ L( ôP/ôlnT).I] m " m .1.

= 0.99 ± .05.

Thus within the experimental error it can be concluded that

N(O)J is a constant independent of crystal direction and

hence any parameter involving this product, e.g., the Kondo

temperature, will be isotropie.

To make an estimate of J for the ZnMn alloy, the assump-

tion made by Kondo that the spin and Coulomb scattering terms ., 48

are equal will be made. Making this assumption and evaluating equation III.3 at T = lOK results in a value for

c PM of .5p(lOK). Also, differentiating equation III.3 with respect to InT results in

Thus

J = (ap/alnT)Ef .5p(lOK)3Z

Using an average value for (ap/alnT) and for p(lOK) appropriate to a polycrystalline sample, i.e., p(lOK) =

(ap/alnT)"), the value obtained for J is -0.23 ev. This value is in general agreement with others obtained by various

methods for ZnMn, e.g., B. Paton and W.B. Muir (1968) ob-

tained -0.31 ev. from de Hass-van Alphen measurements.

Another parameter which is of importance in the Kondo

effect is the Kondo temperature, TK• Third order pertur bation calculations predict that

where Tf is the fermi tempe rature (= 1.2 x 1050K for zinc, -Il calculated assuming a fermi energy of 1.64 x 10 ergs.);

Œ is a constant of the order of unitYi and the other quantities

are as previously defined. Some typical values for a are:

Œ = 1/2 determined by perturbation techniques, such as 49

by Nagaoka (1965), Hamaan (1967), Abrikosov (1965), Kondo (1966); a = 2/3 Doniach (1968), Heeger and Jensen (1967};

a = 1/3 App1ebaum and Kondo (1968). Using the perturbation value, a = 1/2, resu1ts in a Kondo temperature of .05°K which is comparable to the value of .2°K measured from susceptibi1ity by Newrock, Serin, Boato (1969). 111.3 Further Discussion To make a numerica1 estimate of J an assumption con-

cerning the relative sizes of the Coulomb and spin scattering had to be made since neither can be determined by

experiment (Kondo's expression for PM' equation 111.4, 1eads to a value for PM which is exceeding1y sma11 and hence it must be assumed that it is numerica11y incorrect). The assumption made was the one suggested by Kondo taking PM = PA· It is recognized, however, that there are various theories which indicate .that 13 = PA/PM is sorne number not necessari1y equa1 to unity. H. Rohrer (1968) indicated

that for AuMn, 13 - .7 and varies with temperature. Changing

13 from 1 to .7 for ZnMn resu1ts in decreasing the value obtained for J by 15%. In addition; the experimenta1 data obtained for the ZnMn single crysta1s indicates that

(PM)~ > (PM)" whi1e (PA)" > (PA)~. That this is not unreason able can be seen by examining the theoretica1 expression for

the spin scattering term PM. Both Kondo (1964) and Nagaoka

(1965) find that PM is proportiona1 to J2 and since J~ > J" it • o 50

follows that (PM)~ > (PM)'" i.e., the spin scattering is stronger perpendicular to the c-axis than it is paraI leI to the c-axis. On the other hand it is not unreasonable to

expect that the Coulomb scattering off magnetic impurities will be the same as for nonmagnetic or normal impurities. For

the zinc it was foundthat PlI > p~. If this is attributed,

at least in part, to coulomb scattering off any residual

impurities then it follows that (PA)" > (PA)~ and this is what was observed for the ZnMn single crystals. Thus the

assumption made that PA = PM to evaluate J is only an app roximation.

Another interesting point is that in order for the

lattice resistivity, PL' to contribute significantly to the total resistivity, thereby producing the resistance minimum, the lattice coefficient A (PL = AT 5 ) in the zinc manganese (64 ppm) alloy must increase by a factor of about a thousand from that in pure zinc. Garland and Bowers (1968) find that

for indium, A is a function of the purity of the metal and increases rapidly as the purity decreases. Thus it is con ceivable that for 64 ppm manganese in zinc the observed change in A is not impossible. 51

CHAPTER IV e CONCLUSIONS

An experiment was conducted to determine whether there

exists any anisotropy in the strength of the spin or exchange

scattering in an anisotropic dilute magnetic crystal. This

was done by measuring the temperature dependence of the re

sistivity of single crystals of zinc and zinc rnanganese alloy

(64 ppm) along the principal axes of the axially syrnmetric

crystal.

In the pure zinc single crystals below 7°K an anisotropy

was found with the resistivity parallel to the c-axis being

greater than that perpendicular to the c-axis by 4%, i.e.,

p"/p~ = 1.04 ± .02. From this it can be concluded that the

effective masses of the conduction electrons at the fermi

surface parallel and perpendicular to the c-axis differ by

4%, i.e., m*"/m*~ = 1.04. In the zinc manganese single crystals a srnall anisotropy

was found with the spin scattering perpendicular to the c-axis

being larger than that parallel to the c-axis by 8%, i.e.,

J~/Jn = 1.08 ± .04where J is the direct or s-d exchange in tegral. Within the experimental error it was also determined

that the product N(O)J was isotropie where N(O) is the density

of states per atom per spin at the fermi surface. Thus it

appears thatthe anisotropy in J is compensa.ted for by an

anisotropy in the density of states.

Estimating an average value for the exchange constant J 52

appropriate to a polycrystalline sample as done by Kondo the value obtained for J is -0.23 electron-volts. Using this value of J and the perturbation expression for the

Kondo temperature, a value for TK of .05°K is obtained. 3 53

APPENDIX

JOSEPHSON TUNNELLING

A Josephson Junction is a device consisting of two

superconàuctors separated by a thin insulating barrier, as in figure 18a. When there is no magnetic field present, the maximum current that can tunnel across the barrier without any resulting voltage is given by

(A.I) l = l o sin ~ •

In equation A.I, 10 and ~ are defined as follows:

(A.2)

(A.3)

In equation A.2, ~l and P2 are the density of electron states in the superconductors land 2 respectively, while K

is a measure of the correlation between electron pairs in the

two superconductors. In order for tunnelling to be possible, the correlation lengths (a measure of the distance over which

electrons can be said to see and interact with each other) of the superconductors must overlap. As the barrier becomes thicker, the overlap of electron pair wavefunctions between the superconductors decreases resulting in a smaller possible super-current, until, for a thick enough barrier, no tunnelling is possible. AlI these parameters concerning overlap and barrier thickness are contained empirically in the pro- portionality factor K. For a given junction, the number K 54

,...INSU LATOR ~. ~ 'SUPERCONDUCTOR ~ SUPERCONDUGTOR .~ . A B ~

FIG l8~

;, --- 0; - }PENETRATION DEPTH

/I/Hf /1/1111. II. ï//Jl{J'///1 fi) B

-- -I~- -->.C -~I--- 2

FIG l8b

S CHE MA tl C 0 F A

SINGLE JOSEPHSON JUNCTION s 55

will be a co~stant.

The electrons, correlated in pairs in a superconductor,

can be looked upon as having a wavefunction per pair repre-

sented by

{A.4}

where 8 is called the phase angle of the electron pairs. AlI

electron pairs in a bulk superconductor have the same phase

angle. As the electron pairs from superconductor 1 tunnel

across the barrier to superconductor 2, their phase angle 81 must change by exactly the right amount so that it equals the

phase angle in the second superconductor. The change in e2

phase, ~, is given by equation A.3, and is called the phase

difference across the barrier.

Theoretically, l , defined by the constants of the o junction, determines the maximum possible super-current that

the junction can support, while the phase difference, ~,

across the barrier determines the fraction of l that the o junction does support under the operating conditions. In

actual practice, a current l, less than or equal- to l 0 , is driven through the junction, and the phases adjust themselves

to give a phase difference so that A.l is satisfied.

The situation is altered if a magnetic field is present

in the barrier as shown in figure l8b. The magnetic field

will cause the electron pairs to follow a curved path, here

represented approximately by the contours Cl and C . It can 2 o be shown (Adkins 1967) that '6 56

~ _ ~' = 2e lA.ds CA .. 5) 'Ttc '1' c

where the closed contour C is the sum of the contours Cl and

C in the superconductors and of the contour in the insulator 2 which is assumed negligible in dimension compared to the

penetration depths (the thickness of superconductor required

to reduce the magnetic field to Ile of its value outside the

superconductors): ~ and ~' are the phase differences across the barrier where the contour C intersects the barrier; A is the vector potential due to the maanetic field: and dS is an

element of length along the path of inteqration C.

Evaluating this integral one finds

~ ~' 27T4>14> (A. 6) - = o

where 4> is the total flux enclosed in the contour C and

4> = hc/2e o

is the flux quantum for superconductors and has the numerical

value of 2 x 10-7 gauss-cm2 •

By differentiating, equation A.5 can be rewritten as

_2ed ~= B (A. 7) dy -nc

where y is measured along the length of the barrier; and d

is the barrier thickness plus the two penetration depths.

Equation A.7 states that in the presence of a magnetic

field, each point of the junction can support a current de-

termined by its position along the barrier. The total current, 7 57

Icritical' that the junction can support is found by summing the currents that each local section can support, i.e.,

I .. = fIoSin~(Y)dY cr~t~cal barrier

Evaluating this integral, it can be shown that

sin(1TB/B ) o Icritical = Io (1TB/B) (A. 8) o In equation A.8, Io is the maximum current that the total junction can support with no magnetic field present, while B is the field required to produce a change of one o flux quantum in the barrier and penetration depths of the superconductors. Thus one expec·ts a periodic variation in the critical current as the magnetic field is varied (figure

19), the period of oscillation being B . o Figure 21 shows the schematic diagram of a device con-

sisting of two junctions, "a" and "b". A current, Itotal' en tering from the left, divides into two parts Ia and Ib' flowinq through junctions "a" and "b" respectively. An interference

between the two junctions resu1ts at the point Q, caused by the difference in phase of the two currents through the sep- arate paths. This is a result of the magnetic flux enc10sed

in the loop from point P to point Q and back to P in a con- tinuous circular path. In going from P to Q along the upper path, the change in phase is given by (R.P. Feynman, 1966) A.dS ~ Phasep~ = ~a + ~c2ef upper (A.9) 'Cl 58

1CRITICAL

MAGNET) C FI ELD . ( GAUSS)

MAGNETIC FIELD EFFECT ON A

SI NG LE JOSEPHSON JUNCT ION

FIt;19

-1 O. 1 0

MAGNETIC FIELD (MILLIGAUSS) MAGNETIC FIELD EFFECT ON A

DOUBLE JOSEPHSON JUNCTION

FIG 20 59

a .NSULATOR ,-SUPERCONDUC ...." ___".- _ __ ,,.t TOR 'CI. ~-- '\ 1 .----- ...... , J t 1 , \ __ _-,1_ .. ~1 " Q l' T OTAL ":).-..-..... -...,. ,. .. " ~ \ ®B 1 e· t, , \ 1 '---t"b 7- - ...... -- ...... ""

b SCHEMATIC OF A DOUBLE ·JOSEPH SON JUNCTION

FIG 21 o 60

with a similar expression for the lower path. In equation

A.9, ~a represents the change in phase across the barrier

"a" with no magnetic field present whlle the second terrn adds

the phase change due to the presence of the magnetic field.

Since the phase at Q must be independent of the path

taken (all electron pairs in a bulk superconductor have the

same phase), one has

2e l Â.dS ~b - ~a = l'l.c 'J'PQP

= 2ecI> 1"lc

= 21T~ (A.10) cI> o

where again cI> is the total magnetic flux enclosed by the superconductors.

The total, or critical, current that can be supported

with no voltage appearing is, assurning two equal junctions,

i. e. , l , = o

= l sin~ COS1T: o 0 0

l sin~ cos1TB/B (A .11) = 000

where ~o is a constant phase angle dependent on the average current actually flowing through the junctions. For a double junction device, the non-superconducting . are a enclosed between the two junctions is much larger than the 61

normal area enc10sed by a single junction. Thus it takes a smaller field B to produce a change of one flux quantum o

~o than it does for a single junction. This results in a much faster rate of change of l "t" l with magnetic field cr~ ~ca (see fig. 20) for the double junction device as compared to a single junction.

In an actual double junction device, the two junctions are seldom equivalent. This means that total cancellation is never cornpletely achieved and also the oscillations are never perfectly periodic or symmetrical. Typical experimental curves for a double junction device are given in figures 6a and 6b. 62

BIBLIOGRAPHY

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