(SANITIZED)UNCLASSIFIED SOVIET RESEARCH PAPER ON LOW TEMPERATURE PHYSICS(SANITIZED)

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• Document Creation Date: December 22, 2016
• Document Release Date: January 6, 2012
• Sequence Number: 1
• Publication Date: October 29, 1962
• Content Type: REPORT

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THE TUNNEL EFFECT ON .SINGLE-CRYSTAL TIN
-ZAVA P ITS KY.USSR
The author studied the tunnel effect between single crystals of tine
and tine films. The thickness of the films was loss than 1000 R. The
current voltage and current- dV characteristics of Sn fiin Sn oxide -
Sn crystal sandwich were measured. The sandwich resistances are 0.5 + 50
ohm/mm2.
From the B.C.S. theory /4, 5/ it may be secn/1 3/ that for two
suporcon,luctors there is a noticeable jump in the tunnel current at oV =
41 + d 21 even at finite temperatures.
The experimental results for different orientations of crystals are
shown in Figs.la, lb, lc. The results shown in Fig. la agree satisfactorily
with the B.C.S. theory.
Another type of characteristics give Figs. lb
and lc. Here several separated jumps in the current- voltage or o(v)
-voltage characteristics (6 *= n) and several minima in dV are observed.
d
The transition from la type to lb, lc typo is observed when the normal to
the surface of tin crystals deviates from the 10011 orientation. From
experiments /2, 3/ we know that in the case of thin films only one jump in
the V curve is observed. Hence, we may conclude that several jumps
observed in prescnt experiments (see Figs. lb, lc) are dui: to the properties
of the. single-crystal tin.
At present only speculations on the origin of the characteristics of
second type are possible. Recent experiments /6-8/ demonstrated the presence
of several parts of the Fermi surface of tin. It is possible that these
parts have in the supercon_'ucting state different energy gaps. Then the
jump in the current or ((v) should be at eV =p 1 + /, 2 where 2 is the
energy gap of some electron group which play an important part in the
tunnel current for a given orientation of the crystal.
From the jump of 6' (v) characteristics we found, in addition to
2 = 0,56 f 0,58 meV which is particular to every orientation, a 2 = 0,45
meV,d 2 = 0,65 meV for several orientations.
The above results are obtained at temperatures 1036?K. At higher
temperatures the second type jumps are not observed only in the interval about
0.3? from Tc. In Fig. 2 the temperature dependence of d, 2 is shown for
specimens of different orientation. Those data are in excellent agreement
with the B.C.S. theor,;.
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References
1. Giaever I., Megerle K. Phys. Rev., 122, 1101 (1961).
2. Zavaritsky N.V., Zhurn. Exp. Toor. Phys., lil, 657 (1961).
3. Giaever I., Hart H.R., Jr., Megerle K. Phys. Rev., 12 9I1 (1962).
4. Bardeen J., Cooper L.N., Schrieffer J.R. Phys. Rev., 106, 1175 (1957).
5. Bogolyubov N.N., Tolmachev V.V., Shirkov D.V. A new tboory of method in
the suporconduction.
Moscow. (1958).
6. Gold G.E., Priestly M.G. Phil. Mag., 5, 1089 (1960).
7. Khaikin M.S., Zhurn. Exp. Toor. Phys., 1/ , 27, A, 59 (1962).
8. Gantmakher V.S. Report to the IX Low Temperature Physics Conference.
Leningrad. (1962).
Captions
Fig. 1. Tunnel effect for single-crystal-tin specimens of different
orientation. The angles of the normal to the surface of tin
crystals 7 and the [0013 direction are a - 9 = 22?, 5 -.9 = 60?,
V characteristics, --r-- dV
d
Fig. 2. &1 +A 2 temperature dependence for specimens of different
orientation. - - - B.C.S. theory with 2 Q = 1,12 meV at T = 0? t2).
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al a4
FIG.2
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MAGNBTOACOUSTIC OSCILLATIONS AND FERMI SURFACE
P.A.Besugly, A.A.Gal.ldn, A.I.Pushidn
The anisotropy of magnetoacoustic oscillations for wave vector
directions of sound wave q along the principal PA V [zoo] , tIII]
crystallographic directions has been investigated on aluminum specimens
at 4.2?K, ultrasonic frequency 183 Mc/s and 223 Mc/s and in magnetic
fields of 2500 oersteds. Fxpurimental results agree with Fermi surface
for second zone, deduced for aluminum by Harrison on the base of nearly
free electron ;.rudel, representing its size and shape, and at the same
time point out at the absence of sharp intersections at the surface of
second zone.
INTRODUCTION
The oxperihental investigations of oscillation anisotropy of
ultrasonic absorption coefficient are of great interest because they
allow to deteroine the oxperir.rental diameter of Fermi surface for
electrons in a metal and for some cases completely restore its shape/I/.
Here one should not forgot, that the funomona of geometrical resonance
will be observed, if the condition ql ? qr j I (!. -mean free path of
electrons in metal, r-radius of cl. ctrcn orbit) is fulfilled. Thus, the
number of oscillations and hence the reliability of extreme Fermi surface
dimensions are given by l value. This means that the study of oscillation
anisotropy of iltrascnic absorption coefficient has to be made with single
crystal metals of high purity, using ultrasonic frequency of possibly high
frequency.
This paper is devoted to oa netcaccustic effects in aluminum. At
present two papers devoted to experir. untal study of magzotoaeoustic
effects in this metal are Imovrn. In the work of Morse and Bohm/2/ the
oscillations of sound absorption coefficient were not observed,
first time the oscillations cf ultrasonic absorption coefficient in
aluminum were observed by Roberts"", 3who studied absorption coefficient
dependence of magnetic field for longitudinal waves at 10 to 100 Mc/s
frequency. The shortcoming of these studies, which, as it is shown by
Roberts, are in qualitative agreement with Fermi surface model, proposed
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by Harriso,/5/, is that the value of limiting pulse was determined with
small accuracy, as not more than two insufficiently sharp oscillations were
observed by the author. It was interesting therefore to carry out
investigations on aluminum specimens of higher purity, increasing at the
same time the ultrasonic frequency.
EXPERIMENTAL PROCURE AND SPECIMENS
The investigation of magnetic field dependence of ultrasonic wave
absorption coefficient was carried out by a pulse method, described
earlier/6/. The experiments were accomplished by using longitudinal
sound at T=4.2?K in magnetic field up to 2500 oersteds at two frequencies:
183 Mc/s and 223 Mc/s on aluminum specimens for which
R2 3 K = 14000 - 20000
R 4.2 K
As the sound absorption coefficient in aluminum at helium temperatures
and frequencies used is fairly large, the aluminum specimens were used in
the form of disks of 10 mm diameter and about 2 mm thick. The application
of comparatively thin specimens involved a delay element, the latter being
used to separate the last sound pulse from the probe pulse. A bar of
crysalline quartz of square section has 8 mm side and 10 millimeters
length, which together with aluminum specimen was arranged between the
transmitting and receiving quartz was used as a delay element.
Following specimens were used: specimen I with a deflection of the
normal to the disk surface from the {IIO0 crystallographic direction not
exceeding 2? and specimen 3 with the deflection of the normal to the disk
surface from [IIII crystallographic direction not exceeding 2?. The sound
a rave vector q coincided with the direction of the normal to specimen plane.
The velocities of longitudinal sound wave distribution in aluminum for
different orientations were taken from Roberts' paper/3/.
The magnetic field dependence of absorption coefficient of longitudinal
ultrasonic wave in aluminum specimen was recorded by means of two-
coordinate recorder/6/.
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RESULTS
The studies of absorption coefficient oscillation anisotropy 61r. were
carried out when the waves vector q was directed along one of crystal axes
- [110], CI00d and (III), and the magnetic field vector fl was rotating
around q. For the safety of good reproduction of results the Qt' (H)
dependence by a given direction of sound distribution and a given
direction of magnetic field vector $ was recorded twice. In all cases
the obtained oscillation curves appeared identical.
I. q along (IIO] ; ' J = 183 Mc/s.
The particular feature of the results for this orientation is the
availability of a large number of oscillations (to 15 oscillations) for
a number of directions of magnetic field vector H , that indicates at
long mean free path of carriers. The second circumstance, indicating
at the large ql value (for some directions ql) 200), is the mode of
dependence oft (H): absorption coefficient value at saturation is
essentially greater than that without megnetic field.
A large number of sharp oscillations with a good periodicity in a
reversed field were observed q [tid] , H [iiol orientation; i.e. for such
direction of magnotic'field R towards wave vector q, for which in Roberts'
exporiments/3/ the oscillatory behavior of absorption coefficient was not
mentioned. The dependence mode of e((H) (here and further4( is expressed
in arbitrary units) for this direction is given on fig.I.
With the change of the direction of magnetic field vector H the
oscillation period in reversed field changed: at first the decrease of
period was observed, then its increase was observed. Fig.2. gives the
run of c;,< (H) dependence for the case, when the magnetic field vector FR
made an angle of 35G with [II01 direction. The simple comparison of
oscillation curves in figs.I and 2 shows that the.oscillation period in
the second case is larger than in the first one.
Simultaneously with the short period oscillations the long period
oscillations are observed by the directions of magnetic field, making an
angle of 0? - 25? with (OOI] direction. Because of superposition of
oscillations of different periods, the interpretation ofcx (H) dependence
curves appears fairly difficult, that is why in the mentioned directions
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it was possible to estimate periods in the reversed field only for long
period oscillations not exceeding 10 per cent. Long period oscillations
are also observed in the case when magnetic field vector makes an angle
of 15? - 25? with [ii0) direction. Under these conditions however, it
appeared possible to determine the value o 1 only for short period
oscillations .
The distinct oscillatory behavior of absorption coefficient is also
observed for-* [iioj, fl [OOI) orientation, for which in Roberts'
experiments"3~ the oscillations were not mentioned. The particular feature
of this orientation was absence of short oscillations. For this direction
of the field with a period o NH = (6, 4 t 0,3).l0 qe were distinctly
observed. Thu 0- (ii) dependence made for this case is given in fig.3.
Let us compare experimental data obtained with Fermi surface model
for aluminum, proposed by Harrison. Aluminum belongs to metals with
face-centered cubic lattice, for which the first Brillouin zone has a form,
shown in fig.4a by dotted line. According to Harrison"'", 45the first
Brillouin zone, completely filled with electrons, is surrounded by pockets
of holes of second zone, the surface boundaries of which are shown in
fig.4a by solid lines. The third zone (see fig.4b) has an interrelated
set of arms with varying cross sections.
is connected with
As it is known/I'V the oscillation period ? (H)
extreme distances R to Fermi surface (R is normal to q and fl) by a simple
(I)
2` ei
where = L (~ Flank constant),
2n
- the length cf sound wave,
v.. - electron charge,
c, - velocity of light.
If now, using ratio (I), the K values are expressed in Ko units
(K = M , where () 0 = 4.04*10--8 cm)$ found by oscillations periods in
reversed field for each direction H, they will represent the distance in
wave numbers space from the center of Brillouin zone to Fermi surface of
corresponding zone.
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-5,
Fig.5 gives the projection of Brillouin zone onto the planes normal
to (I10) axis, where the form of central section of Fermi surface for
second zone by a plane, also normal to CIIO) direction, is given on
corresponding scale.
In the same figure the values, obtained from experiment based on
ratio (I) are given. The attention is drawn to the facts that the
short period oscillations, for which the larger K values are responsible,
the form of Fermi surface central section for given direction q on the
whole represent well, thus supplying proof of Harrison model. As to
long period oscillations, for which the angular dependence of projection
of extremal diameter onlll0J plane is given in fig.6, the question about
their origin needs further more profound analysis. Probably for these
oscillations non-central sections in the vicinity of zone boundary in
LII00 direction are responsible.
II. q along CI001, = 183 Mc/s
The reliable oosillatory behavior of absorption coefficient is found
practically for all directions of magnetic field R, normal to wave vector
q. Unlike to previous case however, when the sound was directed along
(IIOj axis, the number of observed oscillations was not great (1+-5
oscillations). The second particular feature of oscillations,,obsorvod
for the given wave vector q, is their comparatively large period. In
fact, if at frequency ' = 183 Mc/s for the case when sound wave vector
was directed along [110 axis, the oscillations periods reproducing the
central section of Fermi surface second zone by rotating A from 0 0 to
180?, changed in the range of 2.03'10 4 - 3.01+'10 oe -I then in our
case the oscillations period changed in the range of 6.I'I0' -
7.350I0 4oe-I, i.e. the period was 2,5-3 times larger. It means, that
for these oscillations non-central orbits are responsible.
At the same time it should be noted that for the orientations in the
vicinity of q [I00], R (001 and 71 (I06, R._ [OT0J orientations with long
period oscillations short period oscillations were observed. Naturally,
because of superposition of oscillation of different periods, the accuracy
of a M I determination for short period oscillations was not good enough
(5-10 per cent).
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-,6w
As an illustration of oscillatory dependence of absorption coefficient,
when q is directed along [I00j axis, a record of O( (H), when A maims an
angle of 500 with [OOI) direction, is given in fig.7.
Fig.8 gives the central section of Brillouin zone by the plane, normal
to (I00) axis, at which the central sections of the second zone surface and
two other sections, situated from the central section along the perpendicular
at the distance of 0,75K0 and 0,90 0 (Ko = uo) are represented in proper
scale. As it is seen from the figure the short period oscillations,
observed in a small interval of angles, represent well the dimensions of
central suction in 10I0] and (00I~ directions, and short period oscillations
mainly represent the form of section at K = 0.9K0, thus indicating the
absence of sharp intersections in accordance with completed Harrison's
theor/5/.
Magnetoacoustic effects in aluminum in the direction of sound along
[IOOj axis were studied at two frequencies: 183 and 223 Mc/s; the
values obtained at those two frequencies being in good agreement.
III. q along fIIII ; = 223 Mc/s
As in above mentioned case, the trustworthy oscillating behavior of
absorption coefficient was practically observed for all directions of
vector H in the angular interval 00 1800. Because the number of
oscillations was not large in this case the accuracy in determination of
K (sue eq.(I)) was about 10 per cent. The measurement at a given
direction of wave vector q were carried out at frequencies 183 Mo/s and
223 Mc/s. In view of the fact that the values of measurements at these
frequencies gave a good agreement, and taking into account that the
oscillations of absorption coefficient at 'A = 223 Mc/s appeared more
distinct; further the comparison with the form of the surface, proposed
by Harrison, will be based cn results, obtained at 223 Mc/s.
The shape of Brillouin zone along (III-1 directions is given in fig.9.
The shape represents the central suction of Fermi surface by III, plane
for second zone and two non--central sections at the distance of 0.6K0 and
0.7K0 from central section. Values given in the same figure and
calculated after measured oscillation periods on the base of oq.(I) in
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-7-
the range of experimental error are in good agreement with the form of
section, corresponding to level 0.6Ko.
D I S C U S S I O N
The above comparison of experimental values, obtained in this paper,
with the shape of Fermi surface for aluminum, proposed by Harrison on the
base of nearly free electrons model, shows, that both dimensions of second
zone and its shape can be well reproduced by values of magneto-acoustic
measurements. The discrepancy, as a rule, is in experimental limit of
error, with the exception of sharp edges of Fermi surface; the results of
present study testify on behalf of completed calculation of Harrison"";
5the latter introducing no essential changes in the dimensions of second
zone Fermi surface, lead to rounding its sharp edges. Unfortunately it
was not possible to carefully study it in just these parts of Fermi surface.
At those directions of wave vector q and magnetic field vector I!, that had
to give information about the dimensions of Fermi surface in the directions
of sharp intersections, the oscillations of absorption coefficient proved
to be bad. Apparently it's due to small electron state density in
indicated directions.
In this paper it was possible to state the character of anisotropy
of short period oscillations, representing the central section of Fermi
surface of second zone for wave vector q directions along cIIOJ and (I00]
axes more or less in detail. Also we succeeded in clearing up the
character of oscillation anisotropy with 2.5.3. times larger period which
represents non-central sections of second zone Fermi surface. At the
same time for some directions of q and R the oscillations with still
larger period, which were earlier mentioned in our short ncte/8/, were
studied. Tc estimate their period with enough authenticity and the more
so to study the character of anisotropy of their period for the present
is not possible, although to find them is very interesting. The study
of anisotropy of period of these most long period oscillations (which is
a hard but evidently not a hopeless task) will, probably, throw some light
on the structure of third zone.
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Explanations to the figures to the per
"MAGNETOACOUSTIC OSCILLATIONS OF FERMI SURFACE IN AluMINUM"
by P.A.Besugly, A.A.Galkin and A.I.Pushlcin.
Fig.I. Record of magnetic field dependence of absorption coefficient
at q along (II0] and 2 along [IIO] ; I = 183 Mc/s.
Fig.2. Record of magnetic field dependence of absorption coefficient
at along (IIOj and A direction, making 35? angle with rII0);
y = 183 Me/s.
Fig-3.
Record of magnetic field dependence of absorption coefficient
at q along (II01 and R along [0013 ; 'I = 183 Mc/s.
Fig.4.. Fermi surface in aluminum after Harrison :
a) Regions of hol.;s in second zone;
b) Regions of electrons in third zone*
Fig.5. The projection of Brillouin zone onto the plane, normal to
[II0] axis and the form of central section of Fermi surface
for second zone, 0 --o - measurements data (0 - 2-3 per cent
3
accuracy, 5-10 per cent accuracy) Roberts' results
Fig.6. Angular dependence of projection of extremal diameter onto the
long-period oscillations
1110] plane, obtained from 6H1
measurements.
Fig.7. Record of magnetic field dependence of absorption coefficient
at q along. [100 and H direction, forming 50? angle with COOI] ;
~ = 183 Me/s.
Fig.8. The central section of Brillouin zone by the plane, normal to
(ICO( ,-.xis, on which the contours of sections of Fermi surface
for second zone are shown. The levels are given in Ko units.
values from long period oscillations measure-
ments (Q -Iabout 5 per cent accuracy) --?.-- - about 10 per cent
I
accuracy) d1, lad, - values from short period oscillations
measurements 223 Mc/s ? - 183 Mc/s) f--1 -1 - Roberts'
results"3/.
Fig.9. The projection of Br-illouin zone onto the plane, normal to [III],
in whioh the contours of section of Fermi surface for second zone
are drawn. The levels are given in K units.
Roberts' results131
.
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/
/
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Bffi'FIR MES
Z 71, 1959.
2. R.W.Morse, H.V.Bohm, Proce of the Forth International Conf? on
Low Temperature Physics and Chemistry, Madison, Wisconsin, 1957.
3. B.W. Roberts, Phys.Rev., 119, 1889, 1960.
4. W.A. Harrison, Phys. Rev., 116, 555, 1959.
5. W.A. Harrison, Phys. Rev., 118, 1882, 1960; 118, 1191, 1960.
6, 6, 199, 1960.
7. M.H.Cohen, M.J.Harrison, W.A.Harrison, Phys.Rev., 1179 937, 1960.
8. 42, 84, 1962.
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FIG.I.'
FIG.2.
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FIG.3.
fIG.4.
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FIG.5.
-ZO CsK- I
D skcnw IO st*cw
10
?a
?0
60 30' 00 30 000
Cj B 9onb 110]
FIG.6.
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F IG.7.
FIG.8.
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[110]
FIG.9.
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STAT
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