Magnetoquantum oscillations in the resistance of metallic point contacts: Influence of nonequilibrium phonons

0.45 MB·

by  N. L. Bobrov

Magnetoquantum oscillations in the resistance of metallic point contacts: Influence of
nonequilibrium phonons
N. L. Bobrov1,2,∗, J. A. Kokkedee1, N. N. Gribov1,2, I. K. Yanson2, A. G. M. Jansen1, P. Wyder1
1High Magnetic Field Laboratory,
Max-Planck Institut fur Festkorperforschung and Centre National de la Recherche Scientifique,
F-38042, Grenoble, Cedex 09, France
2B.I. Verkin Institute for Low Temperature Physics and Engineering,
of the National Academy of Sciences of Ukraine,
prospekt Lenina, 47, Kharkov 61103, Ukraine
6
E-mail address: [email protected]
1
0 (Dated: January 5, 2016; Published Physica B 218, 42(1996))
2
The amplitude of magnetoresistance quantum oscillations of Al and Be point contacts in a mag-
n
netic field parallel to the contact axis has been studied as a function of voltage applied over the
a
J contact. Itwasfoundthatforonegroupofcontactstheoscillationamplitudenonmonotonouslyin-

2 creaseswiththebiasvoltageincrease,whilefortheotherpartofthecontactsadecreaseofamplitude
was observed. The scattering of electrons with nonequilibrium phonons and also phonon-phonon

]
l collisions will be discussed as the possible reasons of the observed effects.
l
a
h PACSnumbers: 71.38.-k,73.40.Jn,74.25.Kc,74.45.+c
-
s
e
m I. INTRODUCTION where the constant C has a meaning of the ratio of the
harmonicsatT =0intheabsenceofscatterers. Fromthe
.
t
a The resistance of metallic point contacts at low tem- theoryfornoninteractingparticlesitfollowsthattheDin-
m
peraturesshowstheoscillationsasafunctionofmagnetic gle temperature can be affected by any electron scatter-
- field, which are due to the Landau quantization of the ing process, including electron-phonon collisions [3]. At
d
n conduction electron energy spectrum [1, 2]. An increase the same time, with many-body interactions taken into
o
intemperaturereducestheamplitudeofoscillations. The account, the scattering of electrons on phonons seems to
c
[ corresponding reduction factor is have no effect on the Dingle temperature [3]. The study

1 R = 2π2nkBT/¯hωc ; (1) of the influence of electron-phonon collisions on the am-
v T sh(2π2nk T/¯hω ) plitudeofmagnetoquantumoscillationsintraditionalex-
B c
5 periments is limited to metals with low Debye energy, in
6 where n is the harmonic number in the oscillations and
which phonons can be excited at a tem, perature low
1 ω =eB/m∗cthecyclotronfrequency. Elasticscattering
c
0 enough to observe the effect. Metallic point contacts
of the electrons by impurities also causes the decrease
0
present a unique opportunity to study the influence of
. of the oscillation amplitude nearly the same as resulted
1 nonequilibrium phonons generated in the contact on the
0 fromthetemperatureincreasefromactualvalueT upto
quantum oscillations. Just by applying the bias over the
6 T = T +x. The reduction factor (the Dingle factor)
eff
1 contact,thenonequilibriumphononscanbegeneratedin
has the form
:
v the constriction with any possible energy up to eV [4].
i Inthisworkwestudytheeffectofelectron-phononscat-
X
r RD =exp(−2π2kBnx/¯hωc), (2) tering on the amplitude of the resistance oscillations of
a
where x = ¯h/2πk τ is the Dingle temperature, and τ is Al and Be point contacts in a magnetic field parallel to
B
the elastic lifetime of the electrons. As follows from Eqs. the contact axis.
(1) and (2) the ratio between the first and the second
harmonic of the oscillations is
II. EXPERIMENTAL DETAILS
A /A =Cch(2π2k /¯hω )exp(π/ω τ), (3)
1 2 B c c
Pointcontactswereformedbetweenedgesoftwosingle
crystal electrodes with the same crystallographic orien-
∗Correspondingauthor. tations. The applied magnetic field was always parallel
2
FIG.1: FragmentsofmagnetoresistanceoscillationsofanAl
point contact of R =1.41 Ω for two different bias voltages.
0
to the contact axis. All the experiments were carried
out at the temperature 1.3 K. The first and the sec-
ond derivatives of the I − V curve were measured by
standard modulation technique. To determine the en-
FIG. 2: (a) Point contact spectrum of an Al point con-
ergy dependence of the amplitude of magnetoresistance
tact of R = 1.41 Ω and relative magnitude of the first
0
oscillationsthefollowingprocedurewasused. Forseveral harmonic of the resistance oscillations versus bias voltage;
fixed bias voltage dV/dI(B) dependencies were recorded ∆R/R ≈5×10−4 at V =0 and B =9.5 T.
0
inthesameintervalofmagneticfield(usuallyinbetween (b) Ratio between the amplitude of the first and the sec-
8 and 10 T). Fig. 1 shows the fragments of dV/dI(B) ondharmonicofthemagnetoresistanceoscillationsversusbias
curvesforanAl pointcontactfortwobiasvoltages. The voltage.
amplitude of oscillations A and its second harmonic A
1 2
wasobtainedbymeansofFourieranalysis. Foronegroup
component of the transport velocity of electrons from
of contacts the voltage dependence A (eV) was found to
1
the extremal cross section of the Fermi surface and cor-
vary nonmonotonously with applied voltage with simi-
respondinglytoanincreaseofquantumoscillationsinthe
laritiestothepointcontactspectrumofelectron-phonon
point contact magnetoresistance. At the same time, the
interaction. Such an effect was observed for low Ohmic
scattering of electrons on nonequilibrium phonons in the
Alpointcontacts(Fig. 2(a))andBecontacts(Fig. 3(a)).
boundariesofthepointcontactwillcausetheincreaseof
High Ohmic Al point contacts show a decrease of the
the Dingle temperature and thus a decrease of the oscil-
amplitude of oscillations in their magnetoresistance with
lationamplitude. Theresultingeffectwillbedetermined
the bias voltage increase (Fig. 4(a)).
by the balance between these two processes.
It is obvious that the purity of the metal in the re-
III. DISCUSSION gion adjacent to the constriction area is an important
factor affecting the oscillations of the electron density
Phonons generated in point contact are the result of of states. As follows from Eq. (2) the oscillation am-
the energy relaxation of electrons injected from the con- plitude has nonlinear dependence on the concentration
tact. In analogy with elastic impurity scattering, the of the impurity scatterers. In the case of a large value
scattering of electrons on these nonequilibrium phonons of ω τ (low impurity concentration) it saturates and ex-
c
will influence the amplitude of magnetoquantum oscilla- ponentially drops at small values of ω τ. Hence, if the
c
tions. As with the elastic scattering [5], phonon-induced concentration of defects in the point contact boundaries
scattering within the constriction region will cause the is low, the small increase in the number of the electron-
randomization of the momenta of electrons. This ran- phonon scattering events leads to insignificant decrease
domization will lead to an effective increase of the z- of the oscillation amplitude. However, if the impurity
3
FIG. 3: (a) Point contact spectrum of an Be point con-
FIG. 4: (a) Point contact spectrum of an Al point con-
tact of R = 8.66 Ω and relative magnitude of the first
0 tact of R = 8.73 Ω and relative magnitude of the first
0
harmonic of the resistance oscillations versus bias voltage;
harmonic of the resistance oscillations versus bias voltage;
∆R/R0 ≈2×10−3 at V =0 and B =9.5 T. ∆R/R ≈3×10−3 at V =0 and B =9.5 T.
0
(b) Ratio between the amplitude of the first and the sec-
(b) Ratio between the amplitude of the first and the sec-
ondharmonicofthemagnetoresistanceoscillationsversusbias
ondharmonicofthemagnetoresistanceoscillationsversusbias
voltage.
voltage.
concentrationintheboundariesishighenough,asimilar sistance oscillations is observed (see Fig. 4).
increaseinthenumberofelectron-phononcollisionsmay As for Be, the increase of the oscillation amplitude
dramatically affect the oscillations. with bias rise was observed for all contacts (Fig. 3).
Taking into account all the factors mentioned above This is probably due to high rigidity of the Be crystal
one can expect an increase of the amplitude of magne- lattice, which prevents the spreading of the deformation
toresistance oscillations in relatively large (compare to to the boundaries during the contact formation. These
the diffusive mean free path for phonons) low Ohmic deformations are concentrated near the surface, forming
contacts with pure boundaries. Nonequilibrium phonons theeffectivereflectors,whichreturnthephononsbackto
generated in such contacts are accumulated within the the contact.
constriction area and play an effective part in the Thegivenmodelcannotonlyexplainthemonotonous
isotropization of the electron momenta. Purity of the part of the voltage dependence but also its similarity to
boundariesisofgreatimportancebecausetheoscillations the electron phonon interaction spectrum. The group
aremainlytheresultofquantizingconditionsthere. Fig. velocity of the phonons generated by the electron flow
2(a) shows the example of A (eV) dependence for one of depends on their energy and is minimum for phonons
1
low Ohmic Al point contact. The oscillation amplitude with energies close to the maxima of the phonon density
dramatically increases with the bias voltage increase. A of states. This means that the phonon energy relaxation
low value of the ratio A /A at eV = 0 (Fig. 2(b)) in- length will decrease stepwise at voltages corresponding
1 2
dicates the high purity of the boundaries of this contact. to the characteristic phonon frequencies. Therefore, one
Inthecaseofsmall(highOhmic)contactsthegenerated should expect the similarity of A (eV) with the first
1
phonons leave the constriction area. If, additionally, the derivative dV/dI(eV) of the I −V curve, which is not
boundariesofthecontactarenotpureenough(highvalue the case in our experiments. However, in these specula-
of A /A ), the decrease of the amplitude of magnetore- tions the role of phonon-phonon collisions was not taken
1 2
4
into account. intheregionadjacenttotheconstrictionarea. Asforthe
Thestepwisedecreaseofthephononenergyrelaxation energy dependence of A /A we assume the existence of
1 2
length and correspondingly their group velocity at volt- two mechanisms, which can be responsible for the be-
ageshigherthanoroftheorderoftypicalphononenergies haviour observed:
leadstoasignificantincreaseofthefrequencyofphonon-
phonon collisions. This will result in the thermalization 1. Adecreaseof(quasi-)elasticmeanfreepathofelec-
of phonons, which may have the following consequences: trons in the contact boundaries due to the scatter-
ing of electrons on nonequilibrium phonons.
1. Phonon-phonon collisions will decrease the num-
ber of high-frequency phonons, which are the most 2. Redistributionoftherelativecontributiontothein-
efficient electron scatterers because of their small formation about electron mean free path from the
groupvelocities,whilethethermalizedphononswill areaswithdifferentconcentrationsofimpurities. If
leave the contact area. theimpurityconcentrationishomogeneousoverthe
entire contact region, the ratio A /A will increase
2. An increase of the total number of phonons in the 1 2
withthebiasrise. If,however,theconcentrationof
contact boundaries, resulting from thermalization
impurities decreases with the distance to the con-
leadstoastrongersuppressionofoscillationsinthis
tact centre, the opposite effect will be observed.
area.
Hence, one can expect that for bias voltages higher Certainly, coexistence of both mechanisms is possible.
than or of the order of characteristic phonon energies, Itisobviousthatthefirstofthemwillleadtoanincrease
theriseofA (eV)willslowdownorevenadecreasemay of the A /A ratio only (see Eq. (2)).
1 1 2
be observed. Insummary,wehavestudiedtheenergydependenceof
Nowconsiderthebiasdependenceoftheratiobetween theoscillationamplitudeinmagnetoresistanceofmetallic
the first and the second harmonics of the oscillations point contacts in a magnetic field parallel to the contact
A /A (eV). FromEq. (3)itcanbereadilyseenthatthe axis. Itwasfoundforthefirsttimethatthisdependence
1 2
magnitudeofA /A dependsonthevalueoftheelectron shows a nonmonotonous behaviour and in some cases its
1 2
mean free path in the region of metal where the quan- shapeissimilartothepointcontactspectrumofelectron
tization of electron density of states takes place. Due phonon interaction. The additional scattering of quan-
to geometry of experiments (magnetic field direction is tized electrons on the phonons generated by the flow of
parallel to the contact axis) the ratio A /A at eV = 0 electronsthroughthecontactmaybethepossiblereason
1 2
contains the information about electron mean free path of the observed effect.
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