Dependence of the amplitude of magnetoquantum oscillations of the metallic point-contact resistance on the bias voltage

0.98 MB·

by  N. L. Bobrov

Dependence of the amplitude of magnetoquantum oscillations of the metallic
point-contact resistance on the bias voltage
N. L. Bobrov1,2, N. N. Gribov1,2, J. Kokkedee2, A. G. Jansen2, P. Wyder2, I. K. Yanson1
1B.I. Verkin Institute for Low Temperature Physics and Engineering,
of the National Academy of Sciences of Ukraine,
prospekt Lenina, 47, Kharkov 61103, Ukraine
2High Magnetic Field Laboratory,
Max-Planck Institut fur Festkorperforschung,
and Centre National de la Recherche Scientifique,
6
F-38042, Grenoble, France
1
0 Email address: [email protected]
2 (Dated: January 5, 2016; Published (Fiz. Nizk. Temp., 21, 68 (1995)); (Low Temp. Phys., 21, 51 (1995))

n
Magnetoquantum oscillations of resistance as functions of the potential difference applied across
a
J thecontactarestudiedformetallicpointcontactsmadeofAl andBe. Theamplitudeofresistance

1 oscillations in a magnetic field increases with voltage and is identical to the EPI spectrum of the
point contact for one group of contacts, and with the bias voltage across the contact for another.

]
l Theincreaseintheoscillationamplitudeaswellasitsdecreasehasanonmonotonicdependenceon
l
a energy. The scattering of electrons by nonequilibrium phonons and the Fermi-liquid effects in the
h nonequilibrium electron system are considered as the possible reasons behind the observed effects.
-
s
e PACSnumbers: : 71.38.-k,73.40.Jn,74.25.Kc,74.45.+c
m
.
at I. INTRODUCTION where n is the harmonic number in the oscillation spec-
m trum, and ω =eB/m∗c.
c
- InamagneticfieldB,theallowedelectronstatesliein In addition, in view of the elastic scattering of elec-
d
n the k-space on tubes called Landau tubes. These states trons by impurities,1 the momentum relaxation time be-
o are determined by the condition of quantization of the comesfinite. Accordingtotheuncertaintyprinciple,this
c
[ cross- section of a tube by a plane perpendicular to the leads to a broadening of the Landau levels and hence to

directionofthemagneticfield. Ifwetakeintoconsidera- the same extent of decrease in the oscillation amplitude
1
v tion the tube with the largest cross-sectional area which as that caused by an increase in temperature from the
4 partly lies inside the Fermi surface (FS), its filled part actual value T to T = T +x. The corresponding re-
eff
9
will decrease with increasing B and disappear at infinite duction factor (the Dingle factor), which has the same
0
0 velocity as the tube comes in contact with the FS. Such meaning as the ratio of the actually observed amplitude
0 sharp decreases in the population density, which occur of oscillationsto the amplitude whichwouldbe observed
.
1 periodically in the reciprocal field 1/B, lead to oscilla- in the absence of impurities, has the form
0
tions of the free energy and magnetization (de Haas-van
6
1 Alphen effect dHvA). The frequency F of these oscilla- Rτ =exp(−2π2kBnx/¯hωc), (3)
v: tions is described by the Lifshits-Onsager relation
Xi where x = ¯h/2πkBτ is the Dingle temperature, and τ is
F =(c¯h/2πe)A, (1) the carrier lifetime.
r
a It follows from the theory of noninteracting particles
where A is the area of the extremal Fermi surface cross-
that all processes of electron scattering, including the
section.
electron- phonon collisions, should make a contribution
Anincreaseinthetemperatureblurstheboundarybe-
to the Dingle temperature [1]. If higher-order contribu-
tween filled and unfilled states, and decreases the oscil-
tions to the electron- phonon interaction (EPI) are dis-
lationamplitude. Thecorrespondingreducingfactorhas
the form
2π2k nT/¯hω
R = B c ; (2) 1 Byimpurities,wemeananystaticdefectsthatscatterelectrons
T sh(2π2k nT/¯hω ) elastically.
B c
2
regarded, the effect of phonons boils down just to a de- Intheprevailingtechnologyforthecreationofpressure
crease in the amplitude. The magnitude of the effect is point contacts, the elastic mean free path l , of electrons
i
determined in this case by the EPI function α2(ω)F(ω) neartheconstrictionisusuallyofthesameorderofmag-
for the investigated metal, temperature, and magnitude nitudeasthecontactdiameterd,andmuchsmallerthan
of the magnetic field. However, a consideration of the the mean free path at the banks. The contact resistance
many-particle effect shows that the scattering of elec- for zero bias voltage in the circular aperture model is
tronsbyphononsdoesnotleadtoanincreaseintheDin- defined by Wexler’s interpolation formula [4]
gletemperature[1]. Additionalscatteringofelectronsby
16ρl ρ
phononsuponanincreaseintemperatureiscompensated R = i +β(l /d) (4)
0 3πd2 i d
by a decrease in the mass m∗ =m (1+λ) renormalized
0
duetoelectron-phononinteraction(λistheEPIparame- Here, ρli = pF/e2n = 3/2N(εF)vFe2 is constant for the
ter),whichmakesthisquantitynearlyequaltothe”bare” givenmetal,εF, pF, vF -Fermienergy,ρtheresistivity,
mass m0. A departure from this formula towards higher N(εF)theelectrondensityofstatesattheFermisurface,
amplitudes is observed in strong fields, and the required and β(li/d) is a nonmonotonic weakly varying function
quantity Bincreaseswithtemperature(phononenergy). ofli/dwhosevalueisclosetounity: β(0)=1,β(4.48)(cid:39)
It must be considered that, although EPI renormalizes 0.6828 (minimum of the function), β(∞) = 9π2/128 (cid:39)
the cyclotron mass determining the temperature atten- 0.694.
uation of the amplitude, it does not affect the mass ap- The first term in formula (4) describes the Sharvin
pearing in the Dingle factor, i.e., the correct expression component of the point contact resistance, which does
(3) contains the ”bare” mass m [1]. not depend on the electron mean free path and is de-
0
termined by the shape of the Fermi surface of the given
The effect of EPI on the amplitude of dHvA oscilla-
metal. The second term is the Maxwell component of
tions can be studied in traditional experiments only for
resistancewhichdependsonthepurityofthemetalnear
a limited number of metals (e.g., Hg) with an anoma-
the constriction.
lously low Debye temperature, in which phonons can be
If the point contact is placed in a varying magnetic
excited at low temperatures when the oscillation ampli-
field,itsresistanceacquiresanoscillatorycomponentdue
tudeislargeenoughforobservation. Pointcontactswith
to quantization of the electron energy spectrum. These
a size of several tens or hundreds of angstroms provide
oscillations may be associated with the Maxwellian or
a unique possibility of studying the effect of nonequilib-
Sharvin component of resistance and depend on the pa-
riumphononsgeneratedinacontactontheamplitudeof
rameters of the metal. In the contacts formed by con-
oscillations of the electron density of states at the Fermi
ventional metals, the Farmor radii of electrons are much
surface on the banks.
larger than the typical values of the contact diameter in
Thesimplestmodelofcontactisacircularaperturein fields up to 20 T. In the case of Be, for example, these
an infinitely thin opaque partition separating two metal- quantities are comparable only for the largest contacts
lic half-spaces. Owing to the small size d of the point with d (cid:39) 500 ˚A and in the strongest field 10 T, i.e.,
contact relative to the energy mean free path l of elec- for r = m∗v /eB = 3000 ˚A (the value m∗ = 0.17m
ε B F 0
trons, the electrons are divided into two groups whose was used in the estimates). Hence magnetoquantum os-
Fermi energies differ by the applied potential difference cillations of resistance of a metallic point contact are
eV [2, 3]. In this case, the electrons from the group with mainly associated with the fulfillment of the quantiza-
a higher Fermi energy may go over to the group with a tion conditions at the banks near the constriction. The
lower energy by emitting one phonon as a rule. Multi- oscillations of the electron density of states at the banks
phonon processes are also possible, although their prob- lead to the oscillations of the Sharvin component of the
abilityismuchlower. Thus, ifavoltageisappliedacross point-contactresistance. Thisisthemaindistinctionbe-
thecontact,phononswithallpossibleenergiesrightupto tween such oscillations and the oscillations of the resis-
eV willbegeneratedinthecontactregion. Sincel (cid:29)d, tance of point contacts formed by semimetals [5], which
ε
only a small fraction of total power is liberated in the are associated with the quantum oscillations of the col-
immediate vicinity of the contact, and the temperature lision integral [1] (oscillations of the Maxwellian compo-
of the banks remains practically constant, viz., equal to nentofresistance)andareanalogoustotheconventional
the temperature of the helium bath. Shubnikov-de Haas effect in bulk conductors.
3
the Sharvin component of the point-contact resistance.
The presence of elastic scatterers at the contact banks
leadstoanincreaseintheDingletemperatureandhence
to a decrease in the oscillation amplitude.
Thefirst experimentson theinvestigation oftheeffect
ofnonequilibriumphononsontheamplitudeofthepoint-
contact resistance oscillations were reported in Refs. [5]
and [7]. While describing the experimental results, the
authors ofRef. [5] reduced the effect ofphonons to over-
heating of the electron gas in the contact region, while
twoalternativemechanismsofdampingwereproposedin
Ref. [7], viz., an increase in the Dingle temperature due
to the electron- phonon scattering, or the Joule heating
in the contact region.
In the present work, we study the effect of nonequilib-
rium phonons on the quantum magnetic oscillations in
Be and Al. For one group of contacts (Al and Be), the
amplitude of resistance oscillations in a magnetic field
increases with voltage and is identical to the EPI spec-
trumofthepointcontact. Forotherpoint-contacts(Al),
themagnetoresistanceexhibitsanincreaseintheoscilla-
FIG. 1: Scattering of electrons moving in quantized orbits
tionamplitudewithincreasingpotentialdifferenceacross
bynonequilibriumphonons(markedbystars)accumulatedat
thecontact. Thez-componentofelectronmomentumandthe the contact. Both the increase in the oscillation ampli-
ability to make an effective contribution to current increase tude and its decrease have a nonmonotonic energy de-
afterscattering. Thecase(a)correspondstoasmallnegative pendence.
v component and case (b) to zero v component.
z z
II. EXPERIMENTAL TECHNIQUE AND
Thefactthattheeffectofimpuritiesneartheconstric- PROCESSING OF RESULTS
tion is opposite to the effect of impurities at the banks
of the contacts in the case of a magnetic field oriented Point-contacts were formed between the edges of two
parallel to the contact axis is paradoxical and leads to similarly oriented single-crystal electrodes of Al or Be.
an increase in the amplitude of the contact resistance Thesemetalswerechosenbyusmainlybecausetheiren-
oscillations[6]. Indeed,thecontributiontothemagneto- ergy spectrum contains groups of electrons with small
quantum oscillations comes from the regions of extremal effective masses, which makes it possible to observe the
cross-section of the Fermi surface by the plane p =const oscillations in relatively weak magnetic fields. In the
z
(thez-axisisparalleltothecontactaxisandtothemag- case of Be, the contact axis coincides with the crystal-
netic field direction). In ballistic contacts, such states lographic axis c, while for Al it is parallel to the (110)
correspond to low values of the component v of the axis. The magnetic field was always applied parallel to
z
transport velocity of the carriers. Since the resistance the contact axis. The preliminary treatment of the elec-
of a point-contact is determined by the z-component of trodes included their chemical polishing in a mixture of
the transport velocity of electrons, the amplitude of os- acidsforremovingthedefectlayerformedduringelectric
cillationsoftheSharvincomponentofresistanceissmall. erosion cutting. For preparing Al contacts, the electro-
If the contact region contains elastic scatterers, the mo- forming [8] and displacement techniques [9] were used
menta of the electrons arriving in this region are dis- to the same extent. The Be contacts were formed pre-
tributed isotropically along the directions (Fig. 1). This dominantly by the electroforming technique which en-
results in an increase in the z-component of the veloc- sured their noticeably higher mechanical stability. We
ity of electrons from the extremal FS cross-section, and did not detect any correlation between the method of
hence in an increase in the amplitude of oscillations of preparing a point contact and the form of the A (eV)
1
4
FIG. 3: Fragments of recording of magnetoresistance oscilla-
tionsofanAlpointcontactwithR =1.41Ωfortwodifferent
0
bias voltages.
spectrum of a Be contact, the oscillating component of
the magnetoresistance, and its Fourier spectrum. The
PC spectrum d2I/dV2(V) is obtained from the exper-
imental formulas d2V/dI2(V) and dV/dI(V) by calcu-
lations and has a well-defined structure in the interval
40-80 meV, which is associated with the EPI function
α2 (ω)F(ω). The magnetoquantum oscillations of resis-
pc
tance with a natural frequency 985 T correspond to the
third electron band (cigar) and are known from the de
Haas-van Alphen effect investigations [10]. The Fourier
spectrumalsocontainsthesecondharmonic. Thefollow-
ing procedure was used for determining the dependence
of amplitude of magnetoresistance oscillations of the PC
on bias voltage. For a fixed bias voltage across the con-
tact, the oscillations R (B) = dV/dI(B) were recorded
d
in the same interval of magnetic fields (8.5−10 T as a
rule). Figure 3 shows fragments of the recording for two
FIG. 2: (a) Point-contact spectrum of a Be point contact,
differentbiasvoltagesappliedtoanAl−Alpointcontact.
R =16.24 Ω.
0
(b) Oscillations of the resistance of a Be point contact in a For Al contacts, the investigated oscillations correspond
magnetic field at zero bias voltage B (cid:107) c (cid:107) z, R = 16.24 Ω; totheγ-pocketofthethirdbandwithafrequency290T
0
∆R/r ≈2·10−2 at B ∼9 T. [11]. Two recordings were made for each bias voltage,
0
(c)I andF aretheintensityandthefrequencyofoscillations. viz., in increasing and decreasing magnetic fields. After
recording of the curves for the entire range of the values
ofV,arecordingwasmadeatV =0andcomparedwith
dependence. The first and second IVC derivatives were the initial recording to ensure that the sample remained
recorded by the standard modulation technique, and the unchangedinthecourseofmeasurements. Contactswith
data were registered directly on a computer. The follow- differentresistances(from0.6to20Ω),backgroundlevels
ing characteristics were recorded during measurements: (30 to 85%) and levels of blurring of phonon singulari-
d2V/dI2(V); dV/dI(V); dV/dI(B)| . The tem- ties in the spectrum were studied. The complete set of
V=const
perature was maintained at 1.3 K throughout the mea- characteristics was obtained for five Al contacts and five
surements. By way of an example, Fig.2 shows the PC Be contacts. The results of experimental measurements
5
were subjected to Fourier analysis for determining the
amplitudes of oscillations in a fixed interval of magnetic
fields for the entire set of curves. Since the character-
istics dV/dI(B) (V=const) were recorded for a constant
valueofthemodulatingcurrent,andthedifferentialresis-
tance of the contact increases (up to 10% in the interval
of Debye energies) upon an increase in the bias voltage
across the contact, the obtained results were reduced to
a fixed modulating voltage and normalized to the value
obtainedforzerobiasvoltageacrossthecontact. Figures
4-6 showthetypicalresultsofsuchcalculations. Forone
groupofpointcontacts,theamplitudeA ,ofmagnetore-
1
sistance oscillations increases with eV, and their energy
dependenceisidenticaltotheEPIspectrumofthepoint
contact (Figs. 4, 5). For another group of contacts (Al
contacts), the amplitude decreases monotonically as a
functionofvoltage(Fig. 6). Thepossiblereasonsbehind
such dependences will be discussed in the next section.
III. DISCUSSION OF EXPERIMENTAL
RESULTS
Phonons are generated as a result of the relaxation of
electronsinjectedfromthecontactbyspontaneousemis-
sionprocesses. Theresultingnonequilibriumdistribution
ofphononsdependsonthevoltageappliedtothecontact.
Liketheeffectofelasticallyscatteredimpurities,thescat-
teringofelectronsbynonequilibriumphononsaffectsthe
amplitude of quantum oscillations of the PC magnetore-
sistance in different ways depending on their position.
Thescatteringofprocessesoccurringintheregionofthe
constriction lead to an increase in the amplitude, while
scattering at the banks decreases its resultant value. We
FIG. 4:
shallnowdiscussthedependenceoftheenergyandcoor-
(a) Point contact spectrum of an Al point contact with
dinatedistributionofphononsonthecontactparameters
R =1.41 Ω.
and the effect of this distribution on magnetoquantum 0
(b) Relative change in the fundamental amplitude of resis-
oscillations.
tance oscillations of a point contact in a magnetic field as a
Apart from the defects that weakly affect the elastic function of the applied bias voltage: ∆R/R ≈ 5·10−4 for
0
meanfreepathofphononsbutstronglyscatterelectrons, V =0 and B =9.5 T.
the spring-type contacts also contain, as a rule, a large (c) Dependence of the amplitude ratio of the of the funda-
number of line defects. These defects affect the momen- mental and the second harmonics as a function on eV.
tum mean-free path l , of electrons weakly, but consid-
i
erably shorten the elastic mean free path lph of nonequi-
i
librium phonons generated by the electron flow through d2I/dV2 (eV) for eV > ¯hω which is comparable with
D
the aperture [12]. If lph < d, the additional scattering the spectral part in order of magnitude. A nonequilib-
i
of electrons by these nonequilibrium phonons accumu- rium phonon distribution is established in the contact
lated in the contact leads to the emergence of a back- with an effective temperature that depends on coordi-
ground (the energy-independent second IVC derivative natesandphononreabsorptioncoefficient. Thisdistribu-
6
FIG. 5:
(a) Point contact spectrum of an Be point contact with FIG. 6:
R =8.66 Ω. (a) Point contact spectrum of an Al point contact with
0
(b) Relative change in the fundamental amplitude of resis- R0 =8.73 Ω.
tance oscillations of a point contact in a magnetic field as a (b) Relative change in the fundamental amplitude of resis-
function of the applied bias voltage: ∆R/R ≈ 2·10−3 for tance oscillations of a point contact in a magnetic field as a
0
V =0 and B =9.5 T. function of the applied bias voltage: ∆R/R0 ≈ 3·10−3 for
(c) Dependence of the amplitude ratio of the of the funda- V =0 and B =9.5 T.
mental and the second harmonics as a function on eV. (c) The ratio of the amplitudes of the fundamental and the
second harmonics as a function of eV.
tion differs from the Planck distribution in that it con-
tains a sharp edge at h¯ω = eV [12]. In this case, the
nonequilibrium phonon distribution function is defined where
by the expression
Nω(r)= eV2−¯hω¯hωΘ(eV −¯hω)qω(r), (5) qω(r)4π1Λ2ω (cid:90) drexp(−RR/Λω)q(r(cid:48)); (6)
7
3000 ˚A, lBe ∼1500 ˚A, l =v τ ;
εD εD F εD
ωD
2π (cid:90)
τ−1 = g(ω)dω
εD ¯h
0
atDebyeenergies[13]. Hencealargepartoftheelectrons
loses its energy at the banks and not in the contact. In
other words, the main source of nonequilibrium phonons
atthebanksis”hot”electronsinjectedfromthecontact.
These phonons with randomly oriented momenta scatter
electrons (including the electrons with extremal cross-
sections of the Fermi surface) at the banks. The fraction
ofphononsarrivingatthebanksfromthecontactregion
is insignificant (see Fig. 7).
Thus,wearriveatacompleteanalogywithimpurities:
scatteringofelectronsbyphononsinthecontactleadsto
FIG. 7: Distribution of nonequilibrium phonons over the z-
axis in the units of contact diameter as a function of their an isotropization of the electron momenta. This leads to
diffusive mean free path Λ /d, calculated by formula (6). anincreaseinthez-componentofthevelocityofelectrons
ω
Allthecurvescorrespondingtoz=0arenormalizedtounity; fromextremalFScrosssectionsandhencetoanincrease
Λω/d=0.5 (curve 1); 1 (2); 2 (3); 5 (4); 10 (5); and 1000 (6). in the amplitude of oscillations of the contact resistance
in a magnetic field. At the same time, the scattering
R=|r−r(cid:48)|; of electrons by phonons generated by hot electrons at
(cid:20) (cid:16) (cid:17)2(cid:21) the banks lowers the amplitude of quantum oscillations
q(r)= 1 1− 1− Ω(r) ;
2 2π of the density of states. The resulting dependence of
(cid:16) (cid:17)1/2
Λ = lphlph/3 (ω −ω)1/2 the amplitude of point-contact resistance oscillations on
ω i ε D
eV will be determined by the balance between these two
Here Θ(x) is the Heaviside theta function; lph is the in- processes.
ε
elastic mean free path of phonons which has the same The amplitude of quantum oscillations of the density
order of magnitude as the inelastic mean free path of of states at the banks depends nonlinearly on the con-
electrons in the case of Debye energies and has the same centration of the scatterers [see formula (3)], attaining
energy dependence, Ω(r) is the solid angle at which the saturation for large values of ω τ and attenuating expo-
c
aperture is seen from the point r, and Λ is the diffusive nentially for small values. Hence if the concentration of
ω
relaxation length over which a nonequilibrium phonon staticdefectsscatteringelectronsatthebanksislow,the
loses its energy. slight increase in the number of electron scattering acts
Thus, for a fixed value of lph, the diffusive relax- bynonequilibriumphononsgeneratedby”hot”electrons
i
ation length Λ of phonons decreases with increasing will lead to an insignificant decrease in the amplitude
ω
phonon energy. Figure 7 shows a family of curves cal- ofquantumoscillations. However,iftheconcentrationof
culated from formula (6) and describing the distribution staticdefectsatthebanksisquitehigh,asimilarincrease
of phonons along the z-axis (in the units of contact di- in the number of acts of electron scattering by nonequi-
ameter) as a function of the parameter Λ . It can be libriumphononsmayradicallydecreasetheamplitudeof
ω
seen that high-energy phonons having the smallest value oscillations at the banks. An increase in the contact size
oftheratioΛ /dareaccumulatedneartheapertureand leads to an increase in the time of residence of nonequi-
ω
arestronglyreabsorbedbyelectronswhilelow-frequency librium phonons near the constriction, and hence to an
phonons (with a large value of the ratio Λ /d) dominate increase in the effectiveness with which quantized elec-
ω
at the banks. tronsaredistributedisotropicallyalongthedirectionsas
All the contacts investigated by us lie in the spectral a result of phonon-electron collisions.
regime of current flow, i.e., satisfy the condition d(cid:28)Λ Hence we can expect an increase in the amplitude of
ε
where Λ = (l l /3)1/2 is the diffusive length of energy oscillations with increasingbias voltage in low-resistance
ε i ε
relaxationofelectrons(accordingtoourestimates,lAl ∼ contacts with clean banks. Obviously, the strongest in-
εD
8
crease will be observed for contacts that are ballistic The model presented above can explain not only the
with respect to electrons and diffusive with respect to monotonic part of variation of the amplitude of oscilla-
phonons,sincetheydonotexhibitabackgroundincrease tions with bias voltage, but also its similarity to the EPI
in the amplitude due to scattering of electrons by impu- spectrum. ItwasmentionedabovethatforeV ≤ω ,the
D
rities,andasmallerelasticmeanfreepathofnonequilib- energy of the emitted phonons almost always coincides
rium phonons is responsible for their higher concentra- with the excess energy of electrons, due to a low prob-
tion. ability of multiphonon processes. The group velocity of
phonons generated by the electron flow depends on their
On the other hand, nonequilibrium phonons quickly
energyandminimumforfrequenciesatwhich∂ω/∂q ∼0,
leave the region of contacts that have a high resistance
i.e.,nearthepeaksofthephonondensityofstates. Since
and are ballistic relative to phonons. If, on top of this,
Λ depends on the velocity at which phonons leave the
the contact banks are not very clean, a decrease in the ω
contact, a sharp increase in the phonon reabsorption co-
amplitude of contact resistance oscillations will be ob-
efficientwillbeobservedatcertainenergies,aswellasin
served with increasing eV due to the influence of the
theamplitudeofthepoint-contactresistanceoscillations.
Dingle factor at the contact banks.
Since all electrons lying in the layer eV can take part in
These arguments have been confirmed in the experi- phonon generation, the entire spectrum of phonons with
ments. For high-resistance Al contacts (Fig. 6) with a energies ranging from 0 to eV will be generated for a
relatively low purity of banks near the constriction [this given bias voltage. Hence it can be expected that the
is confirmed by a large value of the ratio A /A of har- dependence of the oscillation amplitude on eV would be
1 2
monics for V =0, see formula (8) below], the amplitude monotoniclikethefirstIVCderivativedV/dI(eV),which
of resistance oscillations always decreases with increas- does not correspond to the experimental results. How-
ing bias voltage across the contact. On the other hand, ever, we have not taken into consideration the role of
low-resistance Al contacts with extremely clean banks phonon- phonon collisions in this discussion.
(anomalously low value of the ratio A /A of harmonics The phonon mean free path lph defined by phonon-
1 2 ph
for V = 0, see Fig. 4), the dependence of the oscil- phononcollisionsdependsonthephononenergyandcon-
lation amplitude on eV is similar to the PC spectrum. centration which, in turn, is determined by the rate of
We obtained the dependences R (H) for the given con- phonon generation (i.e., by eV) as well as the rate at
d
tact in the entire range of magnetic fields (0-10 T) for which they leave the contact. The latter depends on
bias voltages of 0, 20, and 44 mV. Although the os- the group velocity of the phonons being generated, the
cillations began for all three bias voltages in nearly the contact diameter, and the phonon mean free paths Λ
ω
same magnetic fields, the increase in the oscillation am- and lph [14]. Hence, for bias voltages that are higher
ph
plitude with increasing bias voltage across the contact is than or of the same order as the phonon energies, the
more rapid (several times) in high magnetic fields (i.e., phonon-phonon collision frequency increases sharply due
for large values of ω τ). A considerable (more than four- to a stepwise decrease in the diffusive energy relaxation
c
fold) increase in A in strong fields and for large bias length Λ for phonons (similar to the energy relaxation
1 ω
voltages, and its anomalously low value (about an order length Λ for electrons), as well as to a sharp decrease in
ε
of magnitude smaller than the typical value) for V = 0 theirgroupvelocity. Suchcollisionsleadtoathermaliza-
also speaks in favor of a low concentration of impurities tion of phonons, which may have three consequences.
and static defects that scatter electrons directly in the
1. Thedirectionofthemomentumofanelectronmay
contact. Irrespective of the variation of the resistance of
change significantly only as a result of its colli-
Becontactswithincreasingbiasvoltage,weobservedan
sionwithahigh-energyphonon. Hencethermalized
increase in the amplitude of oscillations of R (Fig. 5).
d phononsarelesseffectiveforaconsiderableincrease
This is probably due to an extremely high rigidity of the
inthez-componentofthevelocityofquantumelec-
Becrystallatticeduetowhichthestrainsemergingdur-
trons entering the contact.
ing the contact formation do not move into the bulk of
thematerial,butareratherconcentratednearthesurface 2. The phonon-phonon collisions lead to a decrease
andformeffectivereflectorswhichreturnthephononsto in the number of high-frequency phonons with low
the contact. It is also necessary to use Be single crystals group velocities, which scatter the electrons most
with a high initial purity for preparing contacts. effectively, and the newly formed phonons will
9
rapidly leave the contact region.
3. An increase in the total number of phonons at
the banks as a result of thermalization leads to a
stronger suppression of oscillations in this region
sincecollisionswithphononshavinganyenergyare
effective in ejecting the quantized electrons from
the extremal orbits.
Hence it can be expected that, for bias voltages that are
higher than or of the same order as the characteristic
phonon energies, the rise of the A (eV) dependence will
1
slow down and we may even obtain a descending region.
Another possible factor contributing to the nonmono-
tonicity of A (eV) may be the renormalization of the
1
effective masses of high-energy electrons [15] It was pre-
FIG. 8: Schematic representation of the behavior of the EPI
dicted in Ref. [15] that, in the immediate vicinity of the
parameter λ near the aperture as a function of the applied
contact, the EPI parameter λ responsible for the renor-
voltage, which was predicted in Ref. [15]. The solid and
malization of the effective mass m∗ = m(1+λ) differs
dashedcurvescorrespondtoaninfinitelynarrowphononline
from its equilibrium value λ . This difference is due to
0 and to a phonon band of a finite width with a characteristic
the presence of the nonequilibrium function of electron energy eV = hν . The inset shows the electron distribution
0
distribution in the point contact for eV = 0. For a bal- functionforaballisticpointcontactatthecenteroftheaper-
listic contact, λ will depend on the applied voltage. For ture.
theEinsteinmodelofspectrumofdistributionofphonons
with energy hν , the following expression holds at the
0
center of the contact [15]:
phonon energy should correspond to the point of inflec-
(cid:34) (cid:35)
λ (hν )2 tion on the segment with a negative slope of the curve
λ(eV)= 0 1+ 0 . (7)
2 (hν0)2−(eV)2 A1(eV). Although the exact position of the peak on the
voltage dependence of the oscillations varies from con-
The anticipated behavior of λ(eV) is shown schemati-
tact to contact, the voltages corresponding to the points
cally in Fig. 8. It can be seen that for high voltages
of inflection are usually higher than the values at which
across the contact, the renormalization factor decreases
the phonon peaks are localized in the EPI spectrum of
by half since only half the Fermi surface participates in
a point contact. As regards the observed disparity be-
the renormalization of mass in the case of a nonequilib-
tween the experimental and theoretical results, we note
rium electron distribution function in the point contact
that the theory described in Ref. [15] was developed for
(seeinsettoFig. 8). ForeV =hν ,resonanceisobserved
0 zero magnetic field. It would be interesting to include
in mass renormalization. The dashed curve in this figure
theenergydependenceoftheeffectivemassinthetheory
shows the predicted form of the dependence λ(eV) for
explaining the magnetoquantum oscillations observed in
the case of a blurred phonon line (band) near hν. It can
ballistic point contacts. The ballistic electron transport
be concluded from the theory [15] that the nonmono-
in a point contact makes it possible to study the energy
tonic energy dependence of the effective electron mass dependence m∗(eV) in the absence of thermal heating.
for characteristic phonon frequencies is determined by
the nonequilibrium electron distribution function in the LetusnowconsiderthedependenceoftheratioA /A
1 2
PC for finite bias voltages. ofharmonicamplitudesonthebiasvoltageappliedacross
The energy dependence of the renormalized effective the contact. For a fixed temperature, the ratio of ampli-
mass in a ballistic point contact might be responsible tudes of the first and second harmonics is defined by the
fortheobservednonmonotonicdependenceoftheampli- elastic mean free path l =v τ of electrons in the region
i F
tudeofmagnetoquantumoscillationsontheappliedvolt- where the electron density of states is quantized in the
age. It can be seen from Fig. 8 that the characteristic metal, i.e., at the banks adjoining the contact. It follows
10
from formulas (2) and (3) that the impurity concentration is equally low in all the re-
gions where oscillations take place, the shape of the de-
(cid:18)2π2k T(cid:19) (cid:18) π (cid:19)
A1/A2 =Ccosh ¯hωB exp ω τ . (8) pendenceofA1/A2 oneV willonthewholebesimilarto
c c the EPI spectrum (Fig. 4). If, however, the impurities
The constant C has the meaning of the ratio of ampli- are distributed nonuniformly and their concentration in
tudes of harmonics at T =0 in the absence of scatterers. the oscillation region decreases with distance from the
Thus, as the electron mean free path decreases in the re- center of the contact, or if regions with a low impurity
gion of oscillations, the ratio A /A increases. In view concentration and regions practically free from impuri-
1 2
of the geometry of the experiments (the magnetic field ties exist at the same distance on the contact periphery,
is oriented parallel to the contact axis, while the elec- we shall observe a decrease in the ratio A1/A2 with in-
trons whose number oscillates have a small z-component creasing eV (Figs. 5,6).
ofvelocity)theratiooftheharmonicsforV =0contains
informationabouttheelectronmeanfreepathinthepe-
ripheral regions of the contact that are not far from the IV. CONCLUSIONS
center of the contact, i.e., the regions where the electron
mean free path is long enough for the emergence of os- In this work, we have investigated the amplitudes of
cillations. We assume that the form of the dependence quantum oscillations of Al and Be point contact resis-
of the ratio A /A of the harmonics on the bias voltage tance in a magnetic field oriented parallel to the con-
1 2
applied across the contact is mainly due to two mecha- tact axis as functions of the potential difference applied
nisms: to the contact. It was found for the first time that
this dependence has a nonmonotonic ascending nature
1. A decrease in the (quasi)elastic mean free path of in many cases, and its shape is similar to the EPI spec-
electrons in the region of space carrying informa- trum of the point-contact. Such an effect was observed
tion on the oscillations at V = 0 due to an addi- for Be point contacts and for low-resistance Al point
tional scattering by nonequilibrium phonons. contacts. The additional scattering (in the contact) of
magnetoquantizedelectronsbyphononsgeneratedbyac-
2. A redistribution of the relative contribution to the
celerated electrons may be a possible reason behind the
information on electron mean free paths from spa-
increase in the amplitude of oscillations with voltage.
tial regions with different impurity concentrations.
These processes of scattering of electrons by nonequi-
Of course, both mechanisms can coexist. It is obvious librium phonons increase the contribution to transport
that the first mechanism can lead only to an increase in of charge carriers through the contact for electrons from
the relative intensity of the first harmonic [see formula extremal Fermi surface cross sections. Hence, in con-
(8)]. Thesecondmechanismmayleadtotheoppositeef- trast to the conventional suppression of magnetoquan-
fect. Indeed, it was mentioned above that the amplitude tumoscillationsduetoscatteringbyimpuritiesinabulk
of oscillations depends nonlinearly on the impurity con- metal, the oscillations increase with scattering upon an
centration [formula (3)]. We assume that regions with a increaseintheappliedvoltage. If,however,scatteringby
lowandarelativelyhighconcentrationofimpuritiesexist nonequilibriumphononsextendstothecontactbanks,we
in the vicinity of the point contact. In the current state, observe a familiar decrease in the oscillation amplitude
both types of regions will be exposed to phonons gener- (forhigh-resistanceAlcontacts). ThenonmonotoniceV-
ated at the banks. Such a scattering only decreases the dependence of the oscillation amplitude A (i.e., the ex-
1
amplitude of density of states oscillations insignificantly istenceofapeaknearthephonondensityofstatespeak)
inthecaseofcleanregions,whiletheoscillationsmaybe is probably associated with an energy redistribution of
suppressed considerably in the dirty regions. Hence the phononsduetophonon-phononcollisions,aswellaswith
relative contribution to the resistance oscillations from the possible energy dependence of the renormalized elec-
cleaner regions in space will increase with eV, and the tron mass in the immediate vicinity of the contact.
ratioA /A willdecreaseaccordingly. Consequently,the TheshapeofthedependenceoftheratioA /A ofthe
1 2 1 2
shapeofthedependenceofA /A oneV canprovidein- harmonicsofoscillationamplitudesoneV canbeusedto
1 2
formation about the distribution of impurities near the draw conclusions as to the impurity distribution in the
contact in the region where oscillations are observed. If quantization region. If the impurity concentration is the