the presence of defects
Avotina, Y.S.; Kolesnichenko, Y.A.; Otte, A.F.; Ruitenbeek, J.M. van
Citation
Avotina, Y. S., Kolesnichenko, Y. A., Otte, A. F., & Ruitenbeek, J. M. van. (2006). Signature of
Fermi-surface anisotropy in point contact conductance in the presence of defects. Physical
Review B, 74, 085411. doi:10.1103/PhysRevB.74.085411
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Signature of Fermi-surface anisotropy in point contact conductance in the presence of defects
Ye. S. Avotina,1,2Yu. A. Kolesnichenko,1,2A. F. Otte,2and J. M. van Ruitenbeek2
1B.I. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47, Lenin Ave., 61103, Kharkov, Ukraine
2Kamerlingh Onnes Laboratorium, Universiteit Leiden, Postbus 9504, 2300 Leiden, The Netherlands
共Received 3 April 2006; published 16 August 2006兲
In a previous paper关Avotina et al., Phys. Rev. B 71, 115430 共2005兲兴 we have shown that in principle it is possible to image the defect positions below a metal surface by means of a scanning tunneling microscope. The principle relies on the interference of electron waves scattered on the defects, which give rise to small but measurable conductance fluctuations. Whereas in that work the band structure was assumed to be free-electron like, here we investigate the effects of Fermi surface anisotropy. We demonstrate that the amplitude and period of the conductance oscillations are determined by the local geometry of the Fermi surface. The signal results from those points for which the electron velocity is directed along the vector connecting the point contact to the defect. For a general Fermi surface geometry the position of the maximum amplitude of the conductance oscillations is not found for the tip directly above the defect. We have determined optimal conditions for determination of defect positions in metals with closed and open Fermi surfaces.
DOI:10.1103/PhysRevB.74.085411 PACS number共s兲: 73.23.⫺b, 72.10.Fk
I. INTRODUCTION
The interference of electron waves scattered by single de-fects results in an oscillatory dependence of the point contact conductance G共V兲 on the applied voltage V. This effect origi-nates from quantum interference between the principal wave that is directly transmitted through the contact and the partial wave that is scattered by the contact and the defect or several defects. Such conductance oscillations have been observed in quantum point contacts1–4 and investigated theoretically in the papers.3,5–7
In our previous paper8the oscillatory voltage dependence of the conductance of a tunnel point contact in the presence of a single pointlike defect has been analyzed theoretically and it has been shown that this dependence can be used for the determination of defect positions below a metal surface by means of a scanning tunneling microscope共STM兲. In the model of a spherical Fermi surface共FS兲 the amplitude of the conductance oscillations is maximal when the contact is placed directly above the defect. The oscillatory part of the conductance⌬G for this situation is proportional to
⌬G共V兲 ⬃ sin
冉
2z0冑
kF2
+2meV ប2
冊
,where z0the depth of the defect and kFand m are the Fermi
wave vector and effective mass of the electrons.8 Materials with an almost spherical FS are most suitable for this model. In most metals the dispersion relation for the charge car-riers is a complicated anisotropic function of the momentum. This leads to anisotropy of the various kinetic characteristics.9Particularly, as shown in Ref.10, the current spreading may be strongly anisotropic in the vicinity of a point contact. This effect influences the way the point contact conductance depends on the position of the defect. For ex-ample, in the case of a Au共111兲 surface the necks in the FS should cause a defect to be invisible when probed exactly from above.
Qualitatively, the wave function of electrons injected by a point contact for arbitrary FS共p兲=Fhas been analyzed by
Kosevich.10He noted that at large distances from the contact the electron wave function for a certain direction r is defined by those points on the FS for which the electron group ve-locity is parallel to r. Unless the entire FS is convex there are several such points. The amplitude of the wave function de-pends on the Gaussian curvature K in these points, which can be convex 共K⬎0兲 or concave 共K⬍0兲. The parts of the FS having different signs of curvature are separated by lines of
K = 0共inflection lines兲. In general there is a continuous set of
electron wave vectors for which K = 0. The electron flux in the directions having zero Gaussian curvature exceeds the flux in other directions.10
Electron scattering by defects in metals with an arbitrary FS can be strongly anisotropic.9Generally, the wave function of the electrons scattered by the defect consists of several superimposed waves, which travel with different velocities. In the case of an open constant-energy surface there are di-rections along which the electrons cannot move at all. Scat-tering events along those directions occur only if the electron is transferred to a different sheet of the FS.9
com-mon features of FS geometries to the conductance oscilla-tions: anisotropy of a convex part共bellies兲, changing of the curvature共inflection lines兲 and presence of open directions 共necks兲.
II. THE SCHRÖDINGER EQUATION FOR QUASIPARTICLES
Let us consider as a model for our system a nontranspar-ent interface located at z = 0 separating two metal half-spaces, in which there is an orifice共contact兲 of radius R centered at the point r = 0. The potential barrier in the plane z = 0 is taken to be a delta function,
U共r兲 = Uf共兲␦共z兲, 共1兲
where=共x,y兲 is a two-dimensional vector in the plane of the interface, with r =共, z兲. The function f共兲→⬁ in all points of the plane z = 0 except in the contact, where f共兲 = 1. At the point r = r0 near the contact in the upper half-space, z⬎0, a pointlike defect is placed. The electron inter-action with the defect is described by the potential D共r − r0兲, which is confined to a small region with a
characteris-tic radius rDaround the point r0.
It is known that one can obtain an effective Schrödinger equation for quasiparticles in a metal from the dispersion relation共p兲 共the band structure兲 by replacement of the qua-simomentum p共below for short we write momentum兲 in the function共p兲 with the momentum operator pˆ=បiⵜ.9Here we do not specify the specific form of the dependence 共p兲, except that it satisfies the general condition of point symme-try共p兲=共−p兲. For simplicity we assume that FS has only one sheet; there is only one zone described by the function 共p兲. In the reduced zone scheme a given momentum p iden-tifies a single point within the first Brillouin zone. The wave function共r兲 satisfies the Schrödinger equation with an ef-fective Hamiltonian共pˆ兲,
共pˆ兲共r兲 + 关 − U共r兲 − eV共z兲兴共r兲 = D共r − r0兲共r兲, 共2兲
where U共r兲 is defined by Eq. 共1兲, V共z兲 is the applied
electri-cal potential, and is the electron energy.
We consider a large barrier potential U. In this case the amplitude t of the electron wave function passing through the barrier is t共,pt兲 ⬇ ប共vzin−vzref兲 2iU , 共3兲 where vz in and vz ref
are the z components of the velocity v =共p兲/p of incident electrons共in兲 and electrons specularly reflected by the barrier共ref兲, respectively. Under condition of specular reflection the energy and the component of the momentum tangential to the interface, pt=共px, py兲 at z=0 are
conserved. The components of the electron momentum per-pendicular to the interface, pzin共pt,兲 and pzref共pt,兲 are
re-lated by the equations, 共pt in , pz in兲 = 共p t ref , pz ref兲 = , p t in = pt ref⬅ p t. 共4兲
The velocitiesvzinandvzrefhave the opposite sign
vin· N⬍ 0, vref· N⬎ 0, 共5兲 where N is a unit vector normal to the interface laying in the half-space of the electron wave under consideration 关N =共0,0,1兲 for z⬎0 and N=共0,0,−1兲 for z⬍0兴. We will as-sume that the crystallographic axes in half-spaces z0 are identical. In this case the momenta and velocities for elec-trons incident on the barrier and for those transmitted through the barrier are equal.
In general Eq.共4兲 may have several solutions, i.e., several
specularly reflected states may correspond to an incident state with momentum pzin. Such reflection is called
multi-channel specular reflection.11Below we assume that there is only one reflected electron state.
In the limit of a small probability of electron tunneling through the barrier,兩t兩2Ⰶ1 the applied voltage drops entirely over the barrier and we take the electric potential to be a step function V共z兲=V⌰共−z兲. The reference point of zero electron energy is the bottom of the conduction band in the upper half-space, z⬎0. The conduction band in the lower half-space z⬍0 is shifted by a value eV. We also assume that the applied bias eV is much smaller than the Fermi energy and in solving the Schrödinger Eq. 共2兲 we neglect the electric
po-tential V共z兲. Equation 共2兲 can be solved by using perturbation
theory with the small parameter 兩t兩Ⰶ1.12 In the zeroth ap-proximation in this parameter we have the problem of an impenetrable partition between two metal half-spaces.
We start by solving for the wave function共0兲共,pt; r兲 for
a tunneling point contact of low transparency,兩t兩Ⰶ1 without defects 关D共r−r0兲=0兴. The wave function 0
共0兲共r兲, in zeroth
order in the parameter 兩t兩Ⰶ1, satisfies the boundary condi-tion0共0兲共z=0兲=0 at the interface,
0共0兲共,pt;r兲 = eipt/ប共eipz in
z/ប− eipzrefz/ប兲. 共6兲
Let us consider an electron wave exp共ipr/ប兲 incident on the junction from the lower half-space z⬍0, so that vz⬎0. In
this half-space to first approximation in the parameter t the solution 共0兲共r兲 of the homogeneous Schrödinger equation can be written in the form12
共0兲共r兲 = 0
共0兲共r兲 +共−兲共r兲, z ⬍ 0, 共7兲
where the second term,共−兲共r兲⬀t, describes the changes in the reflected wave as a result of transmission through the contact. The wave function transmitted into the half-space
z⬎0 is proportional to the amplitude t,
共0兲共r兲 =共+兲共r兲, z ⬎ 0. 共8兲
The function 共0兲共, z兲 satisfies the condition of continuity and the condition of conservation of probability flow at z = 0. For small兩t兩 these conditions reduce to
共−兲共,0兲 =共+兲共,0兲, 共9兲
teipt/ប= f共兲共+兲共,0兲. 共10兲
共+兲共,z兲 = t
冕
−⬁ ⬁ dp t⬘
共2ប兲2F共pt− pt⬘
兲e i关pt⬘+pz共+兲共,pt⬘兲z兴/ប , 共11兲 where F共pt− pt⬘
兲 =冕
−⬁ ⬁ de i共pt−pt⬘兲/ប f共兲 ; 共12兲 pz共±兲共,pt⬘
兲 are roots of the equation共pt
⬘
, pz共±兲兲 = 共p兲, 共13兲corresponding to waves with velocities vz共+兲共,pt
⬘
兲⬎0 and vz共−兲共,pt⬘
兲⬍0.Let D共r兲 be a spherically symmetric scattering potential for a pointlike defect, with a range rD that is order of the
Fermi wavelength F 关DⰆD共0兲, the maximal value of D,
when兩r−r0兩ⰇrD兴. For a pointlike defect 共rD→0兲, the
right-hand side in the Schrödinger Eq. 共2兲 can be rewritten as
D共r−r0兲共r0兲.13This makes it possible to find a solution to
Eq. 共2兲 by means of the Green function G0+共r
⬘
, r ;兲 of the homogeneous equation 共at D=0兲. The wave function scat-tered from the defect,共r兲, can be expressed in terms of the wave function共+兲共r兲 transmitted into the upper metal half-space, 共r兲 =共+兲共r兲 +共+兲共r 0兲 J共r,r0兲 1 − J共r0,r0兲 , 共14兲 where J共r,r0兲 =冕
dr⬘
D共r⬘
− r0兲G0 +共r⬘
,r;兲. 共15兲Because the Green function has a singularity at r→r
⬘
, Eq. 共14兲 is correct if the integral J共r,r0兲 共15兲 converges in thepoint r = r0.
To proceed with further calculations we assume that the scattering potential is small and use perturbation theory in the interaction with the defect. This implies that we take 兩J共r0, r0兲兩Ⰶ1. The wave function solution, that is linear in 兩t兩
and with a contribution to first order in D, is
共r兲 =共+兲共r兲 +共+兲共r
0兲
冕
dr⬘
D共r⬘
− r0兲G0+共r,r⬘
;兲.共16兲 The Green function G0共r,r
⬘
;兲 in zeroth approximation in the parameters D and兩t兩 should be calculated from the wave functions共6兲,G0+共r,r
⬘
;兲 =冕
dp 共2ប兲30共0兲共r兲关0共0兲共r
⬘
兲兴* − 共p兲 − i0 . 共17兲 This is the Green function of the Schrödinger equation in the half-space z⬎0 with hard-wall boundary conditions at z=0. Substituting the wave function0共0兲共r兲, Eq. 共6兲, for z⬎0 in
Eq.共17兲, we find G0+共r,r
⬘
;兲 = 1 iប冕
dpt 共2ប兲2 eipt共 − ⬘兲/ប+ipz共+兲z/ប vz共+兲−vz共−兲 共e−ipz共+兲z⬘/ប − e−ipz共−兲z⬘/ប兲, z ⬎ z⬘
. 共18兲For 0⬍z⬍z
⬘
one should make the replacements z↔z⬘
andpz共−兲↔−pz共+兲 in Eq.共18兲; pz共±兲are given by Eq.共13兲.
The main contribution to the integral in Eq. 共16兲 comes
from a small region near the point r
⬘
= r0. Far from the point r = r0共兩r−r0兩ⰇrD兲 the solution 共16兲 takes the form共r兲 =共+兲共r兲 + gG 0 +共r,r 0;兲共+兲共r0兲, 共19兲 where g =
冕
dr⬘
D共r⬘
− r0兲 共20兲is the constant of electron-impurity interaction.
III. POINT-CONTACT CONDUCTANCE
The electrical current I共V兲 can be evaluated from the elec-tron wave functions,, of the system through14
I共V兲 = 2e 共2ប兲3
冕
dpIp⌰共vz兲关nF共兲 − nF共 + eV兲兴. 共21兲 Here Ip=冕
−⬁ ⬁ dxdy Re共*vˆ z兲 共22兲is the density of probability flow in the z direction for the momentum p, integrated over a plane z = x const, nF共兲 is the
Fermi distribution function, and vˆ is the velocity operator, vˆ =共pˆ兲/pˆ. For the definiteness we choose in Eq.共21兲 eV
⬎0. At low temperatures the tunnel current is due to those electrons in the half-space z⬍0 having an energy between the Fermi energy,F, and F+ eV, because on the other side
of the barrier z⬎0 only states with 艌F are available.
After performing the integration over a plane at zⰇz0,
where the wave function共19兲 can be used, we find the
den-sity flow共22兲 becomes
Ip=兩t共,pt兲兩23បR4具共vz共+兲兲2典共兲 +兩t共,pt兲兩 2g2R4 ប Re
冕
dpt⬘
共2ប兲2 ivz共+兲共pt⬘
兲e−ipt⬘0/ប vz共+兲共pt⬘
兲 − vz共−兲共pt⬘
兲⫻共e−ipz共+兲共pt⬘兲z0/ប− e−ipz共−兲共pt⬘兲z0/ប兲
⫻
冕
−⬁ ⬁ dp t⬙
共2ប兲2e i关pt⬙0+p共+兲z 共pt⬙兲z0兴/ប, 共23兲具¯典=
冕
p= dSp 兩v兩¯冕
p= dSp 兩v兩 , 共24兲scaled by the velocity, 兩v兩=兩p兩. In Eq. 共23兲, pz共+兲共pt兲 and
pz共−兲共pt兲 are given by Eq. 共13兲 and共兲 is the electron density
of states per unit volume.
Taking into account Eq.共23兲 we can calculate the
current-voltage characteristics I共V兲. The conductance G共V兲 is the first derivative of the current I共V兲 with V and in the limit T
→0 we obtain
G共V兲 = I
V= e
2共兲具I
p典F+eV. 共25兲
After integration over Sp共24兲, Eq. 共25兲 should be expanded
in the parameter eV /FⰆ1.
IV. ASYMPTOTICS OF THE WAVE FUNCTION AND THE CONDUCTANCE
In this section we find the wave function at large distances from the contact, rⰇF, and an asymptotic expression for
the conductance in the limit of a large distance between the defect and the contact, r0ⰇF and a small contact radius,
RⰆF, whereF is the characteristic electron Fermi
wave-length. For R→0 the function F in Eq. 共12兲 takes the form8
F共pt− pt
⬘
兲 =R2, 共26兲and the wave function共11兲 can be written as 共+兲共,z兲 = t共,p t兲R2
冕
−⬁ ⬁ dp t⬘
共2ប兲2e i关pt⬘+pz共+兲共pt⬘兲z兴/ប . 共27兲 Let us consider the integral⌳共r,兲 =
冕
−⬁ ⬁ dp t⬘
共2ប兲2e i⌫共pt,r兲, 共28兲where⌫共pt, r兲 is the phase accumulated over the path
trav-eled by the electron between the contact and the point r,
⌫共pt,r兲 =
1
ប关pt+ pz共+兲共pt兲z兴. 共29兲
This kind of integral appears in the expressions for the wave function关Eqs. 共27兲, 共18兲, and 共19兲兴 and for the conductance
关Eqs. 共23兲 and 共25兲兴. At a large distance, rⰇF, the exponent
under the integral in Eq.共28兲 is a rapidly oscillating function
and the integral can be calculated by the stationary phase method 共see, for example, Ref. 15兲. The stationary phase
points pt= pt
共st兲are defined by the equation
冏
⌫pt
冏
pt=pt共st兲= 0. 共30兲
With Eq.共29兲 we find
+ z
冏
pz 共+兲共p t兲 pt冏
pt=pt共st兲 =− z冏
vt 共+兲共p t兲 vz共pt兲冏
pt=pt共st兲 = 0. 共31兲 For rⰇFthe asymptotic value⌳as共r,兲 of the integral 共28兲is given by ⌳as共r,兲 = cos 2បr
冑
兩K0兩 exp冋
i⌫0+ i 4 sgn冉
2p z px共st兲2冊
共1 + sgn K0兲册
. 共32兲 Here ⌫0共,r兲 = ⌫共pt共st兲,r兲 共33兲is the phase共29兲 in a point defined by Eq. 共31兲, K共,p兲 is the
Gaussian curvature of the surface of constant energy 共px, py, pz兲=, and cos共r兲=z/r is the angle between the
vector r and the z axis. At the stationary phase points the curvature K共,p兲 can be written as
K0共,n兲 =
冉
1 兩v兩2兺
i,k=x,y,z Aiknink冊
pt=pt共st兲 , 共34兲 where Aik= det共m−1兲mik−1共p兲 is the algebraic adjunct of the element
mik
−1共p兲 = 2
pipk
共35兲 of the inverse mass matrix m−1;16 n
iare components of the
unit vector n = r / r. Note that for an arbitrary FS mik共p兲 in the
point pt= pt
共st兲depends on the direction of vector r. It follows
from Eq.共31兲 that the velocity at the stationary phase point
pt共st兲 is parallel to the radius vector r =共, z兲.
If the curvature of the FS changes sign, Eq.共31兲 has more
than one solution pt= pt,s共st兲共s=1,2, ...兲. In that case the value
of the integral共28兲 is replaced by a sum over all points pt,s共st兲, which in the limit of large distances is
⌳共r,兲 ⬇
兺
s
⌳sas共r,兲, 共36兲
with ⌳sas共r,兲 given by Eq. 共32兲 for each stationary phase
point s. It may also occur that Eq. 共31兲 does not have any
solution for given directions of the vector r, and the electron cannot propagate along these directions. These two energy surface properties result in complicated patterns of the dis-tribution of the modulus of the wave function:共1兲 For direc-tions for which Eq. 共31兲 has several solutions a quantum
interference pattern of the electron waves with different ve-locities should be observed.共2兲 When Eq. 共31兲 has no
solu-tion for the selected direcsolu-tion of the vector r classical mosolu-tion in this direction is forbidden and the wave function is expo-nentially small.
For large values r , r
⬘
ⰇFthe asymptotic behaviors of theGreen function, Eq.共18兲, and of the conductance, Eq. 共25兲 at
r0ⰇF, can be found analogous to the evaluation of the
mo-mentum must be taken in the stationary phase point. For the partial wave scattered by the defect that is moving towards the interface pz共+兲共pt兲 in Eq. 共29兲 must be replaced by
pz共−兲共pt兲. In this case the stationary points have a group
ve-locity v共−兲 directed from the point r0 towards the contact.
From the central symmetry of the FS,共p兲=共−p兲, it follows that the two stationary phase points for the function⌫共pt, r兲
are antiparallel, p共+兲共st兲= −p共−兲共st兲⬅p共st兲.
Next, we derive an asymptotic expression for the wave function 共19兲, with r,兩r−r0兩ⰇF, for a symmetric
orienta-tion of the FS with respect to the interface, so that vz共+兲= −vz共−兲, pz共+兲= −pz共−兲and mik
−1
= 0, if i⫽k. Under these conditions the wave function共19兲 takes the form
共r兲 ⬇共+兲共r兲 + gi 4ប 共+兲共r 0兲
兺
s冉
1 vz共+兲共pt,s共1兲兲 ⌳s as共 −0,z + z0;兲 − 1 vz共+兲共pt,s共2兲兲 ⌳s as共 −0,兩z − z0兩;兲冊
, 共37兲 with 共+兲共r兲 ⬇ tR2兺
s ⌳s as共r,兲. 共38兲In Eq.共37兲 the velocity vz共+兲is taken in the stationary phase points pt共st兲= pt,s共1,2兲corresponding to the directions of the vec-tor with coordinates共−0,兩z±z0兩兲.
We have assumed for the Gaussian curvature, Eq. 共34兲,
that K0⫽0 in the stationary phase points pt
共st兲. For those
points at which K0= 0 the integral共28兲 diverges. This means
that the third derivative of the phase ⌫共pt, r兲 共29兲 with
re-spect to ptmust be taken into account. In Sec. V this is done
for a model FS having cylindrical symmetry with respect to an open direction. Here we only note that the amplitude of the wave function共+兲共r兲 共38兲 in a direction of zero Gaussian
curvature is larger than for other directions, and decreases more slowly as compared to the ⬃1/r dependence of Eq. 共32兲. This results in an enhanced current flow near the cone
surface defined by the condition K0= 0.10 The same effect
also appears in the second term of the wave function共37兲,
that describes the scattered wave: the amplitude of this par-tial wave is maximal in the directions of zero Gaussian cur-vature.
If the FS has a flat part, i.e., Eq.共31兲 holds at all points of
a FS region of finite area Sfl, the associated electron waves propagate in the metal without decrease of their amplitude.10 For the flat part the dispersion relation共p兲 can be presented as 共p兲=v0p, where v0= const is the electron velocity. For
such FS the asymptotic value of the integral共28兲 is
⌳fl as共r,兲 = v0zSfl 兩v0兩共2ប兲2 exp
冉
ir ប兩v0兩冊
. 共39兲When the distance between the contact and the defect is large, r0ⰇF, we obtain the conductance of the tunnel
junc-tion, using Eq. 共23兲 and the asymptotic expression for the
wave function共37兲, G = G0
冉
1 − g 2ប2具共vz共+兲兲2典F共F兲 ⫻兺
s,s⬘ Re⌳s as共r 0,F,eV兲 ⫻Im ⌳s⬘ as共r 0,F,eV兲冊
, 共40兲where G0is the zero-bias共eV→0兲 conductance of the junc-tion without defect,
G0= e23R4ប具兩t兩2典F
2共
F兲具共vz共+兲兲2典F. 共41兲
In deriving Eq. 共40兲 we have assumed that eVⰆF and r0
ⰇF. Therefore, all functions of the energy in Eq. 共40兲 can
be taken at =F, except for the phase ⌫0共,r兲. When eV
ⰆF, ⌫0共r0,F+ eV兲 ⬇ ⌫0共F兲 + ⌫0 F eV, ⌫0 F ⬃ 1 F r0 F 共42兲 and when the product共eV/F兲共r0/F兲Ⰷ1 clearly the
conduc-tance共40兲 is an oscillatory function of the voltage V. Note
that if the inequality eVⰆFis not satisfied, the value of the
conductance共41兲 as well as the amplitude of its oscillations
depend on V. The periods of oscillations are defined by the energy dependence of the function⌫0共,r0兲 and they remain
the same for any voltage. Below presenting formulas for the conductance in different cases we do not expand arguments of oscillatory functions in the parameter eV /F. The obtained
results properly describe the total conductance at eVⰆFand
also can be used for the analysis of periods of oscillations at
eV艋F. In Eq.共40兲 the ⌳s
as共r
0,F, eV兲 denotes the function
共32兲, in which ⌫0=⌫0共r0,F+ eV兲. Equation 共40兲 shows that
for a tunnel junction of small size the amplitude and period of conductance oscillations depend on the local geometry FS in those points for which the velocity is directed along the vector r0.
In the case of a convex FS there is only one stationary phase point satisfying Eq.共31兲 which allows simplifying the
expression for the conductance共40兲. The oscillating part of
the conductance,⌬G, can be written as ⌬G G0 = g 2具共vz共+兲兲2典F共F兲共2ប兲3r02
冉
关vz共+兲共,pt兲兴2兺
i,k=x,y,z Aikn0in0k ⫻sin关2⌫0共r0,F+ eV兲兴冊
pt=pt共st兲 , 共43兲where n0= r0/ r0. The phase of the oscillations in the
conduc-tance, 2⌫0, is determined by the phase that the electron
2⌫0共r0,兲 =
2
បp共st兲r0, 共44兲
where 2兩p共st兲兩 is the chord connecting the two points on the
surface of constant energy for which the velocities are anti-parallel and aligned with the vector r0.
If the direction from the contact to the defect coincides with a direction of electron velocities for a flat part of the FS, we should use asymptotic expression 共39兲 for the function
⌳flas共r0,兲 when calculating the conductance 共40兲. For this
case the oscillating part of the conductance,⌬G, is given by ⌬G G0 = gSfl2vz02 具共vz共+兲兲2典F兩v0兩2共F兲共2ប兲4 sin
冉
2共F+ eV兲r0 ប兩v0兩冊
. 共45兲 Note that r0/兩v0兩=z0/vz0. In the case when the FS has a flatpart, the amplitude of the conductance oscillations for those special directions does not depend on the distance between the contact and the defect.
In the next sections we shall consider two models of an-isotropic FSs. The first model, of an ellipsoidal FS, illustrates the main features of the conductance oscillations for metals with a convex FS, i.e., with positive Gaussian curvature. The FS of the second model has the shape of a corrugated cylin-der, which illustrates the effects of sign inversion of the cur-vature and the presence of open directions of the FS. These two models allow us to obtain dependencies of the conduc-tance on the applied voltage and on the defect position in analytical form.
V. ELLIPSOIDAL FERMI SURFACE
For an ellipsoidal FS the Schrödinger Eq.共2兲 can, in fact,
be solved exactly in the limit R→0 and the wave function 共19兲 and the conductance 共25兲 can be found for arbitrary
distances between the contact and the defect. For this FS the dependence of the electron energy on the momentum p is given by relation 共p兲 =1 2i,k=x,y,z
兺
pkpi mik ; 共46兲where pi are the components of the electron momentum p,
1 / mik are constants representing the components of the
in-verse effective mass tensor m−1. The tensor m−1 can be di-agonalized to the form 兵m−1其
ik= mi−1␦ik so that the
momentum-space lengths
冑
2Fmicorrespond to the semiaxesof the FS ellipsoid共p兲=F.
For the ellipsoidal FS in absence of a defect共zeroth ap-proximation兲 the wave function共+兲共r兲 共27兲 can be obtained
by integration of Eq.共27兲 over momentum, 共+兲共r兲 = t共,p t兲
冑
2共兲3/2zR2 ប3冑
det m−1 ei⌫0关1 − i⌫ 0共r兲兴 ⌫03共r兲 , 共47兲 where ⌫0共,r兲 = r ប冑
2i,k=x,y,z兺
iknink, 共48兲 and t共,pt兲 = ប iUmzz冑
2mzz − mzz兺
i,k=x,y pipk mik +冉
mzz兺
i=x,y pi mzi冊
2 . 共49兲 Here,ikare the elements of the mass tensor, which is theinversive to the tensor m−1共m−1= I, with I the unitary ten-sor; ik= Aik/ det m−1兲. The Green function 共18兲 takes the
form G0+共r,r
⬘
;兲 = 2冑
2 ប3冑
det m−1冢
exp冉
i冑
⌫02共r − r⬘
兲 +8mzzzz⬘
ប2冊
冑
⌫02共r − r⬘
兲 + 8mzzzz⬘
ប2 −exp关i⌫0共r − r⬘
兲兴 ⌫0共r − r⬘
兲冣
. 共50兲Using Eqs.共47兲 and 共50兲 we obtain the wave function 共19兲 to
first approximation in the strength of the impurity potential. The modulus of this wave function is illustrated in Fig.1for a plane normal to the interface passing through the impurity and the contact.
The conductance in the limit of low temperatures, T→0, is obtained from Eqs. 共23兲 and 共25兲 by integration over all
directions of the momentum p and integration over the space coordinatein a plane z = const共z⬎z0兲, retaining only terms FIG. 1. Gray-scale plot of the modulus of the wave function in the plane x = 0 for an ellipsoidal FS. The shape of the FS is defined by the mass ratios mx/ mz= 1, my/ mz= 3, and the long axis of the ellipsoid is rotated by/4 around the x axis, away from the y axis. The coordinates are measured in units ofz=ប/
冑
2mzz. The posi-tion of the defect is r0=共0,5,15兲. An interference pattern is visibleto first order in g 共i.e., ignoring multiple scattering at the impurity site兲, Gell共V兲 = G 0 ell
再
1 −12 2gz 0 2共2 F兲3/2 ប5冑
m zzdet共m−1兲 1 ⌫0 4共 F,r0兲 ⫻冋
1 2冉
1 − 1 ⌫02共F,r0兲冊
sin 2⌫0共F+ eV,r0兲 + 1 ⌫0共F,r0兲 cos 2⌫0共F+ eV,r0兲册
冎
, 共51兲 with⌫0共,r兲 given by 共44兲.The amplitude of the conductance oscillations is maximal when⌫0is minimal. For a fixed depth z0 this minimum
oc-curs when the defect position 0 is in the point 00 with
respect to the point contact at r = 0, where
00= z0
冉
mzz/mzx
mzz/mzy
冊
. 共52兲
The minimal value of the phase then becomes
⌫00=⌫0共F+ eV,00,z0兲 =
1
បz0
冑
2共F+ eV兲mzz. 共53兲The phase⌫00corresponds to the extremal value of the chord of the FS in the direction normal to the interface. G0ellin Eq. 共51兲 is the conductance in the absence of a defect 共g=0兲,
G0ell= 2e 2R4 F 3 9ប3U2
冑
det共m−1兲冑
m zz . 共54兲Figure2 shows a plot of the oscillatory part of the
con-ductance, ⌬G=Gell共0兲−G 0
ell, Eq. 共51兲, for the contact as a
function of the position of the defect,0, in the limit of low
voltage, V→0.
For the ellipsoidal model FS the wave function and con-ductance have been obtained exactly, within the framework of the model. For large distances r and r0they transform into
the asymptotic expressions, Eqs. 共37兲 and 共40兲. We do not
present the asymptotic form explicitly but it agrees to within a term proportional to⌫0−4to the exact from, Eq.共51兲. In Fig. 3we compare the results for the calculations of the conduc-tance by using the exact 共51兲 and asymptotic 共40兲
expres-sions. The figure confirms that for relatively small distances 关Fig. 3共a兲兴 the asymptotic formula still qualitatively de-scribes the conductance very well and that for larger dis-tances 关Fig. 3共b兲兴 the two results are in a good agreement. The parameters for the FS in Figs.2and3 are the same as those for Fig.1.
VI. OPEN FERMI SURFACE
The second model FS we want to discuss has the form of a corrugated cylinder共Fig.4兲, which is open along the
direc-tion p储, 共p兲 = p⬜ 2 2m+1sin 2p储b 2 , − b 艋 p储艋 b, ⬎ 1, 共55兲 where 2/ b is the size of the Brillouin zone, and m is an effective mass. We further impose that the momentum per-FIG. 2. Dependence of the oscillatory part of the conductance,
⌬G, as a function of the position of the defect0in the plane z
= z0for the same shape and orientation of the ellipsoidal FS as in Fig.1. The coordinates are measured in unitszF=ប/
冑
2mzzFandthe defect sits at z0= 5. The figure shows that⌬G is an oscillatory function of the defect position that reflects the ellipsoidal form of the FS and the oscillations are largest when the defect is placed in the position00, defined by Eq.共52兲.
FIG. 3. Comparison of the oscillating part of the conductance for an ellipsoidal FS calculated in the point0共52兲 of the maximum
pendicular to the symmetry axis of the FS remains finite,
p⬜max艋 p⬜艋 p⬜min, 共56兲 where
p⬜max共兲 =
冑
2m, p⬜min共兲 =冑
2m共 − 1兲 共57兲are the maximal and minimal radii of the cylindrical surface, respectively. As a consequence of rotational symmetry the Gaussian curvature K 共34兲 of the surface depends only on
p⬜. The central part of the surface 共belly兲 has a positive curvature K⬎0 while the ends near the Brillouin zone boundary共necks兲 have negative curvature. In the direction perpendicular to the symmetry axis there are two partial waves propagating with different parallel velocities, v1 and v2, belonging to the parts of FS having opposite sign of K. Rotating away from the perpendicular direction towards the axis the two solutions persist but the two corresponding points on the FS move closer together until they merge at the curve defined by K = 0, the inflection line. For directions be-yond this angle 共i.e., for ⬍c in Fig. 4兲 no propagating
wave solutions exist. On the inflection line a unique solution with velocity vcis found. For⬎cthere are two stationary
phase points p⬜s共s=1,2兲 that satisfy Eq. 共31兲, corresponding
with two different velocities v共+兲共p⬜s兲 directed along the ra-dius vector r. The larger value, p⬜2, belongs to the belly of the FS共K⬎0兲 and the smaller one, p⬜1, belongs to the neck 共K⬍0兲. At the inflection line of the surface we have
冏
2p储共,p⬜兲
p⬜2
冏
p⬜=p⬜0= 0, 共58兲
which defines the value of perpendicular momentum p⬜0. From this condition we obtain
p⬜0=
冑
p⬜maxp⬜min. 共59兲 The cone inside of which no propagating states exist is de-fined by the condition冏
v储v⬜
冏
p⬜=p⬜0= cotc=
b
2共p⬜max− p⬜min兲sgn共v储兲, 共60兲 where the components of the velocityv储andv⬜are given by
v储=
1b
2 sin共p储b兲, v⬜=
p⬜
m . 共61兲
In spite of the simplicity of the model FS共55兲, the
inte-grals in Eqs.共19兲 and 共25兲, cannot be evaluated analytically.
We can only discuss the asymptotic behavior for r0ⰇF.
Qualitatively this result should also be valid for r0⬎F. For
the directions that have two stationary phase points, having opposite signs of the Gaussian curvature, Eq.共40兲 acquires
the form Gop= G0op
冋
1 + g cos 2共n 0兲 共2ប兲3ប具共v z 共+兲兲2典共 F兲r02 ⫻兺
s,s⬘=1,2 1冑
兩K0共s兲共F,n0兲K0共s⬘兲共F,n0兲兩 ⫻cos冉
⌫0共s兲共F+ eV,r0兲 + 2共1 − s兲冊
sin冉
⌫0 共s⬘兲共 F+ eV,r0兲 + 2共1 − s⬘
兲冊
册
, 共62兲 where G0op is the conductance of the contact without defect given by Eq. 共41兲, ⌫0共s兲共,r兲=⌫共pt,s共st兲, r兲, and K0共s兲共,n兲= K共pt,s共st兲,兲.
The appearance of the conductance oscillations depends strongly on the orientation of the FS with respect to the interface. Below we will consider two specific orientations, having the axis of the FS either perpendicular or parallel to the interface.
A. Direction of open FS perpendicular to the interface When the isoenergy surface is open along the contact axis
z the components of the momenta in Eq. 共55兲 are p⬜
=
冑
px2
+ py
2
and p储= pz. In this case the conductance of the
clean contact共without defect兲 becomes
G0op=
e2R4m214b2
8ប3U2 . 共63兲
From Eq.共31兲 the stationary phase points for the iso-energy
surface are p⬜s2 =1 2
冋
p⬜max 2 + p ⬜min 2 − 4 b2cot 2+共 − 1兲s冑
冉
p⬜max2 + p⬜min2 − 4 b2cot 2冊
2 − 4p⬜max2 p⬜min2册
. 共64兲 The angle = arccos共z/r兲 is defined by the direction of the radius vector r. The Gaussian curvature K0and the phase⌫0in the points共64兲 are given by the relations
K0共s兲共,n兲 =b 2共− 1兲s sin4 4p⬜s2
冑
冉
p⬜max 2 + p⬜min2 − 4 b2cot 2冊
2 − 4p⬜max2 p⬜min2 , 共65兲 ⌫0共s兲共,r兲 = 1 បp⬜s冑
x2+ y2+ 2z បbarcsin冑
2m − p⬜s2 2m1 . 共66兲 The angle in Eqs. 共64兲 and 共65兲 is contained within theintervalc艋艋/ 2, where thecis given by Eq.共60兲.
The modulus of the wave function is plotted in Fig.5. For the calculation of the wave function, Eq.共37兲, we used
for-mulas 共65兲 for the curvature and 共66兲 for the phase in the
asymptotic expression for the integral ⌳as 共32兲. Although,
strictly speaking Eq.共37兲 is not applicable in the vicinity of
the contact and near the defect, nor inside the classically inaccessible region, Fig.5illustrates the main features of this problem. One observes the interference of the two partial waves with different velocities, the existence of a forbidden cone, the anisotropy of the waves scattered by the defect, and the enhanced wave function amplitude near the edge of the forbidden cone.
At the inflection lines, where K = 0 andis given by Eq. 共60兲, the square root in Eq. 共64兲 is equal to zero. For this
direction two stationary phase points merge and the electron velocity is directed along the cone of the classically forbid-den region. The asymptotic expression for the conductance 共62兲 diverges at these points, which implies that the third
derivative of the phase, Eq.共29兲, with respect to p⬜must be taken into account. When the vector r0connecting the point
contact to the defect lies along the cone of the forbidden region the conductance oscillations have maximal amplitude and the conductance takes the form
Gmaxop = G0op
冋
1 − Cg m1冉
b2 ប4z 0 5冊
1/3冑
p⬜maxp⬜min共p⬜max− p⬜min兲3sin
冉
2⌫00− 5 6冊
册
, where ⌫00=⌫0共p⬜0,z0兲 = 2z0 bប冏
冉
冑
p⬜maxp⬜min p⬜max− p⬜min + arcsin冑
p⬜max 2 − p ⬜maxp⬜min 2m1冊
冏
=F+eV , 共67兲 and C is a numerical constant, C⯝1.97⫻10−4. The energydependencies of p⬜maxand p⬜min are given by Eq.共57兲.
Figure6 shows a plot of the oscillatory part⌬G=Gop共0兲
− G0op of the conductance Gop共0兲, Eq. 共62兲, as a function of
the lateral position of the defect 0 for a fixed distance z0 from the interface. The oscillation pattern has a dead region in the center, corresponding to defect positions inside the classically inaccessible part of the metal for electrons in-jected by the point contact. This region is defined by the cone 共60兲 and its radius00= z0/ cotcdepends on the depth of the
defect under surface. The oscillations in the conductance are largest when the defect is placed at the edge of the cone0
FIG. 5. Gray-scale plot of the modulus of the wave function in the plane x = 0 for a warped cylindrical FS having the open direction along the contact axis z. The coordinates are measured in units of 储=ប/
冑
2m. The parameters used in the model are 1/=0.9, b冑
2m=5.2, and a defect sits at r0=共0,13,18兲.FIG. 6. Dependence of the oscillatory part⌬G of the conduc-tance, as a function of the lateral position of the defect0 in the plane z = z0. The open direction of the FS is oriented perpendicular
to the interface. The coordinates are measured in units of 储F
=ប/
冑
2mF. The parameters used for the model FS共55兲 are 1/F= 0.9, b
冑
2mF= 6, and a defect sits at a depth of z0= 11. In a central=00. In this case the defect is positioned in a direction of
velocity belonging to the inflection line of the FS and the electron flux in this direction is maximal.
B. Direction of open FS parallel to the interface The second orientation we want to discuss is that with the FS共55兲 having its open direction 共the axis兲 parallel to
inter-face, with p⬜=共px, pz兲 and p储= py. The existence of a
classi-cally inaccessible region for this geometry leads to a strongly anisotropic current density in the xy plane. The expression for the point contact conductance without defect G0op be-comes
G0op=
e2R4共2 − 1兲2
4ប3b2U2 . 共68兲
The expressions for the phase, Eq. 共44兲, and the Gaussian
curvature, Eq.共34兲, now read,
⌫0共s兲共,r兲 = 1 ប
冉
冑
共x2+ z2兲关2m共 − 1s兲兴 + 2兩y兩 b arcsin冑
s冊
, 共69兲 and K0共s兲共,n兲 = 共− 1兲s1b 2sin4 4共 − 1s兲冑冉
1 +2 cot 2 m1b2冊
2 −8 cot 2 m12b2 , 共70兲 respectively. For this geometry we use spherical coordinates, with the angle between the vector r and the y axis. The variables1,2have been obtained from Eq.共31兲,s= 1 2
冋
1 + 2 cot2 m1b2 +共− 1兲 s冑冉
1 +2 cot 2 m1b2冊
2 −8 cot 2 m12b2册
. 共71兲 The first stationary phase point,1, corresponds to a positive Gaussian curvature and the second one, 2, to a negativecurvature. For=c, Eq.共60兲, they become equal,
1=2=
1 2m1
共p⬜max2
−
冑
p⬜max2 p⬜min2 兲, 共72兲and the curvature共70兲 vanishes.
Figure 7, acquired by using Eqs. 共37兲, 共69兲, and 共70兲,
illustrates the interference of waves with different velocities the electrons emerging from the contact and the interference with the waves scattered by the defect.
Figure 8 shows a plot of the oscillatory part ⌬G of the conductance Gop共0兲 as a function of the lateral position of
the defect0for a fixed distance z0from the interface. In this
case two dead regions appear symmetrically with respect to the center of the oscillation pattern along the open direction of the FS. The center of the pattern corresponds to a defect sitting on the axis of the contact, 0= 0, for which 1= 0,
2= 1, sin= 1, and cos= 1. At this point Eq.共40兲 takes the
form
Gop共V兲 = G0op
再
1 + 4gz022ប共p⬜max2 + p⬜min2 兲1b
⫻
冋
p⬜min2 sin冉
2p⬜min共F+ eV兲z0ប
冊
− p⬜max2 sin
冉
2p⬜max共F+ eV兲z0ប
冊
+ 2p⬜maxp⬜min
⫻cos
冉
p⬜max共F+ eV兲z0+ p⬜min共F+ eV兲z0ប
冊
册
冎
.共73兲 VII. DISCUSSION
We have analyzed the oscillatory voltage dependence of the conductance of a tunnel junction in the presence of an elastic scattering center located inside the bulk for metals with an anisotropic FS. These oscillations result from elec-tron waves being scattered by the defect and reflected back by the contact, interfering with electrons that are directly transmitted through the contact. The introduction of aniso-tropic electron movement beams the following implication: several points on the FS may share the same direction of the group velocity vector v whereas other directions for v can be absent. Two nonspherical shapes for the FS have been inves-tigated: the ellipsoid and the corrugated cylinder共open sur-face兲.
Contrary to the case of a spherical FS 共Ref. 8兲 in the
ellipsoidal model共46兲 the center of the conductance
oscilla-tion pattern does not need to coincide with the actual posi-tion of the defect but is displaced over a vector 00 = z0共mzz/ mzx, mzz/ mzy兲. When the STM tip is placed at this
point the oscillatory part of the conductance is given by FIG. 7. Gray-scale plot of the modulus of the wave function in the plane x = 0 for the warped cylindrical Fermi surface with the open direction along the y axis and parallel to the plane of interface. The coordinates are measured in units of 储. The Fermi surface
parameters are 1/=0.9, b
冑
2m=5.2, and the defect position is r0=共0,10,25兲. For this geometry the classically inaccessible region⌬G共V兲 ⬀ sin
冉
2បz0
冑
2共F+ eV兲mzz冊
. 共74兲The oscillation period depends on 1 / mzz, the component of
the tensor of inversive mass共35兲 for motion in the z
direc-tion, and on the depth z0 of the defect. This allows us in
principle to map out the positions of defects, as long as the shape of the FS is known. Apart from the period of the os-cillations, there is also information in the amplitude. Since short periods should correspond to small amplitudes this may be used for a test of consistency. However, quantitatively the amplitude is also influenced by unknown factors such as the defect scattering efficiency. The ellipsoidal FS is exceptional in that the problem can be solved exactly. This allows us to compare the calculation with the asymptotic approximation, and this shows that the approximation works very well for distances larger thanF.
In the case of the corrugated cylinder共55兲 the open necks
cause cones with opening angle 2c 共defined by the
inflec-tion line of the FS兲 to be classically inaccessible. If the ori-entation is such that the open direction is orthogonal to the surface this will result in a dead region with radius z0/ cotc
共60兲 where no conductance fluctuations can be observed.
Thus, by measuring the size of this dead region we directly obtain the position of the defect. The oscillation amplitude will be maximal at the border of the dead region, since the current density will be highest in the direction of the group velocity at the inflection line. In analogy to a hurricane the “eye” is surrounded by a ring of intense currents. Such rings of high amplitude oscillations have already been reported very recently in experiments on Ag and Cu共111兲 surfaces.17
For our model FS, along this border the oscillating part of the conductance is, apart from a phase factor, described by
⌬G共V兲 ⬀ sin
冉
4z0 bប冏
冉
冑
p⬜maxp⬜min p⬜max− p⬜min + arcsin冑
p⬜max 2 − p⬜maxp⬜min 2m1冊
冏
=F+eV冊
, 共75兲 where p⬜maxand p⬜minare the maximal and minimal radii of the surface of constant energy in the direction perpendicular to the axis of the cylinder共57兲, 1is the amplitude ofcorru-gation of the FS, and 2/ b is the size of the Brillouin zone. Again we find that the depth of the defect is determining the oscillation period, so that for given FS parameters this infor-mation can be exploited to investigate the structure of the metal below the surface.
If the open direction is parallel with the surface the high-est amplitude will be found with the STM tip straight above the defect. For 0= 0 the conductance oscillations are
de-scribed by Eq.共73兲. Clearly, the oscillation pattern is more
complicated than that from Eq.共74兲, since there are
contri-butions from the belly as well as from the neck parts of the FS, plus a sum frequency. For small necks the signal will be dominated by the oscillation due to the belly.
Although the two models presented in this paper are still rather artificial, they provide insights that are quite valuable for experimental work in this field. The most prominent con-clusion is that the regular oscillations due to convex parts of the FS, that behave as for the isotropic FS discussed previously,8will often be dominated by signals due to special directions. On any surface that features regions of zero cur-vature, the strongest conductance fluctuations will come from electrons traveling with the group velocity of that re-gion. Not only does this hold for the inflection lines around the necks in the共111兲 direction of, e.g., Cu, Ag or Au, it is also true for the almost flat facets in the共110兲 direction of the same metals. In the case of an inflection line the signal will decay as r−5/3, whereas for a flat facet the signal does not decay at all. This effect can be exploited for imaging defects up to much larger depths than previously estimated. The par-ticular shape of the FS for nearly all metals contain many detailed features that will allow us to check the validity of the conclusions drawn from the measured data.17
ACKNOWLEDGMENTS
The authors thank M. Wenderoth for communicating his unpublished results. One of the authors 共Ye.S.A.兲 is sup-ported by a grant of the European INTAS Young Scientists program 共Grant No. 04-83-3750兲 and one of the authors 共Yu.A.K.兲 was supported by the European Erasmus Mundus program on Nanoscience. This research was supported partly by the program “Nanosystems, nanomaterials, and nanotech-nology” of National Academy of Sciences of Ukraine. FIG. 8. Dependence of the oscillatory part⌬G of the
conduc-tance, as a function of the lateral position of the defect0 in the
plane z = z0. The open direction of the FS is oriented parallel to the interface along the y direction. The coordinates are measured in units of 储F. The parameters used for the model FS 共55兲 are
1B. Ludoph, M. H. Devoret, D. Esteve, C. Urbina, and J. M. van
Ruitenbeek, Phys. Rev. Lett. 82, 1530共1999兲.
2C. Untiedt, G. R. Bollinger, S. Vieira, and N. Agraït, Phys. Rev. B
62, 9962共2000兲.
3B. Ludoph and J. M. van Ruitenbeek, Phys. Rev. B 61, 2273
共2000兲.
4A. Halbritter, Sz. Csonka, G. Mihály, O. I. Shklyarevskii, S.
Speller, and H. van Kempen, Phys. Rev. B 69, 121411共R兲 共2004兲.
5A. Namiranian, Yu. A. Kolesnichenko, and A. N. Omelyanchouk,
Phys. Rev. B 61, 16796共2000兲.
6Ye. S. Avotina and Yu. A. Kolesnichenko, Fiz. Nizk. Temp. 30,
209共2004兲 关J. Low Temp. Phys. 30, 153 共2004兲兴.
7Ye. S. Avotina, A. Namiranian, and Yu. A. Kolesnichenko, Phys.
Rev. B 70, 075308共2004兲.
8Ye. S. Avotina, Yu. A. Kolesnichenko, A. N. Omelyanchouk, A. F.
Otte, and J. M. Van Ruitenbeek, Phys. Rev. B 71, 115430 共2005兲.
9I. M. Lifshits, M. Ya. Azbel’, and M. I. Kaganov, Electron Theory of Metals共Consultants Bureau, New York, 共1973兲.
10A. M. Kosevich, Fiz. Nizk. Temp. 11, 1106共1985兲 关Sov. J. Low
Temp. Phys. 11, 611共1985兲兴.
11V. V. Ustinov and D. Z. Khusainov, Phys. Status Solidi B 134,
313共1986兲.
12I. O. Kulik, Yu. N. Mitsai, and A. N. Omelyanchouk, Zh. Eksp.
Teor. Fiz. 63, 1051共1974兲.
13M. Ya. Azbel, Phys. Rev. B 43, 2435共1991兲.
14I. F. Itskovich and R. I. Shekhter, Fiz. Nizk. Temp. 11, 373
共1985兲 关Sov. J. Low Temp. Phys. 11, 202 共1985兲兴.
15V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics共D. Reidel, Dordrecht, 1981兲.
16G. Korn and T. Korn, Mathematical Handbook 共McGraw-Hill,
New York共1968兲.
17A. Weismann, M. Wenderoth, N. Quaas, and R. G. Ulbrich