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Electron-phonon coupling and phonon generation in normal-metal microbridges
Solid State Communications, Vol. 23, pp. 301 —303, 1977.
Pergamon Press.
Printed in Great Britain
ELECTRON-PHONON COUPUNG AND PHONON GENERATION IN NORMAL-METAL MICROBRIDGES 1.0. Kulik, R.I. Shekhter and A.N. Omelyanchouk Physico-Technical Institute of Low Temperatures, UkSSR Academy of Sciences, Kharkov, USSR (Received 11 March 1977 by E.A. Kaner) of a The oftwo current—voltage characteristic, d21/dV2, smallsecond orificederivative connecting pieces of normal metals is shown to be propoitionalto the function G(~~) = ~2(w)F(w) at ~, = eV, where F(~)is the phonon density of states, and ~2(w) the square of the electron— phonon matth element averaged over the Fermi surface and multiplied by the additional structure factor taking into account the geometry of the orifice. The constriction is shown to work,in a current-carrying state, as a source of non-equilibrium phonons emitted in the immediate vicinity of the orifice. HERE WE PRESENT for the first time a theory of
—f~(1 f 9+q)NqJ&(ep.q
electron—phonon interaction, and its influence on current—voltage characteristics, ofa small orifice connecting two pieces of normal metals. Such constrictions having the smallest diameter of the order 10—100 Awere shown to appear as a result of a “soft” breakdown of high resistance tunnel junctions [1, 2] and in the same work they were also shown to have second derivatives of theirrelated current—voltage depen21/dV2 to be closely to the density of dences, d states of phonons F(w) at c~= eV, or rather, t~the funcfiong(~,)= a2(~)F(~), which is the density of states weighted by the square of the electron—phonon matrix element and averaged over the Fermi surface. As will be shown below, the second derivative is in fact proportional to some other function, G(c,~),which contains an additional form-factor accounting for the geometry of the orifice, and is reasonably close to g(~).The physical mechanism of voltage-dependent resistance of the constriction lies in phonon generation in the vicinity of the orifice. Such phonons could, in principle, be detected by some means. The orifice in question is shown schematically in Fig. 1. The voltage bias V, applied between two sides of the junction, results in a current!flowing through the orifice. At the voltage eV ~D. the progressive phonon emission starts, which results in the current change, that is, the 1(V) dependence becomes non-ohmic. To calculate the dependence, one needs to solve the Boltzmann equation v ~+ eE = ~ (1) ar ap where ‘ph is the electron—phonon collision integral: ,
Iph(P,
r)
=
J
drqwq{ [fp+q(l
fpXNq + 1)
—‘
Ii
—
~q) + [fp...q(I fp)Nq
—
“~
P’s J + ljj ‘sp_q C9 + Wq In case ofa ~ 1, where 2a is the diameter of the orifice, and 1 the mean free path of an electron, the solution may be performed in two stages. At the first stage, one considers the case = 0,f= f~,and solves for the field distribution E = —VØaround the orifice. The field is found from the condition of electroneutrality: .~
p_qft~iYq
eJdrpEf(P~r) —fo(e~)I= 0.
(3)
The result is [31 I =
fo
eV + e~(r)+
-~-
1 ~(p, r)j~
(4)
where n equals + I or 1 depending on whether the velocity of the electron V = 8e/8p falls, or does not fall, within the solid angle ~2(r)at point r, see Fig. 1. The electrostatic potential distribution at eV ‘C 6F is simply v1 1 1 ~°kr) = ~ 12(r) sign z, (5) L ~ J and the resistance of the junction is determined by ~ s R~ = = e ~. (6) V (2irh) S is the area of the orifice, and S~the Fermi surface area. At the second step, one substitutesf~°~ +f~1~ for fin equation (1), thenf(l) is governed by the linearized equation: —
—
v
a
(1)
—~—
af~1~
—i--
+ eE~°~
=
—
eE~0
—~-
+
(7) 301
302
PHONON GENERATION IN NORMAL-METAL MICROBRIDGES
Vol. 23, No.5
£2(r,~)
Fig. 1. The geometry of the orifices. ~ is non-transparent shield separating two metallic half-spaces, M1 and M2. L1 is an example of the trajectory of electron moving through the orifice, and L2 of the trajectory reflecting at the walls. The solution of equation (7) is readily obtained in the form of an integral over the trajectory of electron, which comes through the orifice from z = oo point r, or from again z = +to the reflecting the wall, and then returning poánt r. atCalculating the current density at the orifice (z =0) according to the formula j = 2e f dr 9vf, one obtains the non-elastic component of the current at T= 0:
differing from the latest one by a 0-factor in the integrand, resulting from the specific geometry of constriction, and by additional integration over the volume 2pV~/dr. of metal, d Integrating equation (8) over r, first, and over the surface of the orifice, second, one arrives at the very
—
~,
/i(P)
2e =
—f
‘s2ith)
6
simple relation 21 d
r~js
C ~J dr J ~
=
dc~i(eV—~)j
s
‘dS’
where the function G(~)is introduced according to
w9.9’6(c~. ~~9_9,8)0[v’E &2(r, v)] iv~i.
G(c~~) =
—
(8) The integral f dS9 is taken over the Fermi surface, w9_9’ = (2ir/h) 1M9_9 2 is the square of the electron— phonon element, and the frequency of phonon matrix of branch s having the quasi-momentum q. a .~,,
~~ ~
Vj
f ~Vj
~
—
~
,
.~
~‘
f~a ~~-•l J
Vj
vI
N(0)
r
vI~!p ~J v1 x
I J, (uzvz(O)
VI w~9’ 4~a... fJ ~v f ~
1
Symbol 0 [v’ E12(r, v)] stands for plus unity, for v’ falling within the solid angle ~2(r)= 17(r, v) at the point r = p + yr ( is the vector in the plane of the orifice), or for zçro otherwise. The second derivative of equation (8) with respect to V is similar to the well known function N(0)
(10)
V 1
xJ ~
—8e3a3N(0)G(eV),
—~
J
Z
I
Z
v~
11 The integrals in the numerator of equation(11) are taken over the half of the Fermi surfaces, V~v~ <0 (that is, v~>0, v <0 or v~<0, ~ > 0), unlike the case of function g(w), equation (9). Another difference between g(~)and G(~,)lies in the structure factor of the orifice entering equation (11), namely K(v,v’) = (12) IVVzVVzI
(9)
IVzVzI i~v’— V’v
Vol. 23, No.5
PHONON GENERATION IN NORMAL-METAL MICROBRIDGES
303
To say more about the difference between the spectral function of the constriction (SFC) G(w) and
The total amount of energy transmitted to phonons in the volume — a3 is roughly of the order of!1 V
the spectral function of electron—phonon interaction, one needs some additional information about phonons. In the case of the Debye continuum, integrals (9) and (11) may be readily The well-known 2 at ~ ~evaluated. and zero at w> ~.,D result g(~..,) is Awdirect calculation of integral (11) at On thefor other hand, ~ ‘C ~>Dresults in the formula for SFC:
(a/l)4 V, which is typically, say, 1% of!0 V. The rest of
G(c,)
4 =
(13)
Ba,.
The estimates ofA andB are A X/c~.,LandB X/(4, where A is the dimensionless electron—phonon coupling constant, x — i. So, at least at small w, there is a certain difference between g(c.,) and G(w), but, nevertheless, one may expect that this will not be the case for basic phonon maxima (especially, for optical phonons), where g(~) and G(w) should be similar. The structure appearing in the d21/dV2 dependence is due to phonon emission in a pinhole between two metals, where the current density is maximum. An estimation of the validity of an expansion leading to equation (10) results, as was stated at the beginning of the article, in the inequality a ‘Ci, where 1 is of the order of v~/CL~~i0~ cm at eV~-’CI)D. To be more specific, 1 is the electron—phonon relaxation length which depends on V according to /
!~I~ 1 —~
~D ~eV)
~
a e
V
‘C
~ (14)
at eV ~ ~
the energy is released within the inelastic mean free path of electron, 1’, which is much greater thana (and even greater thanthe transport mean free path, 1’). In conclusion, small orifices of the diameter d — 10—100 A connecting two metals, provide, in a very simple manner, important information related to the electron—phonon interaction in normal metals. The procedure of obtaining such information was shown by Yanson [1,2] to be successful in case of shorted tunnel junctions, and very recently by Jensen and others [41 on a different type of microcontacts. Further development of the technique [1,2] (and reference [4]) may prove to be useful in obtaining new information related to the phonon spectrum of many metals, especially noble and alcaline ones. Besides, the orifice in question are interesting objects as “point” sources of nonequilibriwn phonons. It would be of interest to detect such phonons in a properly designed experiment.
to~thank comments. His pioneerwork on point-contact microbridges had stimulated our attempts to understand the peculiarities of the electron—phonon interaction in constricted geometries. We are thankful to E.A. Kaner for his interest in the work, especially concerningthe question of phonon generation by the orifice. ~
REFERENCES 1. 2. 3. 4.
YANSON LK.,Zh. Exp. Theor. Phys. 66, 1035 (1974). YANSON LX. & SHALOV YU.N., Zh. Exp. Theor. Phys. 71,286(1976); Fiz. Nizk. Temp. (Low Temp. Thys., published in Kharkov), 3,99(1977). OMELYANCHOUK A.N., KULIK 1.0. & SHEKHTER R.L,Lett. Zh. Exp. Theor. Thys. 25,465(1977). JENSEN A.G.M., MUELLER F.M. & WYDER P., Thys. Rev. Lett.).(to be published).
Pergamon Press.
Printed in Great Britain
ELECTRON-PHONON COUPUNG AND PHONON GENERATION IN NORMAL-METAL MICROBRIDGES 1.0. Kulik, R.I. Shekhter and A.N. Omelyanchouk Physico-Technical Institute of Low Temperatures, UkSSR Academy of Sciences, Kharkov, USSR (Received 11 March 1977 by E.A. Kaner) of a The oftwo current—voltage characteristic, d21/dV2, smallsecond orificederivative connecting pieces of normal metals is shown to be propoitionalto the function G(~~) = ~2(w)F(w) at ~, = eV, where F(~)is the phonon density of states, and ~2(w) the square of the electron— phonon matth element averaged over the Fermi surface and multiplied by the additional structure factor taking into account the geometry of the orifice. The constriction is shown to work,in a current-carrying state, as a source of non-equilibrium phonons emitted in the immediate vicinity of the orifice. HERE WE PRESENT for the first time a theory of
—f~(1 f 9+q)NqJ&(ep.q
electron—phonon interaction, and its influence on current—voltage characteristics, ofa small orifice connecting two pieces of normal metals. Such constrictions having the smallest diameter of the order 10—100 Awere shown to appear as a result of a “soft” breakdown of high resistance tunnel junctions [1, 2] and in the same work they were also shown to have second derivatives of theirrelated current—voltage depen21/dV2 to be closely to the density of dences, d states of phonons F(w) at c~= eV, or rather, t~the funcfiong(~,)= a2(~)F(~), which is the density of states weighted by the square of the electron—phonon matrix element and averaged over the Fermi surface. As will be shown below, the second derivative is in fact proportional to some other function, G(c,~),which contains an additional form-factor accounting for the geometry of the orifice, and is reasonably close to g(~).The physical mechanism of voltage-dependent resistance of the constriction lies in phonon generation in the vicinity of the orifice. Such phonons could, in principle, be detected by some means. The orifice in question is shown schematically in Fig. 1. The voltage bias V, applied between two sides of the junction, results in a current!flowing through the orifice. At the voltage eV ~D. the progressive phonon emission starts, which results in the current change, that is, the 1(V) dependence becomes non-ohmic. To calculate the dependence, one needs to solve the Boltzmann equation v ~+ eE = ~ (1) ar ap where ‘ph is the electron—phonon collision integral: ,
Iph(P,
r)
=
J
drqwq{ [fp+q(l
fpXNq + 1)
—‘
Ii
—
~q) + [fp...q(I fp)Nq
—
“~
P’s J + ljj ‘sp_q C9 + Wq In case ofa ~ 1, where 2a is the diameter of the orifice, and 1 the mean free path of an electron, the solution may be performed in two stages. At the first stage, one considers the case = 0,f= f~,and solves for the field distribution E = —VØaround the orifice. The field is found from the condition of electroneutrality: .~
p_qft~iYq
eJdrpEf(P~r) —fo(e~)I= 0.
(3)
The result is [31 I =
fo
eV + e~(r)+
-~-
1 ~(p, r)j~
(4)
where n equals + I or 1 depending on whether the velocity of the electron V = 8e/8p falls, or does not fall, within the solid angle ~2(r)at point r, see Fig. 1. The electrostatic potential distribution at eV ‘C 6F is simply v1 1 1 ~°kr) = ~ 12(r) sign z, (5) L ~ J and the resistance of the junction is determined by ~ s R~ = = e ~
—
v
a
(1)
—~—
af~1~
—i--
+ eE~°~
=
—
eE~0
—~-
+
(7) 301
302
PHONON GENERATION IN NORMAL-METAL MICROBRIDGES
Vol. 23, No.5
£2(r,~)
Fig. 1. The geometry of the orifices. ~ is non-transparent shield separating two metallic half-spaces, M1 and M2. L1 is an example of the trajectory of electron moving through the orifice, and L2 of the trajectory reflecting at the walls. The solution of equation (7) is readily obtained in the form of an integral over the trajectory of electron, which comes through the orifice from z = oo point r, or from again z = +to the reflecting the wall, and then returning poánt r. atCalculating the current density at the orifice (z =0) according to the formula j = 2e f dr 9vf, one obtains the non-elastic component of the current at T= 0:
differing from the latest one by a 0-factor in the integrand, resulting from the specific geometry of constriction, and by additional integration over the volume 2pV~/dr. of metal, d Integrating equation (8) over r, first, and over the surface of the orifice, second, one arrives at the very
—
~,
/i(P)
2e =
—f
‘s2ith)
6
simple relation 21 d
r~js
C ~J dr J ~
=
dc~i(eV—~)j
s
‘dS’
where the function G(~)is introduced according to
w9.9’6(c~. ~~9_9,8)0[v’E &2(r, v)] iv~i.
G(c~~) =
—
(8) The integral f dS9 is taken over the Fermi surface, w9_9’ = (2ir/h) 1M9_9 2 is the square of the electron— phonon element, and the frequency of phonon matrix of branch s having the quasi-momentum q. a .~,,
~~ ~
Vj
f ~Vj
~
—
~
,
.~
~‘
f~a ~~-•l J
Vj
vI
N(0)
r
vI~!p ~J v1 x
I J, (uzvz(O)
VI w~9’ 4~a... fJ ~v f ~
1
Symbol 0 [v’ E12(r, v)] stands for plus unity, for v’ falling within the solid angle ~2(r)= 17(r, v) at the point r = p + yr ( is the vector in the plane of the orifice), or for zçro otherwise. The second derivative of equation (8) with respect to V is similar to the well known function N(0)
(10)
V 1
xJ ~
—8e3a3N(0)G(eV),
—~
J
Z
I
Z
v~
11 The integrals in the numerator of equation(11) are taken over the half of the Fermi surfaces, V~v~ <0 (that is, v~>0, v <0 or v~<0, ~ > 0), unlike the case of function g(w), equation (9). Another difference between g(~)and G(~,)lies in the structure factor of the orifice entering equation (11), namely K(v,v’) = (12) IVVzVVzI
(9)
IVzVzI i~v’— V’v
Vol. 23, No.5
PHONON GENERATION IN NORMAL-METAL MICROBRIDGES
303
To say more about the difference between the spectral function of the constriction (SFC) G(w) and
The total amount of energy transmitted to phonons in the volume — a3 is roughly of the order of!1 V
the spectral function of electron—phonon interaction, one needs some additional information about phonons. In the case of the Debye continuum, integrals (9) and (11) may be readily The well-known 2 at ~ ~evaluated. and zero at w> ~.,D result g(~..,) is Awdirect calculation of integral (11) at On thefor other hand, ~ ‘C ~>Dresults in the formula for SFC:
(a/l)4 V, which is typically, say, 1% of!0 V. The rest of
G(c,)
4 =
(13)
Ba,.
The estimates ofA andB are A X/c~.,LandB X/(4, where A is the dimensionless electron—phonon coupling constant, x — i. So, at least at small w, there is a certain difference between g(c.,) and G(w), but, nevertheless, one may expect that this will not be the case for basic phonon maxima (especially, for optical phonons), where g(~) and G(w) should be similar. The structure appearing in the d21/dV2 dependence is due to phonon emission in a pinhole between two metals, where the current density is maximum. An estimation of the validity of an expansion leading to equation (10) results, as was stated at the beginning of the article, in the inequality a ‘Ci, where 1 is of the order of v~/CL~~i0~ cm at eV~-’CI)D. To be more specific, 1 is the electron—phonon relaxation length which depends on V according to /
!~I~ 1 —~
~D ~eV)
~
a e
V
‘C
~ (14)
at eV ~ ~
the energy is released within the inelastic mean free path of electron, 1’, which is much greater thana (and even greater thanthe transport mean free path, 1’). In conclusion, small orifices of the diameter d — 10—100 A connecting two metals, provide, in a very simple manner, important information related to the electron—phonon interaction in normal metals. The procedure of obtaining such information was shown by Yanson [1,2] to be successful in case of shorted tunnel junctions, and very recently by Jensen and others [41 on a different type of microcontacts. Further development of the technique [1,2] (and reference [4]) may prove to be useful in obtaining new information related to the phonon spectrum of many metals, especially noble and alcaline ones. Besides, the orifice in question are interesting objects as “point” sources of nonequilibriwn phonons. It would be of interest to detect such phonons in a properly designed experiment.
to~thank comments. His pioneerwork on point-contact microbridges had stimulated our attempts to understand the peculiarities of the electron—phonon interaction in constricted geometries. We are thankful to E.A. Kaner for his interest in the work, especially concerningthe question of phonon generation by the orifice. ~
REFERENCES 1. 2. 3. 4.
YANSON LK.,Zh. Exp. Theor. Phys. 66, 1035 (1974). YANSON LX. & SHALOV YU.N., Zh. Exp. Theor. Phys. 71,286(1976); Fiz. Nizk. Temp. (Low Temp. Thys., published in Kharkov), 3,99(1977). OMELYANCHOUK A.N., KULIK 1.0. & SHEKHTER R.L,Lett. Zh. Exp. Theor. Thys. 25,465(1977). JENSEN A.G.M., MUELLER F.M. & WYDER P., Thys. Rev. Lett.).(to be published).