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Field emission in high magnetic fields
TECHNICAL
NOTES
eigenvalue equations[4]
[
REFERENCES
cl-g@$I = 1.
(10)
9
Assuming a spherical Fermi surface for evaluating the sum in equation (IO) we obtain to second order in q
I. KITTELC.,Phys. Reti. 110,836(1958). 2. HEKRING C., Exchange Inreracrions among Ilinerclnr Electrons (Edited by W. G. Rado and H. Suhl), Vol. 4. Academic Press, New York (I 966). 3. RAJAGOPAI. A. K. and JOSHI S. K.. Phys. Lert. 24A, 95 ( 1967). 4. See for instance: BLANDIN A., Thcwp of (‘ondenwd Marfar. IAEA, Vienna ( 1966).
J. Phys. Chem. Solids
~yJ(TL[]_~(,_~~)]
”
I649
(Received
1 (11)
where LI = CI- urn - ZpH, m =.; (/ll, 7 - /lk, ) is the net moment, and Ed, IS the Fermi energy for the up and down bands respectively. Taking now p-H -+Um and using the magnon frequency
the roots of equation ( 1 1) can be written as: J’H
n=
+
q2
&/
q -+[
1”:;3)1 +,p(q),!2m]l/2. (12)
We see from ( 12) that the gap at resonance
is directly coupling.
related
to the electron-phonon
BI.AS AIASCIO ARTURO 1.t)PF.Z Crnrro Aromiw Ruriloclw, Comision Notional dc Encrgkr lnsfiiuto de Fisiccr.
“Dr
Josh
A. Balseiro”.
Unioersidud Nrrcionul de Cuyo, San C‘arlos de Bariloche, Rio Negro -Argentitw
Aldmicu.
pp. 1649- 1650.
Field emission in high magnetic fields
[
-~[,+~(I+;~)]]=
Vol. 3 I,
I7 September
1969)
BEHAVIOR of a field emission current in a magnetic field has been of interest for some time [ I, 21. Blatt [3J was the first to theoretically investigate such behavior for a gas of free electrons at the absolute zero of temperature. His results showed that the current should exhibit oscillations which are periodic in H-‘. Independently, and very shortly after Blatt’s calculation, Gogadze, Itskovich, and Kulik[41 considered the more general case of an arbitrary Fermi surface and non-zero temperature, with the restriction that the electron orbits around the extremal areas are closed and non-self-intersecting. Both papers consider the case for which the Fermi energy is independent of magnetic field, and for which the Fermi energy oscillates with magnetic field. Gogadze et ~I.[41 estimated that observable oscillations in both cases are a possibility only when the current originates from small pockets of electrons with low effective masses. This was essentially the same conclusion reached by Blatt. Numerical estimates of the amplitude of the oscillations from both calculations indicate effects on the order of 2 or 3 per cent at 1 tesla for an electron pocket with an effective mass of - 10 z m,,. At 3 teslas the amplitudes should be nearly 10 per cent of the emission current in zero magnetic field. In addition to the oscillatory terms, Blatt [3] THE
1650
TECHNICAL
predicted a monotonic dependence of emission current on magnetic field. For the case of small pockets and low effective mass a quadratic decrease in current with magnetic field is expected. (Gogadze, et a1.[2] did not discuss this part of the current.) As suggested by Blatt, bismuth should be a suitable substance to use to search for the effects described above. The carriers are located in small pockets and have low effective masses. Several attempts have been made in this laboratory to observe this effect, not only in bismuth, but in zinc, tantalum, and tungsten as well. In no instance has any influence of magnetic field on the emission current been found. Current levels have ranged from IO- ” to 10mHA, and magnetic field strengths up to IO teslas have been used. It is particularly noteworthy that not even the monotonic decrease in current has been detected. In bismuth, for example, where the monotonic term should be quadratic, the current should have decreased by about 4 per cent at I tesla (at which point the oscillations should be on the order of 2 per cent) and by nearly 40 per cent at 3 teslas. (At 3 teslas the oscillations should be about 10 per cent of the zero field current level.) The emission currents that were obtained were sufficiently stable over long enough periods of time that effects as small as + 1 per cent should have been readily detected. Cathodes of bismuth and zinc were prepared from pre-oriented single crystals, and cathodes of tantalum and tungsten were prepared from high purity polycrystalline wire. The emission chamber could be evacuated to a pressure between lo-” and 10mGtorr by a conventional oil diffusion pump and cold trap assembly, and then submerged in liquid helium to be cryopumped to pressures estimated to be at least less than lo-” tot-r. Field desorption techniques were employed to clean the emitting surfaces in high vacuum. Most of the experimental runs were made at temperatures near 1*2”K, which were obtained by pumping on the helium bath covering the vacuum chamber. The chamber itself was optically
NOTES
baffled from its pumping line to prevent backstreaming from the warmer parts of the system. The theoretical investigations mentioned above consider only the case of a longitudinal magnetic field, i.e. emission current parallel to the magnetic field direction. A calculation of the behavior of the current when the magnetic field is perpendicular to the direction of current flow in the case of field emission has apparently not yet been made. Both orientations were tried in the course of these experiments, and the same null results were obtained in every case. Cathodes cut and etched from single crystals of bismuth and zinc were always mounted such that the trigonal axis of Bi and c axis of Zn were parallel to the magnetic field for both the longitudinal and transverse field orientations. At the present time no definite explanation can be given for the failure of these experiments to substantiate the theory. It is felt that the experimental conditions sufficiently met the requirements of the calculations as presented in 131 and [4]. A closer look at the theories is perhaps justified. In particular, the fact that the emitting surface is not an infinite plane but is finite and three-dimensional ought to be investigated. The three-dimensional nature of the cathode implies that electrons with less then the maximum velocity parallel to the magnetic field will still have the same tunneling probability as those for which the velocity parallel to H is a maximum, provided they approach the metal surface perpendicularly. This in turn would cause the total current to be composed of electrons from many Landau levels, and could conceivably wash out any possible magnetic field effects. D. J.
FI.OOD
S.A.
HARDIN
REFERENCES W. R.. BLAIR J. C. and EINSPRUC’H
N. G., Cryogenics’S,
138
( 1965).
VAN OOSTROM A. G. J.. Phillips Res. Rep. Supp/. No. I (1966). BI.ATT F. J., Phys. Rec. 131, I66 ( 1963). GOGADZE G. A., ITSKOVICH I:. I. and KUl.lK 1. 0.. Societ phys. JETP 19.622 f 1969).
NOTES
eigenvalue equations[4]
[
REFERENCES
cl-g@$I = 1.
(10)
9
Assuming a spherical Fermi surface for evaluating the sum in equation (IO) we obtain to second order in q
I. KITTELC.,Phys. Reti. 110,836(1958). 2. HEKRING C., Exchange Inreracrions among Ilinerclnr Electrons (Edited by W. G. Rado and H. Suhl), Vol. 4. Academic Press, New York (I 966). 3. RAJAGOPAI. A. K. and JOSHI S. K.. Phys. Lert. 24A, 95 ( 1967). 4. See for instance: BLANDIN A., Thcwp of (‘ondenwd Marfar. IAEA, Vienna ( 1966).
J. Phys. Chem. Solids
~yJ(TL[]_~(,_~~)]
”
I649
(Received
1 (11)
where LI = CI- urn - ZpH, m =.; (/ll, 7 - /lk, ) is the net moment, and Ed, IS the Fermi energy for the up and down bands respectively. Taking now p-H -+Um and using the magnon frequency
the roots of equation ( 1 1) can be written as: J’H
n=
+
q2
&/
q -+[
1”:;3)1 +,p(q),!2m]l/2. (12)
We see from ( 12) that the gap at resonance
is directly coupling.
related
to the electron-phonon
BI.AS AIASCIO ARTURO 1.t)PF.Z Crnrro Aromiw Ruriloclw, Comision Notional dc Encrgkr lnsfiiuto de Fisiccr.
“Dr
Josh
A. Balseiro”.
Unioersidud Nrrcionul de Cuyo, San C‘arlos de Bariloche, Rio Negro -Argentitw
Aldmicu.
pp. 1649- 1650.
Field emission in high magnetic fields
[
-~[,+~(I+;~)]]=
Vol. 3 I,
I7 September
1969)
BEHAVIOR of a field emission current in a magnetic field has been of interest for some time [ I, 21. Blatt [3J was the first to theoretically investigate such behavior for a gas of free electrons at the absolute zero of temperature. His results showed that the current should exhibit oscillations which are periodic in H-‘. Independently, and very shortly after Blatt’s calculation, Gogadze, Itskovich, and Kulik[41 considered the more general case of an arbitrary Fermi surface and non-zero temperature, with the restriction that the electron orbits around the extremal areas are closed and non-self-intersecting. Both papers consider the case for which the Fermi energy is independent of magnetic field, and for which the Fermi energy oscillates with magnetic field. Gogadze et ~I.[41 estimated that observable oscillations in both cases are a possibility only when the current originates from small pockets of electrons with low effective masses. This was essentially the same conclusion reached by Blatt. Numerical estimates of the amplitude of the oscillations from both calculations indicate effects on the order of 2 or 3 per cent at 1 tesla for an electron pocket with an effective mass of - 10 z m,,. At 3 teslas the amplitudes should be nearly 10 per cent of the emission current in zero magnetic field. In addition to the oscillatory terms, Blatt [3] THE
1650
TECHNICAL
predicted a monotonic dependence of emission current on magnetic field. For the case of small pockets and low effective mass a quadratic decrease in current with magnetic field is expected. (Gogadze, et a1.[2] did not discuss this part of the current.) As suggested by Blatt, bismuth should be a suitable substance to use to search for the effects described above. The carriers are located in small pockets and have low effective masses. Several attempts have been made in this laboratory to observe this effect, not only in bismuth, but in zinc, tantalum, and tungsten as well. In no instance has any influence of magnetic field on the emission current been found. Current levels have ranged from IO- ” to 10mHA, and magnetic field strengths up to IO teslas have been used. It is particularly noteworthy that not even the monotonic decrease in current has been detected. In bismuth, for example, where the monotonic term should be quadratic, the current should have decreased by about 4 per cent at I tesla (at which point the oscillations should be on the order of 2 per cent) and by nearly 40 per cent at 3 teslas. (At 3 teslas the oscillations should be about 10 per cent of the zero field current level.) The emission currents that were obtained were sufficiently stable over long enough periods of time that effects as small as + 1 per cent should have been readily detected. Cathodes of bismuth and zinc were prepared from pre-oriented single crystals, and cathodes of tantalum and tungsten were prepared from high purity polycrystalline wire. The emission chamber could be evacuated to a pressure between lo-” and 10mGtorr by a conventional oil diffusion pump and cold trap assembly, and then submerged in liquid helium to be cryopumped to pressures estimated to be at least less than lo-” tot-r. Field desorption techniques were employed to clean the emitting surfaces in high vacuum. Most of the experimental runs were made at temperatures near 1*2”K, which were obtained by pumping on the helium bath covering the vacuum chamber. The chamber itself was optically
NOTES
baffled from its pumping line to prevent backstreaming from the warmer parts of the system. The theoretical investigations mentioned above consider only the case of a longitudinal magnetic field, i.e. emission current parallel to the magnetic field direction. A calculation of the behavior of the current when the magnetic field is perpendicular to the direction of current flow in the case of field emission has apparently not yet been made. Both orientations were tried in the course of these experiments, and the same null results were obtained in every case. Cathodes cut and etched from single crystals of bismuth and zinc were always mounted such that the trigonal axis of Bi and c axis of Zn were parallel to the magnetic field for both the longitudinal and transverse field orientations. At the present time no definite explanation can be given for the failure of these experiments to substantiate the theory. It is felt that the experimental conditions sufficiently met the requirements of the calculations as presented in 131 and [4]. A closer look at the theories is perhaps justified. In particular, the fact that the emitting surface is not an infinite plane but is finite and three-dimensional ought to be investigated. The three-dimensional nature of the cathode implies that electrons with less then the maximum velocity parallel to the magnetic field will still have the same tunneling probability as those for which the velocity parallel to H is a maximum, provided they approach the metal surface perpendicularly. This in turn would cause the total current to be composed of electrons from many Landau levels, and could conceivably wash out any possible magnetic field effects. D. J.
FI.OOD
S.A.
HARDIN
REFERENCES W. R.. BLAIR J. C. and EINSPRUC’H
N. G., Cryogenics’S,
138
( 1965).
VAN OOSTROM A. G. J.. Phillips Res. Rep. Supp/. No. I (1966). BI.ATT F. J., Phys. Rec. 131, I66 ( 1963). GOGADZE G. A., ITSKOVICH I:. I. and KUl.lK 1. 0.. Societ phys. JETP 19.622 f 1969).