- Email: [email protected]
A model calculation of field emission from 3d bands in nickel
SURFACE
SCIENCE 22 (1970) 277-289 o North-Holland
A MODEL
CALCULATION
Publishing Co.
OF FlELD
EMISSION
FROM 3d BANDS IN NICKEL* BEVERLY
A. POLITZER
and P. H. CUTLER
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802. U.S.A. Received 14 April 1970 Field emission from narrow, 3d bands of fe~oma~etic nickel is discussed. A one-dimensional model, with a triangular surface barrier, is used to calculate the ratio of the transmission coefficient of an electron in a tight-binding, 3d band to that of a free electron of the same energy. The results indicate that, depending on the interatomic crystal spacing chosen, the free electron transmission coefficient is roughly 1 or 2 orders of magnitude greater than the 3d transmission coefficient. In disagreement with previous work, this ratio is independent of the applied electric field at all electron energies.
1. Introduction
Recently, there has been a growing interest in the field emission of spinpolarized electrons from ferromagnetic material+3). In the transition metals, iron and nickel, it has been conjectured that different electron populations at the Fermi energy of the exchange-split, 3d spin bands** will lead to a net polarization of a field emitted beam. To see if this is indeed the case theoretically, it will probably first be necessary to abandon the conventional free-electron (Fowler-Nordheim) picture of field emission4) and, alternatively, to involve the energy band structure of these metals i.e., the narrow 3d bands and their complicated Fermi surface. Two questions naturally arise in the consideration of the 3d bands; namely: (1) How does the tunneling probability of a Bloch state corresponding to a 3d band in a transition metal differ from that of a free electron state?, and (2) What effects do the complicated, non-spherical constant-energy surfaces of the 3d bands have on the field emission current? Attempts to include such band structure effects in field emission theory * Supported in part by U.S. Air Force Office of Scientific Research and Grant Nos. AF-AFOSR 213-66 and 69-1704. ** For the sake of simplicity, and in the spirit of the model calculation to be described, we have identified bands as “pure d” and “pure s”. It is recognized, however, that in general hybridi~tion obscures such classifications. 277
278
B.A.
POLJTZER
AND
P. H. CUTLER
have for the most part been restricted to employing simple approximations to non-spherical constant energy surfaces 5-7). Quite inconsistently however, these same calculations have retained the WKB [or slightly-modeled WKBs)] treatment of transmission coefficient. A notable exception has been the recent work of Gadzuks) on field-induced tunneling from narrow d bands; he derives a d band transmission coefficient by calculating the intuitive, but rather ad hoc, “tunneling matrix elements’* using tight-binding wave functions of an ideatized noble metal *. The problem is that a rigorous treatment of transmission coefficient, appropriate to the real bands of a transition metal, requires a detailed knowledge of the Bloch functions** which are often difficult to obtain or to treat analytically, Fortunately in the case of fee nickel, the original tightbinding calculation of Fletchers) on the unhybridized 3d bands provides wave functions which turn out to be analytically tractable in the calculation of tunneling probabilities. En addition these wave functions yield 3d band shapes and widths which may be readily scaled to agree with results of more reliable calculations recently appearing in the literaturelo-14). In view of this, it was decided to inchtde the more realistic 3d Bloch functions in the investigation of the transmission coefficient problem in ferromagnetic nickel. (Fletcher’s secular determinant may be easily modified to include the exchange splitting of the 3d bands in ferromagnetic nickel.) A one-dimensional model calculation is outlined in section 2 and the results obtained are evaluated numerically in section 3. Though admittingly naive, the model indicates that the free electron transmission coefficient is roughly 1 or 2 orders of magnitude larger than the 3d transmission coefficient for an electron of the same energy. A more realistic estimate of this transmission coefficient ratio awaits the completion of a full three-dimensional treatment, which is now in progress. Finally, the implication of the one-
* Gadzuk’s works) may be compared to the calculation discussed here. First, we use the usual boundary matching method of solution which, within the context of the onedimensional model, is exact. Gadzuk employs a general tunneling Hamiltonian formalism assuming the same triangular surface barrier. He makes approximations in the evaluation of the “tunneling matrix elements”, however, whichessentially reduce his three-dimensional treatment to a one-dimensional one. The position of the surface is taken by Gadzuk to be at the last layer of metal atoms, whereas we assume the surface to be at one-half the interatomic spacing away from the last atom. In addition, Gadzuk uses tight-binding wave functions which are sums over Slater orbitals centered on surface atoms only; on the other hand, our nickel orbitals (radial parts only) are centered on all atoms, and the required sums over all atoms in the one-dimensional crystal are performed exactly. Moreover, there are very striking differences in the results obtained by each of these methods. These are elaborated in section 4 of the text. ** Most rigorously, one requires the Bioch functions of a semi-infinite solid with a boundary.
A MODEL
CALCULATION
OF FIELD
EMISSION
FROM
3d
BANDS
279
IN NICKEL
dimensional treatment and its relationship to field emission from anisotropic crystal of ferromagnetic nickel are discussed in section 4.
a real,
2. Model calculation We take as our model a perfectly periodic, semi-infinite, one-dimensional crystal of nickel. As depicted in the potential energy diagram of fig. 1, the interatomic spacing =a, and the surface defined by z=O is taken at a distance of $a away from the last atom layer present. The metal-vacuum potential barrier is assumed to be triangular, that is, only the applied electric
*--
Fig. 1.
Region I (metal)
V
I
1 Region II Y (vacuum)
l
Sketch of electron potential energy V versus distance z through one-dimensional crystal and vacuum.
field, F, is taken into account. The usual image potential is neglected since it complicates an analytic treatment of the problem and is not expected to significantly alter the comparison of the free electron and 3d transmission coefficient. (The same applies to less important effects such as field penetration and patch potentials.) We summarize the potential energy diagram of fig. 1 as follows: Region I, z < 0 : Region IT, z > 0:
V(z) = V(z + u), V(z) = - eFz.
We write the solution, 11/i,to the Schrijdinger equation in Region I as a linear superposition of two normalized Bloch states: an incident state, 4, and a reflected state, x. We will ultimately take C#Jand x to be Bloch states of an infinite, linear lattice with periodicity a; strictly speaking however, these
280
B. A.POLITZER
should be eigenstates
AND
of a semi-infinite
P. H, CUTLER
crystal with a boundaryr5*is).
Then,
In the above, U,,+(Z) is the periodic part of the Bloch function, with n denoting the band index in the reduced zone scheme and K denoting the reduced wave “vector”. A is a constant amplitude. The expectation values of the electron velocity in the states K and-~ are respectively, d&,,
=v,>o,
i.e., an “incident
=v,
i.e., a “reflected
state”,
dAk k=K and
y-!* dtik
k=-K
state”.
Also, we demand that the incident and reflected states be related by the condition of specular reflection, i.e., E,,, = E,,, -k s E. [E is a negative energy measured from the zero in the vacuum.] Note, however, that eq. (1) arbitrarily does not allow for other reflected states which, although energy-conserving, may belong to bands different from the “incident” band, n, i.e., a “direct interband transition”, possible in the 3d transition metals which have overlapping bands. Recalling that u,, _ t = UT,K, the expectation value of the current operator, J,, is J,(Bloch)
= !;
where the Wronskian
]u,,,]‘[l
- ]#-j
+ Gi
w(un*,,K, a,,,)
cl - ]A]‘] 9
(2)
is defined,
for all wave vectors k (L, is the For free electrons, u,,~ = u,, _k = l/JL, repeating length, containing N atoms, of the one-dimensional crystal lattice). Then for an incident state k = k,, where the subscript “f” refers to free, J,(free)
= 2
[l - IA]‘]. z
In eqs. (2) and (3) above we may define J, = J,’ + J;, where the superscript 7” indicates incident current and 9” indicates reflected current. In both cases the terms cc]A12 are to be associated with the reflected current. It can easily be shown from eq. (2) that J, (Bloch), like the free electron current, JI (free), satisfies the conservation condition, div J, (Bloch) =0, since
A MODEL CALCULATION OF FIELD EMISSION FROM 3d BANDS IN NICKEL
281
un,r
and u,, _x obey the usual Schriidinger equation for the periodic part of the Bloch function171. Region II may be for convenience further divided into two subregions. For z>zO where zO= --E/E’ is the classical turning point, the appropriate outgoing solution to the Schr~dinger equation may be shown ta be
where
Iiere cr~Q.2624 (eV A”)-’ for units in which E is in eV, F in V/L%,z in A, h/m in A2/sec, and k in (1$)-l. C is an arbitrary constant (with units of A*> and J,, are Bessel functions of the indicated order (real throughout their z domain). A solution continuous with the above at z=z, may be written $&J’) = C(~clF)*(~‘)*~f_*(i~‘)
eeinf6 - J*(i& eini6] = Cgfp’),
(5)
where
p’ is real throughout its domain of z, and J* + are complex. The expectation value of the outgoing current denoted by J,, is obtained by using eq. (4):
After some manipulation of the Bessel functionsl*), the Wronskian may be evaluated
and the outgoing current fgf becomes in the case of the triangular barrier
We wish ultimately to calculate a transmission coefficient ratio, i&//TBloch, for an electron having the same energy E in both cases. We account for the different dispersion relations for the free and Bloch electron by writing in accordance with previous notation ~{~~}= E(K) = E( - x). (Recall all energies measured from zero in vacuum.) Then using the definition of the
282
B. A.
POLITZER
AND F. H. CUTLER
transmission coefficient (7) From eqs. (2) and (3), we obtain J; (BIoch)
K
Jt’(free) -=-I_ kf
Lz I%,,(Z
= OV +
where the ratio of the currents has been evaluated at the z = 0 emission plane. Recall that the arbitrary constant, C, in the outgoing current &, eq. (6), is determined by matching eq. (5) and its derivative to either tc/t (free) and d~~(free)/dz or ~,(Bloch) and d~~(Bloch)/dz at z=O. We indicate this continuity requirement by subscripting the “C’s” and writing (9) Performing the matching process in both cases yields,
and
The foliowing notation has been used : g’(O)=-
g(O)=g(z=O),
dg (z)’ dZ, _ , z-o
U”,K(O) = kl,.(z
= 017
%,. W _ h,&)l - --. __ dz dz
. iz=O
Combining eqs. (8), (9>,(lo), and (11) in eq. (71, we have finally [kg
Tfree
kf
TBloch
Ic
(0) t_ g’ (O)]
2& (0) - g (0)
TY!5tFi0)‘2 dz
I
3
(12)
{ff~,~(O)l’
where * superscript indicates complex conjugate of the enc’tosed bracket. Now from the definition of g($) and p’(z) in eq. (S), it is easily verified that g’(0) = - (@F)+ r_l$[J+(ip.,) e-i+n - J_,(i~,)
ei3x],
A MODEL
CALCULATION
OF FIELD
EMISSION
FROM
3d BANDS
IN NICKEL
283
where
In the case of field emission the fields F of interest are 0.2 V/A cF~0.4 V/A. The electron energies are E= E, = - c$, where the work function 4 N 5 eV for nickel. An order of magnitude calculation then yields the result that ,u,a8. This value of p0 allows us to use the asymptotic expansions for large arguments19) for J* +(i& and J, .s(ipO)in evaluating g (0) and g’(0) respectively. One finds
(13)
We now proceed to specialize to the case of 3d wave functions for nickel. Assuming tight-binding wave functions for the 3d Bloch states, the incident state b, is written
c#= d+1 eiKRjt#,(z
- Rj) z eiKzu,,,(z) ,
(14)
Rj where Rj denotes location of the jth atom in the linear chain, j= 1,2, 3 etc., and 4, is an atomic orbital (designated by quantum number n) localized around the R,th atom. Then, one obtains by substituting in eq. (12) for u,,.(O) from eq. (14) and for 9 (0) and g’(0) from eq. (13) :
Tfree k k-i y= [
X 1
(15)
f’
with X=
C
~~(-Rj)#~(-R~)(-aE)
COSK(Rj-Ri)
RI> R’J
+ ,/( - gE)
(a, ( _ Rj)
d&i9
+
4,
( _ R;)
d4n ‘,; Rj) 1
+@n(-Rj)drbn(-RI) --___ dz
dz
and
Here
(p,( -Rj>
d#,fz-Rj)/dz~,=c*
and
d#,(-Rj)/dz
denote
respectively +,(z-Rj)l,=a
and
284
B.A.
POLITZER
AND
P. H. CUTLER
3. Numerical estimate T&T,, To make a numerical estimate of Tfree/T,, using eq. (15) above requires a knowledge of the dispersion relations E versus k for both the tight-binding and free-electron cases. The usual parabolic free-electron dispersion relation is written in the units employed here a(E+V)=k;,
where V (in eV) is the absolute value of the energy (measured from the zero in vacuum) to the bottom of the conduction band. For the tight binding case, E=E(k) must be obtained from a somewhat artificial, one-dimensional energy band calculation. This type of calculation would assume for &(z- Rj) a one-dimensional construct of one or all of the five 3d atomic orbitals for nickel, and would then proceed to calculate overlap and 2 and 3-center integrals for use in the secular determinant. Since such a calculation is quite complicated and is more of pedagogical interest, it will not be redone here. Instead, corresponding pairs of E and k will be picked off three-dimensional energy bands shown in fig. 2. These have been calculated from Fletcher’s secular determinant and have been scaled to the minority spin bands of Conollylo). This should provide a rough, numerical estimate of the desired ratio T,,,,/T,. For this one-dimensional calculation, let us assume that 4, is just the radial part of the 3d atomic wave function for nickel. (The corresponding angular parts, the Y,,‘s, have little meaning here.) The normalized, 3d radial wave function used in the Fletcher calculation is 4,,(r)
= r2 [56 e-5r + 1.3 em”].
(16)
The sketches of $,(r) and d& (r)/dr that are shown in fig. 3 reveal the extent of these localized functions; specifically, in a distance r=+ANi, where A,,=3.5 8, is the interatomic spacing for fee nickel, 4”(r) and d&(r)/dr fall to less than 0.01 of their maximum values. It is this localized property that allows us to perform quite accurately the doubly-infinite sum over permuting products of 4,(z-Rj)lZZo and d$,(zRj)/dzl,=, in eq. (15); convergence better than 1 in lo3 is obtained by including only 3 atoms adjacent to the z = 0 surface. Not unexpectedly one finds that the ratio Tfree/Tdis very sensitive numerically to the “overlap” at z=O of the atomic orbitals and their derivatives, each centered on different atoms. This “overlap”, in turn, depends quite strongly on the interatomic spacing, a, chosen for the model calculation and on the distance of the last atom from the surface at z=O. To illustrate this,
A MODEL
CALCULATION
OF FIELD
fl I I)
EMISSION
FROM
3d
BANDS
IN NTCKEL
c1totct
f 0)
285
(000)
(IO01
(000)
r
X
W
K
f
X
W
K
f
0.1
0.0
-0.1
- 0.2
F uI -o.3x
I
w
L
MAJORITY
SPINS
Fig. 2. Plot of unhybridized, minority 3d energy bands along special symmetry directions in ferromagnetic nickel. These have been calculated from Fletcher’s secular determinant 9) and scaled to minority spin bands of Conoily 10). Note here the zero of energyisat the Fermi levef, whereas in the calculations energies are measured from zero in vacuum.
286
B.A. POLlTZER
AND
P. H. CUTLER
two extreme values were chosen for a: (1) a=A,,=3.5 il and (2) u=*A,,= = 1.75 A. In both cases, the necessary “overlaps” were obtained from eq. (16), and the double sum in eq. (15) was performed using Rj= -+(IZj-- 1) a, j= I, 2 etc. The result, for a= ANi, is a simplified expression of the form 0.00369 + (0.580 x 10-4) cos K&
Fig. 3. Sketch of the normalized, 3d radial wave function #=(r) [eq. (16) text] and its radial derivative d#,Jdr. Note localized nature of & and d#,/dr compared to interatomic spacing in a one-dimensional nickel crystal.
A similar looking expression is obtained for U=~ANi- Using eq. (17), we estimate the numerical value of 7’r,,,/Td for the Fermi energy E, z - 5 eV. To do this, one must refer to a particular 3d tight binding band, E versus k, with the bottom of the band at the energy E= - 7? In the real ferromagnetic nickel crystal only two of the unhybridized, minority spin bands contribute to the Fermi surface in the 100 direction, specifically rlZXZ and G5X5. Folding back these bands to obtain the values of the reduced wave vectors, k=k Fermi,which lie in the first Brillouin zone of the one-dimensional crystal,
A MODEL
CALCULATION
OF FIELD
EMISSION FROM
3d
BANDS
IN NICKEL
287
one obtains : T&T,
N 130,
Tfree/Tde 150,
for ri2X2 band for &X5 band
forE=E,anda=ANi=3.5A,
and LIT, = 7 Tfree/Td = g
i) for F, 2X2 band 3 for &X5 band
for E = Ef and a = tANi = 1.751( e
4. Discussion and conclusions We have described above a one-dimensional model calculation evaluating the ratio of transmission coefficients involved in the field emission through a triangular barrier from a free-electron type band and a tight-binding, 36 band of ferromagnetic nickel. A rough numerical estimate shows the freeelectron transmission coefficient to be between one and two order of magnitude greater than the 3d transmission coefficient, depending on the interatomic spacing chosen. In the ~re~dimensio~al, anitropic metal, this difference between one and two orders of magnitude will probably manifest itself as a difference in the transmission coefficient from different crystallographic planes with varying atomic arrangements. In addition, it appears that the ratio Tfrec/Tdis more signi~~ntly affected by the interatomic spacing than by the particular 3d band chosen and its associated geometrical shape. This is probably, however, just an artifact of the one-dimensional calculation. In particular, it may stem from eliminating the angular dependent Yl,‘s from the orbitals 4,. In the real three-dimensional crystal, it is well-known that the secular determinant is diagonal in special symmetry directions; each of the computed 3d bands in these directions is then characterized by a different &_ basis. One may expect that when an actual field emission direction is related to the spatial symmetry associated with an individual 3d band, the tunneling probability will indeed be very band-dependent. Conversely, the tunneling probability from a particular 3d band should, in a real three-dimensional treatment, be anisotropic with direction, We may briefly contrast the results above with those obtained by Gadzuks). Our quantity Tfree/Tdcan be compared to the reciprocal of his F(E), the ratio of the d-band to s-band tunneling probabilities at a given energy E. For the fields commonly encountered in field emission, Gadzuk calculates that F(E)= I x lOa at E=E,. This number is at least an order of magnitude different from that derived here. More important, however, is a discrepancy regarding the electric field dependence of each of these quantities; whereas
288
B. A. POLITZER
AND
P. H. CUTLER
Gadzuk’s F(E) is a strong function of the electric field [i.e., proportional to the fourth power of the field as can be seen in eq. (23) or the curves in his fig. 41, in this work the analogous ratio T,,,,/T, is independent of field at all electron energies. We believe the origin of this discrepancy to lie in Gadzuk’s initial, erroneous
choice of wave function;
$-e
ik(s)z,
with
k(z) =
i.e., he uses
r$csT
-
z)]
in the classically forbidden region of the triangular surface barrier. For this barrier the well-known solutions to the Schriidinger equation (see section 2) are Bessel or Hankel functions. Furthermore close to the metal surface, which is the region of maximum overlap and therefore of greatest importance in Gadzuk’s calculation, the Hankel functions exhibit a field dependence of exp (l/F); Gadzuk’s wave function, on the other hand, is =exp(F+) in this region. It is well known that the density of states of a narrow, 3d band in a transition metal is up to ten times that of a wide, free-electron type conduction band. Perhaps, one may then expect to find in certain crystallographic directions a field emission current from the 3d bands of nickel which is comparable in size to that of a free electron. Polarization of field emitted electrons from the exchange-split, 3d bands of ferromagnetic nickel may then be hypothetically possible. The verification of this, however, awaits completion of the full three-dimensional treatment.
Acknowledgments The authors gratefully acknowledge the assistance of J. W. Conolly providing tables of his energy bands for ferromagnetic nickel.
in
Note added in proof After completion of the present work, our attention was directed to the paper of Prices’-‘), who considered a similar problem. He used Wannier functions to calculate the transmission probabilities for Bloch waves in a one-dimensional model and found the tunneling probability to be substantially smaller than for free electron wave functions. Since Price did not do any numerical calculations, we can only adduce qualitative agreement between the two approaches.
A MODEL
CALCULATION
OF FIELD
EMISSION FROM
3d
BANDS IN NICKEL
289
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
M. Hofmann, G. Regenfus, D. Scharpf and P. Kennedy, Phys. Letters 25A (1967) 270. W. T. Pimbley and E. W. Miiller, J. Appl. Phys. 33 (1962) 238. G. Oberrnair, Z. Physik 217 (1968) 91. R. H. Good and E. W. Miiller, in: Handbuch der Physik, Vol. 21. Eds. Fltigge (Springer, Berlin, 1956) p. 176. D. Nagy and P. H. Cutler, Phys. Rev. 186 (1969) 651. R. Stratton, Phys. Rev. 135 (1964) A794. F. I. Itskovich, Soviet Phys.-JETP 23 (1966) 945. J. W. Gadzuk, Phys. Rev. 182 (1969) 416. G. C. Fletcher, Proc. Phys. Sot. (London) 65 (1951) 192. L. Hodges, H. Ehrenreich and N. D. Lang, Phys. Rev. 152 (1966) 505. S. Wakou and J. Yamashita, J. Phys. Sot. Japan 19 (1969) 1342. J. Yamashita, M. Fukuchi and S. Wakou, Tech. Report ISSP, No. 75, May 1963. J. W. Conolly Phys. Rev. 159(1967) 415. J. M. Tyler, T. E. Norwood and J. L. Fry, Tech. Report, Louisiana State University. V. Heine, Proc. Phys. Sot. (London) 81(1963) 300. E. A. Stern, Phys. Rev. 162 (1967) 565. H. Jones, Theory of Brillouin Zones and Electronic States in Crystals (North-Holland, Amsterdam, 1962) p. 22. E. Jahnke and F. Emde, Tables of Functions (Dover, New York, 1945) p. 144. Ibid, section VIII. P. J. Price, Transmission of Bloch Waves through Crystal Interfaces, in: The Physics of Semiconductors (Bartholomew Press, Dorking, England, 1962) pp. 99-103.
SCIENCE 22 (1970) 277-289 o North-Holland
A MODEL
CALCULATION
Publishing Co.
OF FlELD
EMISSION
FROM 3d BANDS IN NICKEL* BEVERLY
A. POLITZER
and P. H. CUTLER
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802. U.S.A. Received 14 April 1970 Field emission from narrow, 3d bands of fe~oma~etic nickel is discussed. A one-dimensional model, with a triangular surface barrier, is used to calculate the ratio of the transmission coefficient of an electron in a tight-binding, 3d band to that of a free electron of the same energy. The results indicate that, depending on the interatomic crystal spacing chosen, the free electron transmission coefficient is roughly 1 or 2 orders of magnitude greater than the 3d transmission coefficient. In disagreement with previous work, this ratio is independent of the applied electric field at all electron energies.
1. Introduction
Recently, there has been a growing interest in the field emission of spinpolarized electrons from ferromagnetic material+3). In the transition metals, iron and nickel, it has been conjectured that different electron populations at the Fermi energy of the exchange-split, 3d spin bands** will lead to a net polarization of a field emitted beam. To see if this is indeed the case theoretically, it will probably first be necessary to abandon the conventional free-electron (Fowler-Nordheim) picture of field emission4) and, alternatively, to involve the energy band structure of these metals i.e., the narrow 3d bands and their complicated Fermi surface. Two questions naturally arise in the consideration of the 3d bands; namely: (1) How does the tunneling probability of a Bloch state corresponding to a 3d band in a transition metal differ from that of a free electron state?, and (2) What effects do the complicated, non-spherical constant-energy surfaces of the 3d bands have on the field emission current? Attempts to include such band structure effects in field emission theory * Supported in part by U.S. Air Force Office of Scientific Research and Grant Nos. AF-AFOSR 213-66 and 69-1704. ** For the sake of simplicity, and in the spirit of the model calculation to be described, we have identified bands as “pure d” and “pure s”. It is recognized, however, that in general hybridi~tion obscures such classifications. 277
278
B.A.
POLJTZER
AND
P. H. CUTLER
have for the most part been restricted to employing simple approximations to non-spherical constant energy surfaces 5-7). Quite inconsistently however, these same calculations have retained the WKB [or slightly-modeled WKBs)] treatment of transmission coefficient. A notable exception has been the recent work of Gadzuks) on field-induced tunneling from narrow d bands; he derives a d band transmission coefficient by calculating the intuitive, but rather ad hoc, “tunneling matrix elements’* using tight-binding wave functions of an ideatized noble metal *. The problem is that a rigorous treatment of transmission coefficient, appropriate to the real bands of a transition metal, requires a detailed knowledge of the Bloch functions** which are often difficult to obtain or to treat analytically, Fortunately in the case of fee nickel, the original tightbinding calculation of Fletchers) on the unhybridized 3d bands provides wave functions which turn out to be analytically tractable in the calculation of tunneling probabilities. En addition these wave functions yield 3d band shapes and widths which may be readily scaled to agree with results of more reliable calculations recently appearing in the literaturelo-14). In view of this, it was decided to inchtde the more realistic 3d Bloch functions in the investigation of the transmission coefficient problem in ferromagnetic nickel. (Fletcher’s secular determinant may be easily modified to include the exchange splitting of the 3d bands in ferromagnetic nickel.) A one-dimensional model calculation is outlined in section 2 and the results obtained are evaluated numerically in section 3. Though admittingly naive, the model indicates that the free electron transmission coefficient is roughly 1 or 2 orders of magnitude larger than the 3d transmission coefficient for an electron of the same energy. A more realistic estimate of this transmission coefficient ratio awaits the completion of a full three-dimensional treatment, which is now in progress. Finally, the implication of the one-
* Gadzuk’s works) may be compared to the calculation discussed here. First, we use the usual boundary matching method of solution which, within the context of the onedimensional model, is exact. Gadzuk employs a general tunneling Hamiltonian formalism assuming the same triangular surface barrier. He makes approximations in the evaluation of the “tunneling matrix elements”, however, whichessentially reduce his three-dimensional treatment to a one-dimensional one. The position of the surface is taken by Gadzuk to be at the last layer of metal atoms, whereas we assume the surface to be at one-half the interatomic spacing away from the last atom. In addition, Gadzuk uses tight-binding wave functions which are sums over Slater orbitals centered on surface atoms only; on the other hand, our nickel orbitals (radial parts only) are centered on all atoms, and the required sums over all atoms in the one-dimensional crystal are performed exactly. Moreover, there are very striking differences in the results obtained by each of these methods. These are elaborated in section 4 of the text. ** Most rigorously, one requires the Bioch functions of a semi-infinite solid with a boundary.
A MODEL
CALCULATION
OF FIELD
EMISSION
FROM
3d
BANDS
279
IN NICKEL
dimensional treatment and its relationship to field emission from anisotropic crystal of ferromagnetic nickel are discussed in section 4.
a real,
2. Model calculation We take as our model a perfectly periodic, semi-infinite, one-dimensional crystal of nickel. As depicted in the potential energy diagram of fig. 1, the interatomic spacing =a, and the surface defined by z=O is taken at a distance of $a away from the last atom layer present. The metal-vacuum potential barrier is assumed to be triangular, that is, only the applied electric
*--
Fig. 1.
Region I (metal)
V
I
1 Region II Y (vacuum)
l
Sketch of electron potential energy V versus distance z through one-dimensional crystal and vacuum.
field, F, is taken into account. The usual image potential is neglected since it complicates an analytic treatment of the problem and is not expected to significantly alter the comparison of the free electron and 3d transmission coefficient. (The same applies to less important effects such as field penetration and patch potentials.) We summarize the potential energy diagram of fig. 1 as follows: Region I, z < 0 : Region IT, z > 0:
V(z) = V(z + u), V(z) = - eFz.
We write the solution, 11/i,to the Schrijdinger equation in Region I as a linear superposition of two normalized Bloch states: an incident state, 4, and a reflected state, x. We will ultimately take C#Jand x to be Bloch states of an infinite, linear lattice with periodicity a; strictly speaking however, these
280
B. A.POLITZER
should be eigenstates
AND
of a semi-infinite
P. H, CUTLER
crystal with a boundaryr5*is).
Then,
In the above, U,,+(Z) is the periodic part of the Bloch function, with n denoting the band index in the reduced zone scheme and K denoting the reduced wave “vector”. A is a constant amplitude. The expectation values of the electron velocity in the states K and-~ are respectively, d&,,
=v,>o,
i.e., an “incident
=v,
i.e., a “reflected
state”,
dAk k=K and
y-!* dtik
k=-K
state”.
Also, we demand that the incident and reflected states be related by the condition of specular reflection, i.e., E,,, = E,,, -k s E. [E is a negative energy measured from the zero in the vacuum.] Note, however, that eq. (1) arbitrarily does not allow for other reflected states which, although energy-conserving, may belong to bands different from the “incident” band, n, i.e., a “direct interband transition”, possible in the 3d transition metals which have overlapping bands. Recalling that u,, _ t = UT,K, the expectation value of the current operator, J,, is J,(Bloch)
= !;
where the Wronskian
]u,,,]‘[l
- ]#-j
+ Gi
w(un*,,K, a,,,)
cl - ]A]‘] 9
(2)
is defined,
for all wave vectors k (L, is the For free electrons, u,,~ = u,, _k = l/JL, repeating length, containing N atoms, of the one-dimensional crystal lattice). Then for an incident state k = k,, where the subscript “f” refers to free, J,(free)
= 2
[l - IA]‘]. z
In eqs. (2) and (3) above we may define J, = J,’ + J;, where the superscript 7” indicates incident current and 9” indicates reflected current. In both cases the terms cc]A12 are to be associated with the reflected current. It can easily be shown from eq. (2) that J, (Bloch), like the free electron current, JI (free), satisfies the conservation condition, div J, (Bloch) =0, since
A MODEL CALCULATION OF FIELD EMISSION FROM 3d BANDS IN NICKEL
281
un,r
and u,, _x obey the usual Schriidinger equation for the periodic part of the Bloch function171. Region II may be for convenience further divided into two subregions. For z>zO where zO= --E/E’ is the classical turning point, the appropriate outgoing solution to the Schr~dinger equation may be shown ta be
where
Iiere cr~Q.2624 (eV A”)-’ for units in which E is in eV, F in V/L%,z in A, h/m in A2/sec, and k in (1$)-l. C is an arbitrary constant (with units of A*> and J,, are Bessel functions of the indicated order (real throughout their z domain). A solution continuous with the above at z=z, may be written $&J’) = C(~clF)*(~‘)*~f_*(i~‘)
eeinf6 - J*(i& eini6] = Cgfp’),
(5)
where
p’ is real throughout its domain of z, and J* + are complex. The expectation value of the outgoing current denoted by J,, is obtained by using eq. (4):
After some manipulation of the Bessel functionsl*), the Wronskian may be evaluated
and the outgoing current fgf becomes in the case of the triangular barrier
We wish ultimately to calculate a transmission coefficient ratio, i&//TBloch, for an electron having the same energy E in both cases. We account for the different dispersion relations for the free and Bloch electron by writing in accordance with previous notation ~{~~}= E(K) = E( - x). (Recall all energies measured from zero in vacuum.) Then using the definition of the
282
B. A.
POLITZER
AND F. H. CUTLER
transmission coefficient (7) From eqs. (2) and (3), we obtain J; (BIoch)
K
Jt’(free) -=-I_ kf
Lz I%,,(Z
= OV +
where the ratio of the currents has been evaluated at the z = 0 emission plane. Recall that the arbitrary constant, C, in the outgoing current &, eq. (6), is determined by matching eq. (5) and its derivative to either tc/t (free) and d~~(free)/dz or ~,(Bloch) and d~~(Bloch)/dz at z=O. We indicate this continuity requirement by subscripting the “C’s” and writing (9) Performing the matching process in both cases yields,
and
The foliowing notation has been used : g’(O)=-
g(O)=g(z=O),
dg (z)’ dZ, _ , z-o
U”,K(O) = kl,.(z
= 017
%,. W _ h,&)l - --. __ dz dz
. iz=O
Combining eqs. (8), (9>,(lo), and (11) in eq. (71, we have finally [kg
Tfree
kf
TBloch
Ic
(0) t_ g’ (O)]
2& (0) - g (0)
TY!5tFi0)‘2 dz
I
3
(12)
{ff~,~(O)l’
where * superscript indicates complex conjugate of the enc’tosed bracket. Now from the definition of g($) and p’(z) in eq. (S), it is easily verified that g’(0) = - (@F)+ r_l$[J+(ip.,) e-i+n - J_,(i~,)
ei3x],
A MODEL
CALCULATION
OF FIELD
EMISSION
FROM
3d BANDS
IN NICKEL
283
where
In the case of field emission the fields F of interest are 0.2 V/A cF~0.4 V/A. The electron energies are E= E, = - c$, where the work function 4 N 5 eV for nickel. An order of magnitude calculation then yields the result that ,u,a8. This value of p0 allows us to use the asymptotic expansions for large arguments19) for J* +(i& and J, .s(ipO)in evaluating g (0) and g’(0) respectively. One finds
(13)
We now proceed to specialize to the case of 3d wave functions for nickel. Assuming tight-binding wave functions for the 3d Bloch states, the incident state b, is written
c#= d+1 eiKRjt#,(z
- Rj) z eiKzu,,,(z) ,
(14)
Rj where Rj denotes location of the jth atom in the linear chain, j= 1,2, 3 etc., and 4, is an atomic orbital (designated by quantum number n) localized around the R,th atom. Then, one obtains by substituting in eq. (12) for u,,.(O) from eq. (14) and for 9 (0) and g’(0) from eq. (13) :
Tfree k k-i y= [
X 1
(15)
f’
with X=
C
~~(-Rj)#~(-R~)(-aE)
COSK(Rj-Ri)
RI> R’J
+ ,/( - gE)
(a, ( _ Rj)
d&i9
+
4,
( _ R;)
d4n ‘,; Rj) 1
+@n(-Rj)drbn(-RI) --___ dz
dz
and
Here
(p,( -Rj>
d#,fz-Rj)/dz~,=c*
and
d#,(-Rj)/dz
denote
respectively +,(z-Rj)l,=a
and
284
B.A.
POLITZER
AND
P. H. CUTLER
3. Numerical estimate T&T,, To make a numerical estimate of Tfree/T,, using eq. (15) above requires a knowledge of the dispersion relations E versus k for both the tight-binding and free-electron cases. The usual parabolic free-electron dispersion relation is written in the units employed here a(E+V)=k;,
where V (in eV) is the absolute value of the energy (measured from the zero in vacuum) to the bottom of the conduction band. For the tight binding case, E=E(k) must be obtained from a somewhat artificial, one-dimensional energy band calculation. This type of calculation would assume for &(z- Rj) a one-dimensional construct of one or all of the five 3d atomic orbitals for nickel, and would then proceed to calculate overlap and 2 and 3-center integrals for use in the secular determinant. Since such a calculation is quite complicated and is more of pedagogical interest, it will not be redone here. Instead, corresponding pairs of E and k will be picked off three-dimensional energy bands shown in fig. 2. These have been calculated from Fletcher’s secular determinant and have been scaled to the minority spin bands of Conollylo). This should provide a rough, numerical estimate of the desired ratio T,,,,/T,. For this one-dimensional calculation, let us assume that 4, is just the radial part of the 3d atomic wave function for nickel. (The corresponding angular parts, the Y,,‘s, have little meaning here.) The normalized, 3d radial wave function used in the Fletcher calculation is 4,,(r)
= r2 [56 e-5r + 1.3 em”].
(16)
The sketches of $,(r) and d& (r)/dr that are shown in fig. 3 reveal the extent of these localized functions; specifically, in a distance r=+ANi, where A,,=3.5 8, is the interatomic spacing for fee nickel, 4”(r) and d&(r)/dr fall to less than 0.01 of their maximum values. It is this localized property that allows us to perform quite accurately the doubly-infinite sum over permuting products of 4,(z-Rj)lZZo and d$,(zRj)/dzl,=, in eq. (15); convergence better than 1 in lo3 is obtained by including only 3 atoms adjacent to the z = 0 surface. Not unexpectedly one finds that the ratio Tfree/Tdis very sensitive numerically to the “overlap” at z=O of the atomic orbitals and their derivatives, each centered on different atoms. This “overlap”, in turn, depends quite strongly on the interatomic spacing, a, chosen for the model calculation and on the distance of the last atom from the surface at z=O. To illustrate this,
A MODEL
CALCULATION
OF FIELD
fl I I)
EMISSION
FROM
3d
BANDS
IN NTCKEL
c1totct
f 0)
285
(000)
(IO01
(000)
r
X
W
K
f
X
W
K
f
0.1
0.0
-0.1
- 0.2
F uI -o.3x
I
w
L
MAJORITY
SPINS
Fig. 2. Plot of unhybridized, minority 3d energy bands along special symmetry directions in ferromagnetic nickel. These have been calculated from Fletcher’s secular determinant 9) and scaled to minority spin bands of Conoily 10). Note here the zero of energyisat the Fermi levef, whereas in the calculations energies are measured from zero in vacuum.
286
B.A. POLlTZER
AND
P. H. CUTLER
two extreme values were chosen for a: (1) a=A,,=3.5 il and (2) u=*A,,= = 1.75 A. In both cases, the necessary “overlaps” were obtained from eq. (16), and the double sum in eq. (15) was performed using Rj= -+(IZj-- 1) a, j= I, 2 etc. The result, for a= ANi, is a simplified expression of the form 0.00369 + (0.580 x 10-4) cos K&
Fig. 3. Sketch of the normalized, 3d radial wave function #=(r) [eq. (16) text] and its radial derivative d#,Jdr. Note localized nature of & and d#,/dr compared to interatomic spacing in a one-dimensional nickel crystal.
A similar looking expression is obtained for U=~ANi- Using eq. (17), we estimate the numerical value of 7’r,,,/Td for the Fermi energy E, z - 5 eV. To do this, one must refer to a particular 3d tight binding band, E versus k, with the bottom of the band at the energy E= - 7? In the real ferromagnetic nickel crystal only two of the unhybridized, minority spin bands contribute to the Fermi surface in the 100 direction, specifically rlZXZ and G5X5. Folding back these bands to obtain the values of the reduced wave vectors, k=k Fermi,which lie in the first Brillouin zone of the one-dimensional crystal,
A MODEL
CALCULATION
OF FIELD
EMISSION FROM
3d
BANDS
IN NICKEL
287
one obtains : T&T,
N 130,
Tfree/Tde 150,
for ri2X2 band for &X5 band
forE=E,anda=ANi=3.5A,
and LIT, = 7 Tfree/Td = g
i) for F, 2X2 band 3 for &X5 band
for E = Ef and a = tANi = 1.751( e
4. Discussion and conclusions We have described above a one-dimensional model calculation evaluating the ratio of transmission coefficients involved in the field emission through a triangular barrier from a free-electron type band and a tight-binding, 36 band of ferromagnetic nickel. A rough numerical estimate shows the freeelectron transmission coefficient to be between one and two order of magnitude greater than the 3d transmission coefficient, depending on the interatomic spacing chosen. In the ~re~dimensio~al, anitropic metal, this difference between one and two orders of magnitude will probably manifest itself as a difference in the transmission coefficient from different crystallographic planes with varying atomic arrangements. In addition, it appears that the ratio Tfrec/Tdis more signi~~ntly affected by the interatomic spacing than by the particular 3d band chosen and its associated geometrical shape. This is probably, however, just an artifact of the one-dimensional calculation. In particular, it may stem from eliminating the angular dependent Yl,‘s from the orbitals 4,. In the real three-dimensional crystal, it is well-known that the secular determinant is diagonal in special symmetry directions; each of the computed 3d bands in these directions is then characterized by a different &_ basis. One may expect that when an actual field emission direction is related to the spatial symmetry associated with an individual 3d band, the tunneling probability will indeed be very band-dependent. Conversely, the tunneling probability from a particular 3d band should, in a real three-dimensional treatment, be anisotropic with direction, We may briefly contrast the results above with those obtained by Gadzuks). Our quantity Tfree/Tdcan be compared to the reciprocal of his F(E), the ratio of the d-band to s-band tunneling probabilities at a given energy E. For the fields commonly encountered in field emission, Gadzuk calculates that F(E)= I x lOa at E=E,. This number is at least an order of magnitude different from that derived here. More important, however, is a discrepancy regarding the electric field dependence of each of these quantities; whereas
288
B. A. POLITZER
AND
P. H. CUTLER
Gadzuk’s F(E) is a strong function of the electric field [i.e., proportional to the fourth power of the field as can be seen in eq. (23) or the curves in his fig. 41, in this work the analogous ratio T,,,,/T, is independent of field at all electron energies. We believe the origin of this discrepancy to lie in Gadzuk’s initial, erroneous
choice of wave function;
$-e
ik(s)z,
with
k(z) =
i.e., he uses
r$csT
-
z)]
in the classically forbidden region of the triangular surface barrier. For this barrier the well-known solutions to the Schriidinger equation (see section 2) are Bessel or Hankel functions. Furthermore close to the metal surface, which is the region of maximum overlap and therefore of greatest importance in Gadzuk’s calculation, the Hankel functions exhibit a field dependence of exp (l/F); Gadzuk’s wave function, on the other hand, is =exp(F+) in this region. It is well known that the density of states of a narrow, 3d band in a transition metal is up to ten times that of a wide, free-electron type conduction band. Perhaps, one may then expect to find in certain crystallographic directions a field emission current from the 3d bands of nickel which is comparable in size to that of a free electron. Polarization of field emitted electrons from the exchange-split, 3d bands of ferromagnetic nickel may then be hypothetically possible. The verification of this, however, awaits completion of the full three-dimensional treatment.
Acknowledgments The authors gratefully acknowledge the assistance of J. W. Conolly providing tables of his energy bands for ferromagnetic nickel.
in
Note added in proof After completion of the present work, our attention was directed to the paper of Prices’-‘), who considered a similar problem. He used Wannier functions to calculate the transmission probabilities for Bloch waves in a one-dimensional model and found the tunneling probability to be substantially smaller than for free electron wave functions. Since Price did not do any numerical calculations, we can only adduce qualitative agreement between the two approaches.
A MODEL
CALCULATION
OF FIELD
EMISSION FROM
3d
BANDS IN NICKEL
289
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
M. Hofmann, G. Regenfus, D. Scharpf and P. Kennedy, Phys. Letters 25A (1967) 270. W. T. Pimbley and E. W. Miiller, J. Appl. Phys. 33 (1962) 238. G. Oberrnair, Z. Physik 217 (1968) 91. R. H. Good and E. W. Miiller, in: Handbuch der Physik, Vol. 21. Eds. Fltigge (Springer, Berlin, 1956) p. 176. D. Nagy and P. H. Cutler, Phys. Rev. 186 (1969) 651. R. Stratton, Phys. Rev. 135 (1964) A794. F. I. Itskovich, Soviet Phys.-JETP 23 (1966) 945. J. W. Gadzuk, Phys. Rev. 182 (1969) 416. G. C. Fletcher, Proc. Phys. Sot. (London) 65 (1951) 192. L. Hodges, H. Ehrenreich and N. D. Lang, Phys. Rev. 152 (1966) 505. S. Wakou and J. Yamashita, J. Phys. Sot. Japan 19 (1969) 1342. J. Yamashita, M. Fukuchi and S. Wakou, Tech. Report ISSP, No. 75, May 1963. J. W. Conolly Phys. Rev. 159(1967) 415. J. M. Tyler, T. E. Norwood and J. L. Fry, Tech. Report, Louisiana State University. V. Heine, Proc. Phys. Sot. (London) 81(1963) 300. E. A. Stern, Phys. Rev. 162 (1967) 565. H. Jones, Theory of Brillouin Zones and Electronic States in Crystals (North-Holland, Amsterdam, 1962) p. 22. E. Jahnke and F. Emde, Tables of Functions (Dover, New York, 1945) p. 144. Ibid, section VIII. P. J. Price, Transmission of Bloch Waves through Crystal Interfaces, in: The Physics of Semiconductors (Bartholomew Press, Dorking, England, 1962) pp. 99-103.