Introduction
Recent development of such technologies as electron beam milling and
ion beam lithography enable the
fabrication of structures with dimension as small as a few nanometers. Those
low dimensions are comparable with the de Broglie wave length of a free
electron inside the metal. Because of this, it has become possible to fabricate
some microelectronic devices [1-4] working on the basis of the wave properties
of the electrons.
We will discuss what happens when regular indents, influencing free
electron de Broglie waves, are etched into the surface of a metal.
In the beginning we will review the well-known case of a fermion inside
the potential energy box, and we will investigate how the modification of the
geometry of the boundary of potential energy box will affect the boundary
conditions for the particle wave function, and the resulting solutions for
Schroedinger’s equation. Next, the behavior of fermions inside this potential
energy box will be studied. The results will be extrapolated to the particular
case in which the fermion is an electron and the potential energy box is a
metal. It will be shown that modification of the metal’s surface will lead to a
increase of Fermi energy level, with a resulting decrease of the work function
of the metal.
Finally, the experimental possibility of the fabrication of such
indents on the surface of the thin film of metal is studied. Practical recommendations regarding
dimensions and the shape of indents are given. In addition, the influence of
nonregularities in real metal, such as granolas inside the metal and the
roughness of the surface of the metal, are studied.
Fermions in a potential energy box with an indented wall
We begin with the general case of an elementary particle in the
potential energy box. Assume a potential energy box with one of the walls
modified as shown on Fig. 1. Let the
potential energy of the particle inside the box volume be equal to zero, and
outside the box volume equal to U.
There is a potential energy jump from zero to U at any point on the
walls of the box. Let us assume that five walls of the potential energy box are
plain and the sixth wall is modified as shown on fig.1. The indents on the
sixth wall have the shape of strips having depth of a and width of b. We name
the box shown on fig. 1 “modified
potential energy box” (MPEB) to distinguish it from the ordinary potential
energy box (OPEB) for which all walls are plain.
The behavior of a particle in the OPEB is well known. The Schroedinger
equation for particle wave function inside the OPEB has form [5]:
d2y/dx2+(8p2m/h2)Ey=0 (1)
Here y is the wave function of the particle, m is the mass of the particle, h
is Planck’s constant, and E is the energy of the particle. Equation (1) is
written for the one dimensional case. General solution of (1) is given in the
form of two plain waves moving in directions X and -X.
y(x)=Aexp(ikx)+Bexp(-ikx) (2)
Fig.1 Potential energy box with indented wall. a
is the depth of the indent and b is width of the indent.
here A and B are constants and k is the wave vector
k=[(2mE)1/2]/(h/2p) (3).
It is well
known that in the case of U=¥, the solution for equation (1) is defined by the boundary condition y=0 outside the OPEB as follows:
y=Csin(kx) (4).
Here C is constant. Let the width of the OPEB be L.
Then the boundary conditions y(0)=0 and y(L)=0 will give the solution to Schroedinger’s
equation in the form of sin(kL)=0 from where we get kL=np (n=1, 2, 3,…). We have a well-known discrete series
of possible wave vectors corresponding to possible quantum states
kn=np/L (5)
and according to (3) discrete series of possible
energies En=n2(h2/8mL2).
But
if the potential energy box has a modified wall, what happens to the boundary
conditions for the elementary particle wave function? One more boundary condition should be added. The wave function
should be equal to zero not only at x=0 and x=Lx+a, but also at the
point x=Lx ( Fig.2). Yet
another additional boundary condition is added because the modified wall could
be divided into two parts with equal area. The first part is situated at
distance Lx from the opposite wall, while the second part is
situated at distance Lx+a from the opposite wall. Once U=¥ is true for every point of both parts of the modified
wall, y=0 is true also for every
point of both parts of the modified wall. Because of that we have y(Lx)=0 and y(Lx+a)=0. And, as shown above, y=0 for the unmodified wall, which results in a final
added boundary condition y(0)=0. In total, we have the following three boundary conditions in the
X direction:
y(0)=0, y(Lx)=0 and y(Lx+a)=0 (6).
There is no general solution of (1) which will be true
for any pair of Lx and a, unlike the case of OPEB in which we will
have solutions for any L . To find possible solutions we begin from the last
two boundary conditions in (6). It is obvious that the last two boundary
conditions define possible solutions, just as they do for the OPEB of width of
L=a. The wave function should be zero at points Lx and Lx+a.
Possible solutions are sinusoids having a
discrete number of half periods equal to a (Fig.2). The first boundary
condition from (6) will be automatically satisfied together with the last two boundary conditions only in the case
that
Lx=pa (7)
where p=1, 2, 3, … There will be some solutions also
for the case Lx¹pa. For example in the case Lx=pa/2 we will have solutions
satisfying all three boundary conditions for n=2, 4, 6,… It is obvious that number of solutions
satisfying all boundary conditions (6) will be maximum in the case Lx=pa. Because of that let’s assume Lx=pa, keeping in mind that we have already
maximized possible solutions.
Fig.2 Solutions of Schroedinger equation. Boundary
conditions y(Lx)=0 and y(Lx+a)=0
define solutions as shown on the right side of the figure.
Let’s assume we have MPEB which has
dimensions satisfying condition (7). Then solutions
will be
kn=np/a (8)
just as we had for the OPEB (5) of width L=a. However,
the whole width of the box is replaced by a part of it (Lx+a is
replaced by a). It is interesting to
compare solutions for MPEB (8) and OPEB of width of L=Lx+a (in this
one dimensional case). Solutions for OPEB having width of L=Lx+a
will be
kn=np/(Lx+a)
(9)
and the solution for the MPEB will be (8). There are
fewer possible k-s in the case of the MPEB
compared to an OPEB of the same width. Here we come to a very important
conclusion: Modifying the wall of the potential energy box as shown in Fig.1
leads to a decrease in the number of possible quantum states. More precisely,
altering the potential energy box leads to decrease in the number of possible
wave vectors per unit length on k line (Lx+a)/a times. This last
equation is easily obtained from (8) and (9).
Until
now we were working on the one dimensional case. Let’s return to Fig. 1 and
study the Y and Z dimensions as well. In the case of a<<Lx, Ly,
Lz we will have an
OPEB solutions of equation (1), for Y and Z directions. Both walls for Y and Z
dimension are plain and once a is small enough it will not influence the
solutions of Schroedinger equation for both Y and Z dimensions considerably.
Because of that we will have: kx=n(p/a), ky=n(p/Ly), kz=n(p/Lz) and the volume
of elementary cell in k space will be
Vm=p3/(a Ly
Lz) (10)
Which is again (Lx+a)/a times more than the
volume of the elementary cell in k space for the OPEB Vm=p3/[(Lx+
a) Ly Lz)]. Volume in k space for three dimensional case
changes like linear dimension on k line in the one dimensional case. Because of
that we can easily extrapolate results we have in one dimensional case to the
3D case.
Returning
back to the one dimensional case, let’s conduct the following imaginary
experiment: we have two potential
energy boxes of the same dimensions,
one an OPEB with all walls plain, and another a MPEB with one wall modified.
Posit a large number of fermions. Let’s put an equal number of fermions, one at
a time, in both of the potential energy boxes and observe the wave vector and
energy of the most recently added fermion in both boxes. The first fermion in
both boxes will occupy quantum state k0=0 in ordinary box and km0=0
in MPEB. The second fermion in OPEB will occupy k1=p/(Lx+a) and in MPEB km1=p/a. If we continue adding equal number of fermions to both boxes we
will get kn=np/(Lx+a) for the OPEB and kmn=np/a for the MPEB. It is obvious that the nth fermion will
have (Lx+a)/a times more wave vector in the MPEB than in the OPEB.
Correspondingly, the energy of the nth fermion in the MPEB will be
[(Lx+a)/a]2 times
higher than in the OPEB. This is only
true for the one dimensional case. It is not difficult to prove (it will be
done in next division) that for three dimensional case, the ratio of energies
of the nth pair of fermions will be
(Em/E)=[(Lx+a)/a]2/3 (11).
here Em is the energy of nth
fermion in the MPEB and E is the energy of the nth fermion in the
OPEB. Index n is skipped in formula (11) because the ratio of energies does not
depend on it.
Free electrons in the metal with modified wall
Free
electrons inside the solid state is one of the examples of fermions inside the
potential energy box. The theory of electron gas inside the lattice is well
developed and is based on different
models, most simple of which is the quantum model of free electrons, which
gives excellent results when applied to most metals. It is well known that free
electrons in metal form a Fermi gas. Boundary condition y=0 outside the metal is used in all theories because
in metals the potential energy barrier is high enough to allow that simple
approximation. In the quantum theory of free electrons, cyclic boundary
conditions of Born-Carman
kx=2pn/L (12)
are used instead of (5). Here n=0, ±1, ±2, ±3,… Cyclic boundary
conditions leave the density of quantum states unchanged, and at the same time
they allow us to study running waves instead of standing waves, which is useful
for Physical interpretation. The result of the theory is Fermi sphere in k
space. All quantum states are occupied until kF at T=0. kF
is maximum wave vector inside the metal at T=0 because states with k>kF
are empty.
Now
let’s see what happens when we modify one of the walls of the metal (fig.1). As
shown above, the distance between quantum states in k space in kx
direction will become 2p/a instead of 2p/(Lx+a). The number of quantum states per unit volume
in k space will decrease (Lx+a)/a times. Metal retains its
electrical neutrality, which means that the same number of free electrons, have to occupy separate quantum states
inside the metal. Because the number of quantum states per unit volume in k
space is less than in the case of ordinary metal, some electrons will have to
occupy quantum states with k>kF.
This shows that the Fermi wave vector and the corresponding Fermi energy
level will increase.
We
then calculate the maximum wave vector km at T=0 for metal with a
modified wall (fig.1). Posit that the lattice is cubic, the metal is single
valence, and the distance between atoms is d. The volume of metal box shown on
fig.1 is
V=LyLz(Lx+a/2)
(13).
Number of atoms inside the metal is q=V/d3.
The number of free electrons is equal to q and we have
q=LyLz(Lx+a/2)/d3 (14)
for the number of free electrons. The volume of
elementary cell in k space is
Ve=(2p/a)(2p/Ly)(2p/Lz) (15)
And the volume of the sphere of the radius of km
in k space is
Vm=(4/3)pkm3
(16),
here km is maximum possible k in the case
of modified wall and Vm is the volume of modified Fermi sphere in k
space. Number of possible k=kx+ky+kz
in k space is Vm/Ve. Each k contains two quantum states occupied by two electrons with spins
1/2 and –1/2. Using (14), (15), (16) we will have
(q/2)=(km3aLxLz/6p2) (17),
and for the radius of modified Fermi sphere
km=(1/d)[3p2(Lx/a +1/2)]1/3 (18).
It is well known that the radius of a Fermi sphere kF
for an ordinary metal does not depends on its dimensions and is kF=(1/d)(3p2)1/3. Comparing the last with (18) we get
km=
kF (Lx/a+1/2)1/3 (19).
Formula (19) shows the increase of the radius of the Fermi sphere in the case
of metal with modified wall in comparison with the same metal with plain wall.
If we assume a<<Lx, Ly, Lz formula (19)
could be rewritten in the following simple form:
km=kF(Lx/a)1/3 (20).
According to (3) the Fermi energy in the metal with
the modified wall will relate to the Fermi energy in the same metal with the
plain wall as follows:
Em=EF(Lx/a)2/3
(21).
Formulas (20) and (21) are grounded in pure
Mathematics. Well-known boundary conditions and well-known solutions of
Schroedinger equation, combined with a very unusual geometry for the metal wall
results in an increase of the Fermi level in the metal.
An obvious question emerges: let’s assume that we made ratio Lx/a
high enough for Em to exceed vacuum level. What will happen? If we assume that some electrons have
energies greater than the vacuum level, they will leave metal. The metal, as a
result, will charge positively, and the bottom of the potential energy box will
go down on the energy scale, because metal is charged now and it attracts
electrons. Once the bottom of the potential energy box decreases, vacant places
for electrons will appear at the top region of potential energy box. Electrons
left the metal will return back because of electrostatic force and occupy the
free energy states. Accordingly, Em will not exceed the vacuum
level. Instead, the bottom of the potential energy box will go down exactly at
such distance to allow the potential energy box to carry all electrons needed
for electrical neutrality of the metal. Regarding the work function it is clear
that increasing the ratio of Lx/a will decrease first until it gets
equal to zero. Even with a further increase in Lx/a, the work
function will remain zero. In real metals surfaces are newer ideally plain.
Roughness of the surface limits the increase of fermi level. Limits of
increasing of the fermi level will be
discussed in more details in next division of current article.
It is useful to recall here that analysis was
made within the limits of quantum theory of free electrons. Model of free
electrons give excellent results for single valence metals. As it will be shown
later Gold (which is single valence) is only metal in which described structure
could be practically realized. More developed theories, which take into account
electron-lattice and electron-electron interaction could be used to obtain more
precise results. However results obtained in this work will remain valid within
all theories at least for the region (-p/d )<k<(p/d), where d is lattice constant. Dimensional effects
in semiconductor and semimetals were studied theoretically [6, 7]. Particularly influence of thin film
dimensions on its Fermi level, is
studied in [7].
Problems of practical realization and possible
solutions
An achievable structure, which satisfies the requirements given in this
work is shown by (fig.3). This type of structure could be obtained in the way of depositing of a thin metal
film on the insulator substrate, and then etching the indents inside the metal
film. What are the limitations? If we
return to formulas (2) and (4) we see that plain waves are solutions of the
Schroedinger equation. A standing wave comprises two plain waves moving in the
direction of X and –X. Wave diffraction will take place on the indent.
Diffraction on the indents will lead to the wave “ignoring” the indent, which
changes all calculations above. Consequently,
results obtained are valid only when the diffraction of the wave on the
indent is negligible, or
b>>l1=2a (22)
Here l1=2p/k1 is de Broglie wavelength of
electron with wave vector k1 (n=1 on fig. 2). It is obvious that
(22) will be automatically valid for n= 2, 3, 4…
Another theoretical requirement is that Lx should be
multiple of a (7). In the case (7) is not valid, the number of quantum states
will be less than the number given by formula (8). Decreasing the number of
quantum states will magnify the effect of increasing of EF, but it
will be problematic to control work function decrease without keeping (7) valid
during the metal film
Fig.3 Possible realization of metal with indented
wall. Indents are etched on the surface of thin metal film deposited on
insulating substrate.
deposition stage, as well as during indent etching. On
the other hand if (7) is deliberately kept not valid it will lead to
eliminating of possible quantum states from E=0 to energy level defined by
roughness of the surface (discussed later in this section).
There are some requirements to the homogeneity of the metal film. The
film should be as close to mono crystal as possible. This requirement is
because the wave function should be continuous on the whole length of Lx+a,
which means that the metallic film could not be granular. If the metallic film
is granular, the wave function will have an interruption on the border of two
grains, and the indented wall’s influence on the boundary conditions will be
compromised. It is necessary to note here that lattice impurities do not
influence free electrons with energies E<EF. In order to interact
with an impurity inside the lattice, the electron should exchange the energy
with the impurity in the lattice. That type of energy exchange is forbidden
because all quantum states nearby are already occupied. The mean free path of
an electron, sitting deep in Fermi sea is formally infinite. So the material of
the film can have impurities, but it should not be granular. That type of
requirement is quite easy to satisfy for thin metal films.
The surface of the film should be as plain as possible, as surface
roughness leads to the scattering of de Broglie waves. Scattering is
considerable for the wavelengths of the order or less than the roughness of the
surface. Substrates with a roughness of 5 A are commercially available. Metal
film deposited on such substrate can also have a surface with the same
roughness. The de Broglie wavelength of a free electron in metal sitting on the
Fermi level is approximately 10 A0.
Scattering of the de Broglie wave of electrons having energies E>EF
will be considerable. Consequently, energy states with energies E>EF
will be smoothed. Smoothing of energy levels decrease the lifetime of the
energy state and lead to continuous energy spectrum instead of discrete one.
Fig. 4 shows comparison
Fig.
4 Energy diagrams of some single valence metals on the scale of de Broglie
wavelength calculated as l=2p/k
from (3).
of fermi and vacuum levels of some single valence
metals on the energy scale and simultaneously on the scale of de Broglie
wavelength of the electron calculated from formula (3). It is evident that 5 A
roughness of the surface is enough to eliminate energy barrier (in the case Lx¹pa) for such metals as Cs and Na. The same roughness creates
gap from zero to approximately fermi level in energy spectrum of such metals as
Au and Ag.
It is evident that the depth of
the indent should be much more than the surface roughness. Consequently, the
minimum possible a is 30-50 A. According to (22) the minimum possible b will
be 300-500 A. These dimensions are well within the capabilities of e-beam
lithography and ion beam milling. The primary experimental limitation in the
case of the structure shown on
fig.4 is that the ratio (Lx/a)³5, in order to achieve a work function which is close
to zero. Consequently, the thickness of the metal film should be at least
180-300 A0. Usually films of such thickness still repeat the
substrate surface shape, and the film surface roughness does not exceed the
roughness of the base substrate. However, the same is not true for metal films
with a thickness of 1000 A0 and more, because a thick film surface
does not follow the surface of the substrate. That puts another limit 15³(Lx/a)³5 on the dimensions of the structure, when metal films
are deposited on the substrates. Other possible solutions, such as metal
crystals of macroscopic dimensions like those frequently used for electron beam
microscope cathodes, will not be limited by the same requirements.
And finally there are some limits on materials which could be used for
thin film. Most metals oxidize under influence of atmosphere. Even when placed
in vacuum metals oxidize with time because of influence of residual gases.
Typical oxides have depth of 50-100 A which is considerable on the scale we
discussed. Because of that Gold is only material which could be used in
practice.
Conclusions
It has been shown that modifying the wall of a potential energy box
changes the boundary conditions for the wave function of an elementary particle
inside the potential energy box. New boundary conditions decrease the number of
solutions to Schroedinger’s equation for a particle inside the MPEB. If the
particles are fermions, the decrease in the number of quantum states results in
an increase in the energy of the nth particle situated in the
potential energy box. General results obtained for fermions in the potential
energy box were extrapolated to the particular case of free electrons inside
the metal. Calculations were made within the limit of quantum theory of free
electrons. It was shown that in the
case of a certain geometry of the metal wall,
the Fermi level inside the metal will increase. A controllable increase
in the Fermi level, and the
corresponding decrease of the work function of the metal will have practical
use for devices working on the basis of electron motion, electron emission,
electron tunneling etc. A discussion of the practical possibilities to realize
the modified wall in a metal shows that electron beam lithography and ion beam
milling are capable of practical realization.
Author thanks Isaiah Cox, Nicholas Antoniou, Artemy Martinovski, Boris
Tsekvava and Richard Forbes for useful discussions.
This work has been done for Borealis Technical, assignee of related
patents (US 6,281,514 B1 and US 6,117,344).
References
1.
Jing Kong, Erhan
Yenilmez, Thomas W. Tombler, Woong Kim, Robert B. Laughlin, Lei Liu, C. S.
Jayanthy and S.Y. Wu “Quantum Interference and Balistic Transmission in
Nanotube Electron Waveguides” Phys. Rev. Lett. v. 87, N 10, 106801 (2001)
2.
P. Debray, O. E.
Raicher, P. Vasilopoulos, M Rahmman, R. Perrin and W. C. Mitchell “Balistic
electron transport in stubbed quantum waveguides: Experiment and Theory” Phys.
Rew. B, v. 61, p.10 950, (2000).
3.
L. Rochester, R.
Tarkiainen, M. Ahlskog, M. Paalanen and P. Hakonen “Multiwalled carbon
nanotubes as ultrasensitive electrometers” Appl. Phys. Lett. v. 78, N 21, p.
3295, (2001)
4.
N. Tsukada, A.D.Wieck, and K.Ploog “Proposal of Novel Electron Wave Coupled
Devices” Appl. Phys. Lett. 56 (25),
p.2527, (1990);
5.
L.D. Landau and
E.M.Lifshits “Quantum Mechanics” , Moscow 1963.
6.
V.A.Volkov and T.N.
Pisker “Quantum effect of dimensions in the films of decreasing thickness”
Solid State Physics, 13, p.1360
(1971).
7.
V.N. Lutskii. JETP
Letters 2, p. 245 (1965).