In real space th e.electro n s are confined within the solid. Low -energy b o u n d electro n s c a n n o t participate in conduction unless they are therm ally excited a n d escape from the atom ic core. High-energy ‘quasi-free electrons m ig rate th ro u g h o u t the solid, being constrained by the physical b o u n d ary of the solid only (Fig. 5.8). T he m ovem ent of these higher energy electrons is alm o st unaffected by the periodic po tential of the atom ic cores.
So far we have looked only at the distribution of electrons in real space. We will in tro d u c e later the concept of reciprocal space which can be used to describe the electronic states of a solid in a particularly econom ical and elegant m anner.
E n erg y ba n ds in a s o lid 9 5
P o te n tia l a ro u n d
io n ic c o re P osition x
Fig. 5.8 Schematic diagram of energy levels in a one-dimensional lattice, shown in real space.
96 Bound electrons and the periodic potential 5.4.3 The Fermi energy
What is the highest energy level occupied by an electron when all electron, in their lowest available state1
We have stated that at absolute zero, when the electrons all occupy the lo available energy state, the energy of the highest occupied state is the F level. This energy level separates the occupied from the unoccupied elec
levels only when the electron configuration is in its ground state, th a t is, <
at OK.
The location of the Fermi level in relation to the allowed energy staô
crucial in determining the electrical properties of a solid. M etals always V , part,aly fined .ee-etaro,, band, so tha, ,he Fermi ,eve, corresponds 1 level m the mtddle ol the band and this makes the metals electrical conduct
S m môn?C;SeT ^ e'y ằ ằ ằ
consequently they ate poor e l e c t S c Z ^ ^ 5.4.4 Nomenclature of electron bands
How can the most important enerav hô a L
self-consistent manner? 9> ands be described in a distinct
Energy bands in a solid 97
1 ■&* --- ---
. this band satisfies both In a metal which contains a partially fille_ ,uctjon band (Fig. 5.10). This criteria and so is both a valence band an , lhe 5ands in metals which explains some of the confusing nomenc are sometimes describe as occurs in the literature, in which the free; electrons. In a metal they are
‘conduction’ electrons and sometimes as both.
5-4.5 Effective mass of electrons in band
i ui ue described in ci simple How ca„ the motions o f electron ằ■ôôằ <■ằ ene' s>’
Way? ns in the conduction band
is found experimentally that the ^ ffree electrons we have shown that, is affected by how full the band is. In t e c
h2k 2
stant in this case.
Where m is ôhe mass of ôhe Ê ằ"ô k tsT n g'V e As we shall see shortly, this relationship ^ ^ reIalionship by using t e solid. However, we can maintain
relation,
h2 .2
zm ■
call the effective mass. is where now m* is an adjustable Para^ 1^ ^ relationship between £ ‘*ndjc can
* ean s that any deviations from a p a ^ of the electrons at that point n be expressed as a change in the efc a convenient artifice wmch Espace. Remember that of course eieclrons in bands. The
allows us to describe the behaviour o
;s, it is simply an expression of the c h a n £ e smaller or larger than the free-electron m a s s inge in the mass of the electrons is the in te r a c ti lattice in the material. Collisions betw een d rift!
slow down the acceleration of an electro n w h i effective, or apparent, mass,
tuation is in terms of the curvature o f th e e n e r be interpreted in the following way: u s in g t and wave vector k given in section 4.2 a n d ta k i
\e periodic potential
Wave vector k
ectron energy E with wave vector k. Diagram (c) sho\
ergy band giving large effective mass at interm edia w k as shown in (d). At k values close to ± n/a tl negative.
the second derivative of the energy with respect to the wave vector gives h2/m.
This relationship only holds exactly for a free-electron parabola. When the relationship between E and k is no longer parabolic the deviation can be expressed in term s of a change in the effective mass so that,
d 2E _ h2 d k 2 m*
and consequently the effective mass can be defined by,
Reciprocal space 99
F o r electrnn h ^ curva*u*’e electron levels or electron band in fc-space.
small curvature th* f S 'a'* curvature m* is small, while for bands with that since d2F /d k 2 & 3* e ecfron bands, nt* is large. It is also worth noting when an elertrn 030 f C negat' ve m* can be negative. This simply means that the lattice is r° m St3te ^ l° State ^ + ^ ^ momentum transfer to therefore arm 3 ^ u 3*1 rnomentum transfer to the electron. The electron tnerefore appears to have a negative mass.
effective m f r exam Pie' consider the energy states, as shown in Fig. 5.11. Small mediate k Sin^ * 'n thlS Case’ ,arge R e l iv e masses occur at inter- band in k c C u-6 ect ve mass is determined by the curvature of the energy with hioh 1 *S means that narrow bands necessarily contain electrons effective8 maff n muaSS‘ Conversely wide bands can contain electrons of low ettective mass or high effective mass.
5.5 R EC IPR O C A L OR WAVE VECTOR /c-SPACE
Is there an economical way o f describing all of the allowed energy states in a solid?
Earlier in section 4.2 we introduced the idea of the wave vector k. This arose when we m ade a simple calculation for the solution of the wave equation for free electrons and then for electrons in a square-well potential. A plot of energy
£ against wave vector k was given first in Fig. 4.1. The dimensions of k are reciprocal length. It tells us the spatial periodicity of the wave function, or if you prefer, the num ber of cycles of the wave which occur in a given distance of 2n metres.
We found that for free electrons the energy depended on /c2, and all values of k were allowed. When the electrons are trapped, as in the square-well potential, only certain values of k are allowed in order that the wave functions can meet their boundary conditions.
In the last section we began to plot energy against k because this was useful in determining the effective mass of the electrons. We shall find that when it comes to describing electrons in solids plotting £ against k is a very useful way
of representing the electronic properties of the material. The plot of E ag3>nSt k is known as a reciprocal space plot because the dimensions of k are metre ” *•
When an electron is confined within a solid and experiences the periodicity of the lattice this periodicity affects the relationship between E and k. Another
way of looking at this is that a wave function described by the wave v ecto r k will have different energies depending on the presence and type of the c ry st3 ^ lattice it encounters. We have already noticed, for example, that the interactions between an electron and the lattice alter its effective mass and so d isto rt the
relationship between E and k.
5.5.1 Brillouin zones
How can periodicity o f the lattice be introduced into reciprocal space?
We may think of the Brillouin zones [9] as a method for introducing th e periodicity of the crystal lattice into our model of the electronic stru ctu re o f materials. The Brillouin zone is a region in reciprocal space. Before saying m o re about exactly what this Brillouin zone is, let us consider the effects of a p erio d ic potential on the energy levels of ‘free electrons'.
We know £ = h2k2/2m for free electrons, but when we add the presence o f 3 periodic potential all of this changes. We know from the previous chapter th a t if the energies of the electrons are below the level of the periodic p o te n tia l barriers the electrons will penetrate spatially into these potential barriers, b u t their wave functions will be attenuated the further they penetrate. If the p o te n tia l barriers are much higher than the electron energy, then the wave function will be reflected at these barriers and the electron will be contained entirely w ith in the potential box formed by the barriers.
Now that we have introduced the idea of the electron wave function b ein g reflected by the energy barriers of the periodic potential, let us find the co nditio n y'for this. For Bragg reflection we need
2a sin 0 = ml
where a is the lattice parameter, 0 is the angle of reflection, X is the w avelength of the electrons and n is an integer. For simplicity let us again look at th e one-dimensional lattice. In this case sin 0 = 1 and the Bragg reflection co n d itio n is
2a = nX and X = 2n/k for an electron. So that
k = nn.
a
This then is the condition for reflection of the electron wave function in a onc-dimensional lattice of parameter a. This can easily be generalized to th ree
100 Bound electrons and the periodic potential
dimensions and the concept remains the same. We now have a division of reciprocal or k-space into a number of zones, known as Brillouin zones, at the boundaries of which reflection of the electron wave functions takes place.
The lattice can be divided into a number of Brillouin zones in reciprocal space.
Reciprocal space 101
First Brillouin zone Second Brillouin zone
nth Brillouin zone
— 71 + 71
— to —
a a
- I n - n n 2n
to ---- and - to —
a a a a
r J H ,o and 10
a <* a
Ml
a Since free-electron wave functions arc not reflected, and yet m a solid reflection occurs at the Brillouin zone boundaries, we may reasonably expect that the most severe deviation of the electron energies from free-electron-like behaviour will occur at the Brillouin zone boundaries.
In the plot of E versus k for a one-dimensional weak periodic potential shown in Fig. 5.12, the free-electron parabola is clearly apparent but with some distortion at the zone boundaries. At the k values corresponding to the zone
Energy E
Fig. 5.12 Deformation of a free-electron parabola due to a weak periodic potential.
boundaries transmission of an electron through the solid is prevented. "The incident and reflected electron wave functions form a standing wave. C e rta in energies are therefore forbidden since there are values of E for which th e re is no corresponding value of the wave vector k
5.5.2 1 he reduced-zone scheme
Con the representation o f the electrons in reciprocal space be made more comPa ct by making use o f the periodicity condition?
We can take our one-dimensional plot of energy versus wave vector a n d m a p all sections into the range ~n/a ^ k ^ n/a. This is done by using the periodicity constraint so that,
*1 = *n + G
where kn is the wave vector in the /ith Brillouin zone, k x is the wave vector in the first Brillouin zone and G is a suitable translation vector. Now any P ° j n t in /c-space can be mapped by symmetry considerations to an equivalent p o in t in the first Brillouin zone, but notice that many points from the extended-zone representation can be mapped to the same point in the reduced-zone representation.
This reduced-zone scheme representation allows the entire electron b a n d structure to be displayed within the first Brillouin zone (Fig. 5.13). This i* a very compact representation and has distinct advantages because the electronic states can be displayed in the most economical way in a single diagram o f th e first zone.