H o \ v d o e s th e F e r m i su r fa ce a p p e a r in t h r e e d i m e n s i o n s ?
Having looked b r i e f l y at the problem of mapping the Fermi surface in two-dimensional /(-space we now have a much better idea of what to e x p e c t in three dimension', when we look at the Fermi surfaces of real metals.
The Fermi surface 121
Fig. 6.11 Reduced-zone representation of a two-dimensional distorted Fermi ‘sphere’.
Fig. 6.12 Three-dimensional Fermi surface of copper.
The first metal for which the Fermi surface was completely mapped in /c-space was copper. This work was performed by Pippard [7, 8], The Fermi surface of copper is particularly easy to visualize because it all lies in the first Brillouin zone. From our discussion of the two-dimensional examples above, in which the Fermi surface that was contained entirely within the first zone was circular, we may expect the Fermi surface of copper to be approximately spherical. This is indeed the case, the surface does however have necks extending tow ards the Brillouin zone boundaries. This is caused by Bragg reflection where the Fermi surface comes close to the zone boundary. The shape of the Fermi surface resembles a ‘diving-bell’. This is shown in Fig. 6.12. Both silver and gold have Fermi surfaces with similar shapes to that of copper.
122 Electronic properties of metals
Fig. 6.*. 3 Three-dimensional Fermi surface of aluminium
Fig. 6.14 Three-dimensional Fermi surface of lead.
In divalent calcium, which has of course two outer elections per atom , the Fermi surface extends into the second Brillouin zone. Again, in the extended- zone scheme, the Ferm i surface is approxim ately spherical. However, w hen folded back into the first zone in the reduced-zone scheme it looks quite different, resembling a ‘coronet’. D ivalent beryllium also has a coronet-shaped Ferm i surface, however, since it solidifies with hexazonal symmetry the surface in reciprocal space has six-fold symmetry.
Aluminium, being trivalent, has three outer electrons. Its electron bands are very close to free-electron parabolae, and its Ferm i surface if plotted in the extended-zone scheme is fairly simple in shape, being alm ost spherical and extending into the third Brillouin zone. Again once it is folded back o n to the first zone it takes a very different form which has come to be know n as the
’m onster’. This is shown in Fig. 6.13.
Lead has four valence electrons and its Fermi surface extends into the fourth - Brillouin zone. In this case the Fermi surface in the first zone is extremely complicated forming a ‘pipeline maze’, as shown in Fig. 6.14.
6.3.6 M ethods of determining the Ferm i surface
How can we examine the shape o f the Fermi surface o f a metal?
First we should say why measurements of the Fermi surface are im p o rtant at all. Recall that the classical D rude metal of electrons in solids failed over the prediction of the specific heat capacity of metals. This was because even the free electrons in the conduction band of a metal are not able to absorb therm al energy unless they are within an energy kBT of the Ferm i surface. This m eans that most of the electronic properties of a metal are determ ined by electrons lying at, or just below, the Fermi surface. Clearly the electrons close to the Fermi surface are most im portant in determ ining those properties of a metal which depend on the electrons. We conclude therefore, th at by know ing the details of the Fermi surface we can make predictions about m any of the
properties of a metal.
There are a num ber of different m easurem ents that can be made which give information about the Fermi surface. For a detailed description of these consult Ashcroft and M ermin [9] who have devoted an entire chapter to m ethods of measuring the Fermi surface. We list here only the most im portant techniques used:
• de H aas-v an Alphen effect
• magnetoacoustic effect
• ultrasonic attenuation
• anom alous skin effect
• magnetoresistance
• cyclotron resonance
• positron annihilation.
The Ferm i surface 123
124
Electron velocity (Cyclotron resonance)
Electronic properties of metals
Linear dimension (Magnetoacoustic effect)
Curvature (Anomalous skin effect)
X \ / /^ E x tre m a l area
Region of contact / V \ / / / (de Haas-van A lp
(Magnetoresistance) \ \ ____ / / / effect)
/ f^uniber of electrons
,n a "slice“ (Positron annihilation) nh vsic' cnrface from various p y Fig. 6.15 Information that can be obtained about the Fermi su
measurements as described by Mackintosh [4].
surface These ^ ves different information abo u t t h e F e r m i
(41 p,Cted ln Fl'g- 6.15, based on a figure given by M ack in to sh S y| ^ ^
is used to probe theVerm^" is the most imPortant tech n iq u e which (i.e. largest cross-s^r \ suriaceFrom these measurements the e x tr e m a l area magnetoacoustic effect'enL^6^ °f the Fermi ‘sPhere’ can be found The Fermi ‘sphere' to be calc I ° ^ ,mear dimension (ie - ,arSest diameter) of the ultrasonic attcmiat ^ ated' *n tbeabsenceofa magnetic field the conventional in this case the 3 S°®'ves information about the Fermi surface, however complicated ° mlerPre,ation in terms of Fermi surface geometry is more
The
the Fermi sua f°US C^ect’ wb'cb can be used to measure the c u rv a tu re of surface meô f 3Ce 0 "sPace *s one of the oldest techniques used fo r Ferm i tion of the arcments_,j da,es hack to the work of Pippard [7, 8j. T h e penetra- classical s k i n e d .'nt0 a solid a( hi8her frequencies deviates fro m the becomes depended eq,Ua,'0n and 11 can be sh°wn that the field p en etratio n fc.g. the c u r v a t u r e . T m V ° n1CCr,a"1 fea,ures of the Fermi surface geom etry
urvature, at suffiaemly high frequencies.
The Fermi surface 125
Magnetoresistance measurements, that is the dependence of electrical resis
tance on magnetic field, can be used to find the region of contact of the Fermi surface with the Brillouin zone boundary since the magnitude of this contact affects the conductivity. Cyclotron resonance, the circular motion of a charged particle moving in a plane normal to a magnetic field, can also be used to investigate the Fermi surface. Specifically, it can be used to find the electron velocity on the Fermi surface.
Finally, positron annihilation can be used to find the number of electrons in a two-dimensional slice through the Fermi ‘sphere'. When the material is bombarded with positrons the electrons annihilate the positrons, yielding two photons. The momentum of the emitted photons can be used to determine the momentum distribution of the electrons in the metal. This can then be used to indicate the number of electrons in a given slice through k -space.
6.3.7 The de H a a s-v a n Alphen effect
How d o e s t h e d i f fe r e n tia l susceptibility o f a m e t a l d e p e n d on th e a p p l i e d f i e l d str ength?
The de Haas-van Alphen effect is the most important technique used for obtaining information about the Fermi surfaces of metals. At low temperatures and under high magnetic fields (typically H > 5 kOe or 400 kA m 1) it was ^°^irJ that the differential susceptibility d M / d H of metals was dependent on the field
strength in an oscillatory manner. . .
When the differential susceptibility is plotted against \/H this periodic dependence is shown most clearly, although often two or more periods are superimposed. Similar behaviour has been observed in the conductivity and the
magnetostriction. The former is known as the Shubnikov-de Haas effect- T w o methods are widely employed to measure these de Haas-van Alphen o s c ilia tio n s.
One uses a torque magnetometer and simply measures the oscillations in angular position of a sample of the metal as the field strength H and m a g n etiz atio n M
are increased.
The second method uses field pulses and measures the voltage in d u c e d in a flux coil wound on the sample. Since the voltage from the flux coil will be
V — N ( d B ; d t ) ^ p 0 N ( d M / d t ) and the rate of change of magnetic field ( d H / d t )
is known, then it follows that,
126 Electronic properties of metals
v s, d Md H
V - P o N ---
d H d t
and so ( d M / d t f ) can be calculated. Onsager [11] has shown that the p e rio d ic ity of the oscillations in (1///), measured in Oe~ *, is given by,
A ( 1 /H) =
h >4ô,.
where A cu is the extremal ‘area’ in reciprocal space of the Fermi su rface in a plane normal to the magnetic field. So for a spherical Fermi surface A cxt = n k F- From measurements of the oscillations of ( d M / d H ) against (1 / / / ) it can be seen that the extremal area is easily calculated from the periodicity, since 7t, e
and h are all well-known constants
/tcxl = --- •
h A ( \ / H )
This^enables the extremal area of the Fermi surface to be determined in different directions.
r e f f r f n c e s
1- Drude, P (1900) Ann. der Physik, 2, 566.
2. Fermi, F (1928) Zen. fur Physik, 48, 73.
3. Ziman, J. M. (1963) Electrons in Metals - A Short Guide to the Fermi Surface, T a y l°r and F rancis, London.
4. Mackintosh, A. (1963) Sei. An 209, 110.
5. Cracknell,A P and Wbng, K. C. (1973) The Fermi Surface, Clarendon Press, Oxford.
6. Shockley. W. (1937) Phys. Rev., 52, 866.
7- Pippard, A B. (1957) Phil. Trans. Roy. Soc., A250, 325.
8- Pippard A B. (I960) Rep. Prag. Phys. 23, 176.
9. Ashcroft. N W and Mermin, N. D. (1976) Solid State Physics, H o lt, R i n e h a r t &
Winston, New York, p 76.
10. De Haas. W. J and van Alphen, P. M (1930) Proc. Neth. Roy. Acad. Sci.y 33, 1106- IL Onsager ] (1952) Phil. Mag., 43, 1006.
Exercises 127 FU R T H E R r e a d i n g
Chambers, R. G. (1990) Electrons in Metals and Semiconductors. Chapman and Hall, London.
Dugdale, J. S. (1976) The Electrical Properties of Metals and Alloys, Edward Arnold and Sons, London.
Lehmann, G. and Ziesche, P. (1990) Electronic Properties of Metals, Elsevier, Amsterdam.
MacDonald, D. K. C. (1956) Electrical conductivity of metals and alloys at low tempera
tures in Handbook of Physics, ed. S. Flügge, Springer, Berlin.
Moruzzi, V. L., Janak, J. F. and Williams, A. R. (1978) Calculated Electronic Properties
° f Metals, Pergamon, Oxford.
Mott, N. F. and Davis, E. A. (1971) Electronic Processes in Non Crystalline Materials, Clarendon Press, Oxford.
Mott, N. F. and Jones, H. (1936) The Theory of the Properties of Metals and Alloys, Oxford University Press.
EXERCISES
Exercise 6.1 Brillouin zones in a two-dimensional lattice. M ake a plot of the first two Brillouin zones of a rectangular two-dimensional lattice with unit vectors along the x and y directions of a = 0.2nm , and b = 0.4nm. Give its dimensions in m ' 1;- calculate the radius of the free electron Fermi sphere if the atom has valence 1; draw this sphere on the first Brillouin zone; and show the electron band structure for both the first and second energy bands, assuming there is a small gap at the zone boundary.
Exercise 6.2 Number o f k-states in reciprocal space. Show that the num ber of different /c-states in the reciprocal space of a simple cubic lattice is equal to the number of lattice sites.
Exercise 6.3 Fermi energy o f sodium and aluminium. Assuming that the free electron model applies, calculate the Fermi energy of body-centred cubic N a and face-centred cubic Al. The dimensions of the cubic unit cells in the crystal lattices are 0.43 nm and 0.40 nm respectively.
Electronic Properties of Semiconductors
7
This chapter discusses the electron band structure o f semiconductors and shows how the occupancy of the electron energy levels in these materials is fundamentally different from that in metals. The reason for this is that in semiconductors and insulators in their lowest energy state, the electron bands are either filled or empty.
This means that it is very difficult for the electrons to move under the action of an electric field because it would result in an increase in energy, and such energy states are not immediately available. Hence the conductivity is low. Although the energy of the electrons does vary with the wave vector k, a more simplified band structure representation is often used for semiconductors. This is the fiat band' approach which merely represents the allowed energy levels without reference to the corresponding values ofk. This approach, which is adequate for most purposes, is used widely here and in subsequent discussion of the electronic structure of semiconductors. One aspect which the fiat-band model does not represent however is the difference between direct and indirect band gap semiconductors. The direct band gap materials, in which the top of the valence band and the bottom o f the conduction band are located at the same point in k-space, are very important in optoelectronic applications.