Chapter 1

REVIEW OF SEMICONDUCTOR PHYSICS AND DEVICES

INTRODUCTION

This book assumes some elementary knowledge of semiconductor physics and devices, corresponding to that acquired in a typical electrical engineering undergraduate curriculum. At the end of this chapter, we quote a few books in this area, which may be useful in reviewing the topics commonly covered in such a course. Since it might be useful to also have the main results, which we need to quote from elementary semiconductor theory, collected in a convenient place, we will briefly review these in this chapter. This review will also serve the purpose of defining a notation which we can use throughout the book. We will cover the important concepts regarding transport of charge carriers in semiconductors: in particular it is useful to review how the concepts of effective mass, mobility and density of states follow from the energy band concept. We also review the properties of p-n-junctions and Schottky barriers, reverse break­down, and phonons. The expansion of these elementary ideas necessary for the understanding of specific devices will be left to the appropriate later chapters in which these devices are discussed. We prefer this approach to one in which all relevant semiconductor physics would be presented first, on the grounds that the discussion of the physics material can be given a more lively presentation ifit is directly connected with the device whose operation it explains.

ENERGY BANDS

Semiconductor materials, as used in microwave solid state devices, are single crystals, i.e. the atoms of the crystals form a periodic lattice. The materials we find in microwave devices are either elemental semiconductors, such as germanium (used only rarely) and silicon, or compound semiconduc­tors, primarily consisting of a combination of elements from the third and fifth columns in the periodic system of elements, "III-V" - compounds. Examples of the latter are GaA .. and InP. Recently, ternary and tertiary compounds have also come to use, and will be referred to in Chapters 11 and 12. Representative crystal structures are the diamond structure (germanium and silicon, see Fig­ure 1.1) and the zincblende structure (GaA .. and most other III-V-compounds, see Figure 1.2). The most important symmetry directions in both cases are the < 100> directions along the sides ofthe cube, and the < 111 > directions along the body diagonals. (The notation < 100 > is used for all equivalent directions [100], [100], [010], etc.)

S. Yngvesson, Microwave Semiconductor Devices© Kluwer Academic Publishers 1991

2 Microwave Semiconductor Devices

Figure 1.1. The crllstal lattice of diamond, germanium and ,ilicon. From SZE, S.M. (1985). "Semiconductor Device.: Phll,ic, and Technologll, " John Wilell & Son" New York.

T a

1 Figure 1.2. The zincblende lattice, applicable to almo.t all 111- V ,emicon­

ductor •. From SZE, S.M. (1985). "Semiconductor Device.: PhllSic. and Technologll, " John Wilell & Sons, New York.

Chapter 1 3

Electrons in semiconductors (or other solids) move as if they were waves, with a wavelength, ~, and a wave-vector, k, such that

(1.1)

Further, the momentum of the electron can be written in terms of k as:

p = lik (1.2)

Here, Ii = h/21r, and h is Planck's constant.

The development of the electron wave-function for various cases can be found by solving the Schroedinger equation, using the appropriate potential function. Electron waves traveling through a crystal experience the atomic lattice as a series of periodic perturbations, and solid state physics books show how there are solutions to the Schroedinger equation for the periodic potential corresponding to the crystal lattice. As a matter of fact, in a perfect crystal (i.e. one which is perfectly periodic) an electron wave can propagate through the crystal without change. F. Bloch first demonstrated how this can occur. The problem initially appeared puzzling to Bloch, who wrote:

When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in the metal ... By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation (quoted in Kittel (1976), p. 185, by permission).

The form of the wave-function described by Bloch in the above passage is thus the following:

';'(z) = u(z) x exp(jb) (1.3)

The first factor in the wave-function is the so-called Bloch-function, which is periodic with the periodicity of the crystal, i.e.,

u(z) = u(z + Na) (1.4)

Here, N is an integer, and a is the lattice constant in the direction of travel. The Bloch electron wave-function is approximately orthogonal to that of the atom, which explains why the electron can travel so easily past a large number of atoms.

The second factor in (1.3) of course represents a sinusoidal plane wave, which typically varies much more slowly, as also shown in Figure 1.3. The wave-vector in this factor is the one we introduced in (1.1).

Fortunately, we need not solve the Schroedinger-equation from first princi­ples in all cases ofinterest to us. Such solutions are quite well known, and they show that the electron has both allowed and forbidden regions, or "bands", for which solutions exist, and do not exist, respectively. The electron energy can

4 Microwave Semiconductor Device.

Atoms in Crystal

// "'~ a •• • a ..... a •• ••

Figure 1.3. nlu6tration of a Bloch wave function.

periodic • continuation

k=- 2L a

E

periodic

continuation

k=2L a

k

Figure 1.4. nlu.tration of allowed and forbidden energy band .. in a .. emicon­ductor.

Chapter 1 5

be found as a function of the wave-vector k for the principal crystal directions, and plotted as shown schematically in Figure 1.4.

Since this is a book on microwave devices, it would be appropriate to make an analogy between the motion of an electron in a periodic potential, and that of an electromagnetic wave in a periodic structure. The electromagnetic case is discussed by for example (Pozar, 1990), and there are many similarities between that solution and the energy band solutions for electrons in a crystal. Specifically, the microwave periodic structure also shows "pass-bands" and "stop-bands". The well-known Floquet's theorem (Pozar, 1990) results in a wavefunction for the microwave case, which has exactly the same properties as the wave-function quoted above in (1.3) (Pozar, 1990, Sec. 9.1). Another common property of both types of solutions is that the "bands" themselves are periodically repeated along the k-axis (the corresponding variable in the microwave case is /3). The information beyond the first "zone" (from k = -1I"/a to k = +1I"/a in Figure 1.4) is therefore redundant, and need not be plotted. The zone defined above is the so-called "Brillouin-zone", and the values k = ±r / a represent the "Brillouin-zone boundaries". The Brillouin zone boundaries have the simple physical interpretation that for k = 1I"/a, exactly one half wavelength fits into one period of the crystal (or the microwave periodic structure). For this k-value the periodically repeated reflections from each atom add in phase, making propagation through the structure impossible. Hence, the origin ofthe "stop-bands" or forbidden regions in the band diagram of a semiconductor.

In another analogy with the microwave case, we define the group velocity for the electron wave as follows:

1 Vg = r;,(de/dk) (1.5)

(for the microwave case, V,I = dI.J/d/3, which becomes (1.5) if we note that the energy, e ;:= 1U.J for the electron case). The group-velocity is thus proportional to the slope of an energy band. As we might expect, the group velocity goes to zero at the Brillouin zone boundary, in agreement with the fact that the electron is forbidden to propagate through the crystal for this k-value. For a "free" electron, we would further expect that the kinetic energy would be

p2 h2k2 e=-=--

2m 2mo (1.6)

Note that near the extrema, the energy bands have approximately the same parabolic shape as that of the free electron, except that the curvature is in general different. The curvature of the energy bands is related to the concept of "effective mass", defined by:

(1.7)

6 Microwave Semiconductor Devices

large mass

----+----l~ k -----'-----l~ k

Valence Band

Figure 1.5. Comparison of energy bands with different effective ma.fle,.

(we use m: for an electron effective mass, to distinguish it from m;' for a hole, which can be defined analogously, see below). In a thermal equilibrium situation, or if the electrons are not accelerated too strongly, the electrons stay close to the minimum of the band, and a constant m: can be utilized.

The energy of an electron can thus still be written in a manner similar to (1.6), but with mo substituted by m·. Near a minimum in an energy band (assume for example that the minimum occurs at " = 0):

11,2,,2 &= &(k= 0)+-

2m: (1.8)

Figure 1.5 gives a few examples of energy bands with different effective masses. For a band with a maximum (our typical "valence band") there is an energy expression which is similar to (1.8) except for a minus sign in the second term. In this case, we predict a negative effective mass, by using (1.7). The problem is taken care of by discussing the "hole" for this band, and defining the effective mass of the hole as:

m;' = _11,2 [~~;rl (1.9)

Using these definitions, we find that the hole has both a positive mass and a positive charge.

Energy bands for "real" (non-idealized) semiconductors are shown in Fig­ure 1.6. It is a common practise to plot the bands for two different crystal directions in the same diagram, using the fact that the bands are symmetric

Chapter 1

! [111] a

k[OOO] ~[100] a

(al GaA'

7

L X ,~------~----------~,

! (111] a

k(OOO] ~(100] a

(bl Si

Figure 1.6. Principal energy band6 of GaA6 and 6ilicon. The p06itive k-azi6 repre6ent6 the {100J (X) direction6, while the negative azi6 6how6 the band! for the [111J (L) direction6. Reproduced from WANG, S. (1989). "Fundamen­tal. of Semiconductor Theory and Device PhY6ic6," Pentice Hall, Englewood Cliff" N.J., with permi66ion.

around k = o. The examples given are for both a direct bandgap semicon­ductor (GaA8) and an indirect bandgap case (silicon): The direct bandgap material has its lowest minimum in the conduction band at the same k-value (k = 0) as the highest maximum of the valence band, while for the indirect bandgap case, these two extrema do not occur for the same k.

Standard text books also show how one can calculate the density of al­lowable electron states in an energy band, in a particular energy interval dE, assuming unit volume of the material:

(1.10)

(for the conduction band case), and

(1.11)

8 Microwave Semiconductor Devicea

(for the valence band case). Briefly, the derivation is carried out by counting the number of allowed states in k-space in a spherical shell, for which the energy has a given value E. In Chapter 11, we will derive a similar relation for electrons which are confined to moving in 2-dimensional space.

STATISTICAL PROPERTIES OF ELECTRONS AND HOLES

Electrons are fermions, for which the probability of occupation of a par­ticular state obeys the Fermi-Dirac (F D) distribution function. Also, each state (including spin) can only be occupied by at most one electron. The F D function is given by:

1 feE) = d£-£, )/l&sT + 1 (1.12)

Here, eF is the Fermi energy, for which the probability of occupation is always 1/2. In order to arrive at the total density of electrons in the conduction band, we multiply the density of states (1.10) by the F D function, and integrate over energy. The result is to a good approximation:

No

neT) ~ ~(21!'m:kBT/h2)f xe-(£·-£,)/l&sT

The corresponding expression for holes in the valence band is:

N. "

(1.13)

(1.14)

The position of the Fermi level for an intrinsic semiconductor is roughly in the center of the bandgap, or more exactly at:

{ e. + e., } k (N.) eF = --2- + BT x in N~

By multiplying (1.13) and (1.14), we find the general expression:

neT) x peT) = N.N~ x e-(£·-£·)/IosT

(1.15)

(1.16)

Here, e. - e. = eg, the "bandgapn. By applying (1.16) to an intrinsic (un­doped) semiconductor, we also find:

n;(T) = p;(T) = VN.N. x e-(£·-£·)/2IosT (1.17)

A doped semiconductor contains impurities, which are either donors (n-type case) or acceptors (p-type case). Donor and acceptor energies are close to the conduction band and valence band edges, respectively, with typical energy differences (ionization energies) in the range 0.005 to 0.05 eV. At room tem­perature, almost all impurities are ionized, and we can write (the subscript '0' indicates thermal equilibrium):

n? P ~ •

no - Nt!. (1.18)

Chapter 1

for an n-type case (Nd is the acceptor density)

n~ 7L.~N·"-~ • rpO - 0, ·~pO - N.

for a p-type case (N" is the acceptor density).

9

(1.19)

The distribution of electrons versus energy for a couple of typical cases is illustrated in Figure 1.7.

CARRIER TRANSPORT

Charge carriers (electrons or holes) are transported in devices due to forces arising from either an electric field or a concentration gradient. With no ex­ternal driving forces, the carrier executes a random motion due to its thermal energy. When external forces of moderate magnitude are also applied, they superimpose a slow average "drift" or "diffusion" of the carrier on the ther­mal random motion, see Figure 1.8. The average time between changes of the electron momentum is called the momentum relazation time*, T, and is of the order of 10-13 to 10-12 seconds at room temperature. For reasonably low electric fields, the average electron drift velocity is proportional to the field, and we can define the proportionality constant as the mobility:

for electrons, and

eT I'n= -

m' • (1.20a)

eT I'p = m' (1.20b)

h

for holes. The units are cm2/V-sec. We have used e, a positive quantity, to designate the magnitude of the electron charge, and will continue to use this convention throughout the book. Adding up the contributions by all carriers, we find for the current density due to drift in an electric field, E(V/cm):

1ft 1,

f = (~+ P:;;;)E(Afcm2 ) (1.21)

The electrical conductivity follows from (1.21):

tr == J/ E = (nel'n + pel'p) (l1cm)-l (1.22)

IT we add gradients in the carrier concentrations (dn/dz and/or dp/dz), find the total electron or hole currents as follows:

we

- - (dn) I n = nel'nE + eDn dz i (1.23a)

* Since little or no energy may be lost in many collisions ofthe charge carrier with the lattice, the energy relaxation time (T.) is, in general, different from the momentum relaxation time. The energy relaxation time will be introduced in Chapter 2.

10

e e

N (e)

(a)

o 0.5 1.0

F (el

(b)

Microwave Semiconductor Devices

e

n

n (el AND p (el

(e)

Figure 1.7. (a) Demitie, 01 allowed ,tate, in the conduction and valence band, (b) Probability 01 occupation 01 the state, in the ,arne band, (c) The product 01 the densities 01 state" and the probability 01 occupation lor the ,arne band,.

Figure 1.S. The path 01 an electron in a conductor, moving under the influ­ence 01 an electric field.

Chapter 1 11

a '" ,

,. '00 200

2000

.oco >

500

~ 200 ~ ~

::; .00

;;; 50 0

" 20

IMPURITY CONCENTRATION tem- 3 )

Figure 1.9. Mobility of electronl and holel in lilicon, (a) al a function of temperature (b) al a function of impurity concentration. From BEADLE, W.F., TSAI, J.C.C., and PLUMMER, R.D., Ed,. (1985). "Quick Reference Manual for Semiconductor Engineerl," John Wiley (J Sonl, New York, with permiuion.

- - (dP) A Jp = peJ.'pE - eDp dz z (1.23b)

Here, D" and Dp are the diffusion constants for electrons and holes, respec­tively. Another useful equation is the Ein6tein relation, valid for thermal equi­librium conditions at some temperature, T:

(1.24a)

(1.24b)

The mobility in general depends on the temperature and the impurity concentration in the crystal. Some representative curves are shown in Figures 1.9 and 1.10. Note that electrons in GaA. show considerably higher mobility than holes in the same material, and either carrier in silicon. This has been the single most important factor pushing the development of GaA. devices rather than silicon devices for the microwave range.

12 Microwave Semiconductor Device.

le .... ptlt.lu.<r 1 11'1

Figure 1.10 (a). Temperature dependence of the Hall mobility for electron6 in GaA6, for three different den6itie6 (A) 5 x 1013 cm-3 (B) 1015 cm-3 (C) 5 x 1016 cm- 3 • (b) Dependence on impurity concentration of the mobility for electron. and hole. in GaAB. Reproduced from BLAKEMORE, J.M. (1982). "Semiconductor and Other Major Propertie6 of GaA6," J. Appl. Phy.. 53, R123, with permi68ion.

As we shall see further on in this book, almost all practical devices produce electric fields so large that the mobility concept no longer applies. Figure 1.11 illustrates the electric field dependence of the drift velocity of electrons in silicon and GaA... In both materials, the drift velocity reaches a saturation value at fields of a few kilovolts/cm. The concept of the saturated velocity, and the peak in the velocity/field curve in the GaA. case, are very important for the explanation of the function of many microwave devices. The detailed discussion and interpretation of these effects will be delayed to later chapters, in conjunction with the treatment of the operation of particular devices.

The diffusion constant, D, also depends on the electric field, and its field dependence is illustrated in Figure 1.12.

Chapter 1 13

GoAs (E LECTRONS)

~ E 107 2

/VI UtHfL (ELEdRONS) ->- ....

t: .... Si (HOLES) u .... 0 --l W > I- 106 lJ..

/ .-

/ V /~~

0:: Go As (HOLES) 0

/ Ir3~O~ / /

V ~~ /

II 1/ 103 104 10!!i

ELECTRIC FIELD (V/em)

Figure 1.11. Drift velocity ver6U6 electric field for carrier6 in GaAs and Si. From SZE, S.M. (1985). "Semiconductor Device,,: Phy"ic" and Technology," John Wiley tJ Son", New York, with permission.

10 r--------------------------,

4 6 10 12

Electric field. C'(kY/cm)

Figure 1.12. Mea6ured di/Ju6ion coefficient for electron" in GaA. ver"U6 electric field. From GLISSON, T.H., SADLER, R.A., HAUSER, J.R., and LITTLEJOHN, M.A. (1980). "Circuit Effect. in Time-of-Fli9ht DifJu"ivity Mea"urementB," Solid State Electron., f3, 637, with permiuion.

14 Microwave Semiconductor Device.

electron •

hole generation

•

Direct Recombination

Recombination Centers

Annihilation of electronlhole pair via recombination

center

Fig 1.13. Recombination proce"e. in .emiconductor •.

CARRIER RECOMBINATION AND GENERATION

In a semiconductor under thermal equilibrium conditions, the product of the electron and hole densities is given by (see (1.16) and (1.17»:

pn=n~ (1.25)

The equilibrium may be disturbed, for example by absorption of photons, which can create new electron-hole pairs (a "generation process"), so that (1.25) no longer applies. If the semiconductor is allowed to return to equi­librium, this will be accomplished with the help of recombination processes, i.e. processes by which a conduction band electron makes a transition down to the valence band, where it will "fill" one of the hole states. Thus, in the recombination process an electron-hole pair will be annihilated. In thermal equilibrium, both generation and recombination processes still are active, but balance each other.

In direct bandgap semiconductors, recombination proceeds rapidly across the bandgap, a "direct" recombination process. In semiconductors with an indirect bandgap, recombination typically is mediated by so-called recombi­nation centers, which are impurities with states near the center of the energy gap. Introduction of Au impurities in silicon, for instance, can shorten the recombination "life-time" from one microsecond to 0.1 ns. Similar effects can be obtained by irradiation with high-energy particles. In a direct bandgap

Chapter 1 15

material such as GaA8, the recombination life-time is of the order of 1 ns or shorter. The different types of recombination and generation processes are pictured in Figure 1.13.

P-N-JUNCTIONS

A very important property of p-n-jundions for microwave devices is the width, W. Figure 1.14 illustrates the structure of a p-n-junction, and the distribution of electrical charge carriers. In order to find the electric field or the potential in the junction, we need to solve Poisson's equation:

d2V _ Total Charge Density _ e(p(z) - n(z) + ND(Z) - NA(Z» (1.26) dz2 - f - f

Usually, we can apply the depletion approximation, i.e. the junction is assumed to be depleted of mobile charges. The electric field and potential in a junction are given in Figure 1.14. From these we can also calculate the total width of the junction for the important case of an abrupt p+ -n-junction (V is the applied voltage, a negative number for reverse bias)

_ _ (2f(Vo - V») ~ W_z,,- N

e D (1.27)

Vo is the "built-in" potential difference across the junction. Note that the width is essentially proportional to the applied voltage to the power 1/2. If we instead assume a linearly graded junction, this dependence will be with the 1/3 power of the voltage. The depleted region acts as a capacitor, with a differential capacitance defined as:

(1.28)

For the abrupt and linearly graded junction, respectively, the depletion capac­itance then is given by:

Gt!. = ~ [2fe NAND ] i (Vo _ V)-t 2 NA +ND

(1.29)

and

(1.30)

It is common to write Go

Cd = -----=-.---::-=_;_

( v) i or t 1- v.-

(1.31)

In addition to the depletion capacitance, there is also a diffusion capacitance for a p+-n-junction biased in the forward direction, due to charge storage on the n-side. The diffusion capacitance is proportional to the current through the diode.

16

t Electron energy

N N

m.: .. r~::, Ec G~

~G nENd

9: 00 0-donors

G Ev G-...acceptors

o 01~ 0 ! Hole en. ~-! --=--

I , I depletion' region

! x.. "" I I

! , ,

(a)

, , INCREASED

REVERSE BIAS

Microwave Semiconductor Device6

Elecoric Field. E

Il. " Ib ~//" ..... " ..... __ Effect of bias

,/,','/' """"""

---L-L~-----+ ____ ~~~~_x

(b) Potential. V

(e)

Figure 1.14. (a) Geometry and charge di6tribution in a p-n-junction. (b) Electric field di6tribution in a p.n-junction (c) Potential di6tribution in a p-n-junction.

Chapter 1 17

The 1-V-characteristic of a p-n-junction is given by:

(1.32)

The ideality factor, 1/, has a value between 1 and 2 (1.0 applies to an ideal diode). Again, we define the dynamic resistance as 1'.d = 1/(dI/dV). The value for 1'.d is then simply:

1'.d = e(I + Is) kBT (1.33)

The numerical constants work out so that 1'.d can be found as

1'.d = 26 x 1/ ohms/rnA of current (1.34)

If the 1-V-characteristic is plotted in a log-lin plot, we should obtain a straight line. At high currents, the characteristic deviates from this line, when 1'.d be­comes so small that the series resistance due to the material outside the junc­tion proper begins to dominate. The reverse characteristic also deviates from the simple theory, due to recombination-generation processes in the depletion region. These processes give rise to extra electron-hole pairs, which separate due to the high electric field in the depletion region, and thus increase the reverse current.

SCHOTTKY BARRIERS

A Schottky barrier diode consists of a metal anode, and a semiconductor cathode. The semiconductor is either silicon or GaA.. A potential barrier is formed on the semiconductor side, as shown in Figure 1.15. While straight­forward theory would predict the height of the potential barrier to be equal to the difference in work function (energy required to remove the electrons out of the solid) for the metal and the semiconductor, actual junctions have potential barrier heights which are often dominated by effects due to surface states, which are especially prominent in GaAB. As for the p-n-junction, charge is stored near the junction with reverse bias applied, with part of the negative charge stored in the surface states. The capacitance with reverse bias has the same voltage-dependence as that for the abrupt p-n-junction, see (1.29). Since no charge storage occurs in the metal with forward bias applied in the conducting region, there is no diffusion capacitance, and this makes the Schottky barrier diode a much faster diode. It is therefore generally preferred as a microwave detector or mixer device and will be discussed further in Chapter 9. P-n­junctions occur in microwave devices such as IMP ATT devices or tunneling devices, however, but in this case other physical effects also contribute to the operation of the device (see Chapters 3 and 4).

18

Metal

Charge in metal

Microwave Semiconductor Devices

Semiconductor

Depletion Region

Figure 1.15. Energy band and charge diagrams for a Schottky barrier diode.

REVERSE BREAK-DOWN

At a sufficiently high bias voltage with reverse polarity, a p-n-junction diode will show a rapidly increasing reverse-directed current due to either tun­neling (highly doped diodes with low break-down voltages) or avalanche (im­pact ionization) break-down. Referring to Figure 1.14b, we can calculate the reverse break-down voltage due to avalanching for a p+ -n diode. The depletion region is located almost entirely in the n-type material for this case, and the voltage is easily found as:

1 V ~ 2'W x E(max) (1.34)

Chapter 1 19

By substituting W from (1.27), we find the following expression for the break­down voltage, VB:

(1.35)

It has been observed that the break-down occurs at a characteristic peak elec­tric field, Em, which is almost independent of the doping. Equation (1.35) thus shows that the break-down voltage is roughly inversely proportional to the doping on the n-side of a p+ -n diode. When the doping exceeds about 5 x 1018cm-a, the junction will be so narrow that tunneling begins to domi­nate. When impact ionization processes dominate, carriers with energy greater than about 1.5 times the bandgap energy are able to generate new electron­hole pairs.

To discuss the probability of impact ionization qualitatively, we may as­sume that the accelerated carriers can still be described by a distribution func­tion, such as (1.12), but with a higher temperature, the carrier temperature, Te. In Chapter 2 we show that Te is approximately related to the accelerating electric field (E) as (2.6):

(1.36)

where Vs is the saturation velocity and T. the energy relaxation time. We expect the ionization probability (Q) to be proportional to the fraction of the carriers which have an energy greater than the minimum energy required for ionization E,. From the distribution function and (1.36), we then find (Wang, 1989):

cr = Ai X exp (- k:~e) = A x exp (-~) (1.37)

We thus I:onclude from this qualitative discussion that Q depends in the above exponential manner on both the carrier temperature (or the average carrier energy < e » and the electric field. This is in approximate agreement with experimental data, to be reviewed in Chapter 3. Further discussion oftunneling and avalanche break-down effects will be deferred to Chapters 3 and 4.

PHONONS

The concept of a phonon arises from a quantum-mechanical discussion of the vibration modes of the atoms in a solid. Classically, we expect the vibra­tions of the solid to show eigen-modes analogous to the modes of a string. The shortest wavelength that needs to be taken into account, is one for which one half wavelength fits into the spacing between two atoms, the lattice constant, in analogy with the situation for electron waves, discussed in the beginning of this chapter. We thus expect that there will be a cut-off wavenumber, qmax, for the vibrational waves in the solid. A wave of this type, which resembles the vibrations of a string, is called an "acoustic" mode. Two types of acoustic modes exist, with transverse and longitudinal polarization, respectively (the polarization indicates the direction in which the atoms vibrate). The frequency

20 Microwave Semiconductor Device.

Ge Si GaA, 16 16 16

TO 14 14

0.06

N 12 12 0.05 :I: N

~ '0 10 10 TO 0.04 .! ~ LO '" >- 8 8 8 >-" 0.03 eo c TO ~ .

6 6 6 c

~ t.u

0.02 LL 4 4

2 0.01

klkmaK 0 1.0 0 1.0 0 1.0

[lDO)---- [lDO) ---- [lDO)~

Figure 1.16. Mea.ured phonon ,pectrafor Ge, Si and GaA6. Reproducedfrom WANG, S. (1989). "Fundamentab of Semiconductor Theory and Device Phy.ic.," Prentice Hall, Englewood Cliff" N.J., with permission.

(Transverse) Acoustic mode (Transverse) Optical mode

Figure 1.17. nlu6tration of acoustic and optical phonon mode,.

(II) versus wave-number (q) curves for these vibrations of the crystal lattice are called "dispersion curves", and examples of such curves for three common semiconductors are shown in Figure 1.16. The crystal lattice also has another possible vibration mode, the "optical" mode, for which in each unit cell the two constituents vibrate in opposite phase. Figure 1.17 illustrates the basic difference between acoustic and optical modes. We note from Figure 1.16 that

Chapter 1 21

there are two different polarizations for optical modes, as well, and that in general the optical modes have higher frequency than the acoustic ones. The name "optical" mode refers to the fact that the optical modes couple more strongly to optical radiation than the acoustic ones. Note, however, that the frequencies for optical modes, while high (of the order of 1013 Hz) are still about two orders of magnitude lower than those of optical radiation (about 1015 Hz).

While the above ideas can be understood in classical terms, quantum me­chanics dictates (as in the case for photons!) that the energy of each vibrational mode can only take on discrete values, which are multiples of h x v. As the en­ergy of a vibrational mode increases, one thus talks about adding a number of "phonons" to the energy ofthe mode, each phonon possessing an energy of hv. Note that phonons can then be both "emitted" or "absorbed", depending on whether the energy of the mode in question increases or decreases. The phonon is also a property which refers to the lattice as a whole, not to any individual atom. The important phonon effects, which are most relevant to the topic of this book, involve interactions between electrons and phonons, referred to as "scattering" events. We will interpret such "collisions" between electrons and the lattice as giving rise to the emission or absorption of a phonon, while the momentum and the energy of the electron change in such a way as to conserve total momentum and energy. The reader may want to refer again to Figure 1.8 for a schematic illustration of this process from the point of view of what happens to the electron. While the maximum momenta "carried" by phonons and electrons are the same (determined by the maximum wave-vector times h), the maximum energy of a phonon is only of the order of about 0.03 eV (if we use GaA8 as our example). The maximum energy of an electron in most devices is considerably higher than this value, as we will discuss in detail in later chapters.

REFERENCES

KITTEL, C. (1976). "Introduction to Solid State Physics," Fifth Edition, John Wiley &: Sons, New York.

FURTHER READING

PIERRET, R.F., and NEUDECK, G.W. (1983-1990). "Modular Series on Solid State Devices," Volumes I-X, Addison-Wesley, Reading, MA. Of special interest are: Vol. VIII, DATTA, S. (1989). "Quantum Phenomena," and Vol. X, LUNDSTROM, M. (1990). "Fundamentals of Carrier Transport".

SEEGER, K. (1989). "Semiconductor Physics: An Introduction," Fourth Edi­tion, Springer-Vedag, Berlin.

22 Microwave Semiconductor Device6

SHUR, M. (1990). "Physics of Semiconductor Devices," Prentice Hall, Engle­wood Clift's, N.J.

SMITH, R.A. (1961). "Wave Mechanics of Crystalline Solids," Chapman &; Hall, London.

SZE, S. (1981). "Physics of Semiconductor Devices," Second Edition, John Wiley &; Sons, New York, Ch. 1.

WANG, S. (1989). "Fundamentals of Semiconductor Theory and Device Phys­ics," Prentice Hall, Englewood Clift's, N.J.

WATSON, H.A., Ed. (1969). "Microwave Semiconductor Devices and Their Circuit Applications," McGraw-Hill, New York, Chapters 2 through 6.

WOLFE, C.M., HOLONYAK, Jr., N., and STILLMAN, G.E. (1989). "Physi­cal Properties of Semiconductors," Prentice Hall, Englewood Clift's, N.J.