Electrical Con19

PartA|2

2. Electrical Conduction in Metalsand Semiconductors

Safa Kasap, Cyril Koughia, Harry E. Ruda

Electrical transport through materials is a largeand complex field, and in this chapter we coveronly a few aspects that are relevant to practi-cal applications. We start with a review of thesemi-classical approach that leads to the conceptsof drift velocity, mobility and conductivity, fromwhich Matthiessen’s Rule is derived. A more gen-eral approach based on the Boltzmann transportequation is also discussed. We review the conduc-tivity of metals and include a useful collection ofexperimental data. The conductivity of nonuniformmaterials such as alloys, polycrystalline materials,composites and thin films is discussed in the con-text of Nordheim’s rule for alloys, effective mediumtheories for inhomogeneous materials, and theo-ries of scattering for thin films. We also discusssome interesting aspects of conduction in thepresence of a magnetic field (the Hall effect). Wepresent a simplified analysis of charge transport insemiconductors in a high electric field, includinga modern avalanche theory (the theory of luckydrift). The properties of low-dimensional systemsare briefly reviewed, including the quantum Halleffect.

2.1 Fundamentals: Drift Velocity, Mobilityand Conductivity ................................ 20

2.2 Matthiessen’s Rule.............................. 22

2.3 Resistivity of Metals ............................ 232.3.1 General Characteristics . ........................ 232.3.2 Fermi Electrons.................................... 25

2.4 Solid Solutions and Nordheim’s Rule ... 26

2.5 Carrier Scattering in Semiconductors. ... 28

2.6 The Boltzmann Transport Equation ...... 29

2.7 Resistivity of Thin Polycrystalline Films 30

2.8 Inhomogeneous Media:Effective Media Approximation ............ 32

2.9 The Hall Effect .................................... 35

2.10 High Electric Field Transport . ............... 37

2.11 Impact Ionization ............................... 38

2.12 Two-Dimensional Electron Gas ............ 40

2.13 One-Dimensional Conductance . ........... 42

2.14 The Quantum Hall Effect ...................... 43

References ..................................................... 44

A good understanding of charge carrier transport andelectrical conduction is essential for selecting or devel-oping electronic materials for device applications. Ofparticular importance are the drift mobility of chargecarriers in semiconductors and the conductivity of con-ductors and insulators. Carrier transport is a broad fieldthat encompasses both traditional bulk processes and,increasingly, transport in low dimensional or quan-tized structures. In other chapters of this handbook,Baranovskii describes hopping transport in low mobil-ity solids such as insulators, Morigaki deals with theelectrical properties of amorphous semiconductors andGould discusses in detail conduction in thin films. In

this chapter, we outline a semi-quantitative theory ofcharge transport suitable for a wide range of solids ofuse to materials researchers and engineers. We intro-duce theories of bulk transport followed by processespertinent to ultra-fast transport and quantized transportin lower dimensional systems. The latter covers suchphenomena as the Quantum Hall Effect, and QuantizedConductance and Ballistic Transport in QuantumWiresthat has potential use in new kinds of devices. Thereare many more rigorous treatments of charge transport;those by Rossiter [2.1] and Dugdale [2.2] on metals,and Nag [2.3] and Blatt [2.4] on semiconductors arehighly recommended.

© Springer International Publishing AG 2017S. Kasap, P. Capper (Eds.), Springer Handbook of Electronic and Photonic Materials, DOI 10.1007/978-3-319-48933-9_2

PartA|2.1

20 Part A Fundamental Properties

2.1 Fundamentals: Drift Velocity, Mobility and Conductivity

Basic to the theory of the electronic structure of solidsare the solutions to the quantum mechanical prob-lem of an electron in a periodic potential known asBloch waves. These wavefunctions are traveling wavesand provide the physical basis for conduction. In thesemi-classical approach to conduction in materials, anelectron wavepacket made up of a superposition ofBloch waves can in principle travel unheaded in anideal crystal. No crystal is ideal, however, and the im-perfections cause scattering of the wavepacket. Sincethe interaction of the electron with the potential of theions is incorporated in the Bloch waves, one can con-centrate on the relatively rare scattering events whichgreatly simplifies the theory. The motion of the elec-trons between scattering events is essentially free (withcertain provisos such as no interband transitions) sub-ject only to external forces, usually applied electric ormagnetic fields. A theory can then be developed thatrelates macroscopic and measurable quantities such asconductivity or mobility to the microscopic scatteringprocesses. Principle in such a theory is the concept ofmean free time � which is the average time betweenscattering events. � is also known as the conductivityrelaxation time because it represents the time scale forthe momentum gained from an external field to relax.Equivalently, 1=� is the average probability per unittime that an electron is scattered.

There are two important velocity quantities thatmust be distinguished. The first is the mean speedu or thermal velocity vth which, as the name im-plies, is the average speed of the electrons. u is quitelarge being on the order of

p3kBT=m�

e � 105 m=sfor electrons in a nondegenerate semiconductor andp2EF=m�

e � 106 m=s for electrons in a metal, wherekB is Boltzmann’s constant, T is the temperature, EF isthe Fermi energy of the metal, andm�

e is the electron ef-fective mass. The distance an electron travels betweenscattering events is called the free path. It is straightfor-ward to show that the average or mean free path foran electron is simply `D u� . The second velocity isthe mean or drift velocity v d (variables in boldface arevectors) which is simply the vector average over the ve-locities of all N electrons,

v d D 1

N

NX

iD1

v i : (2.1)

With no external forces applied to the solid, the electronmotion is random and thus the drift velocity is zero.When subject to external forces like an electric field,the electrons acquire a net drift velocity. Normally, themagnitude of the drift velocity is much smaller than u

so that the mean speed of the electron is not affectedto any practical extent by the external forces. An ex-ception is charge transport in semiconductors in highelectric fields, where jv dj becomes comparable to u.

The drift velocity gives rise to an electric current. Ifthe density of electrons is n then the current density Jeis

Je D �env d (2.2)

where e is the fundamental unit of electric charge. Forthe important case of an applied electric field E, thesolutions of the semi-classical equations give a driftvelocity that is proportional to E. The proportionalityconstant is the drift mobility �e

v d D ��eE : (2.3)

The drift mobility might be a constant or it might de-pend on the applied field (usually only if the field islarge). Ohm’s Law defines the conductivity � of a ma-terial J D �E resulting in a simple relation to the driftmobility

� D en�e : (2.4)

Any further progress requires some physical theory ofscattering. A useful model results from the simple as-sumption that the scattering randomizes the electron’svelocity (taking into proper account the distribution ofelectrons and the Pauli Exclusion Principle). The equa-tion of motion for the drift velocity then reduces toa simple form

dv d

dtD F.t/

m�

e� v d

�; (2.5)

where F.t/ is the sum of all external forces acting onthe electrons. The effect of the scattering is to intro-duce a frictional term into what otherwise would be justNewton’s Law. Solutions of (2.5) depend on F.t/. Inthe simplest case of a constant applied electric field, thesteady-state solution is trivial,

v d D �eE�m�

e

: (2.6)

The conductivity and drift mobility can now be relatedto the scattering time [2.5],

�e D e�

m�

eand � D ne2�

m�

e: (2.7)

Electrical Conduction in Metals and Semiconductors 2.1 Fundamentals: Drift Velocity, Mobility and Conductivity 21Part

A|2.1

More sophisticated scattering models lead to more ac-curate but more complicated solutions.

The simple expression (2.7) can be used to explainqualitative features of conduction in materials oncea physical origin for the scattering is supplied. For anyscattering site, the effective area is the scattering cross-section S as depicted in Fig. 2.1. The scattering crosssection is related to the mean free path since the vol-ume S` must contain one scattering center. If there areNS scattering centers per unit volume then

`D 1

SNS; (2.8)

or, rewriting in terms of � ,

� D 1

SNSu: (2.9)

Once the cross-section for each physically relevantscattering mechanism is known then the effect on thescattering time and conductivity is readily calculated.

An overly restrictive assumption in the above anal-ysis is that the electron’s velocity is completely ran-domized every time it is scattered. On the other hand,suppose that � collisions are required to completelydestroy the directional velocity information. That isonly after an average of � collisions do all traces ofcorrelation between the initial and the final velocitiesdisappear. The effectivemean free path `eff traversed bythe electron until its velocity is randomized will now belarger than `; to first order `eff D �`. `eff is termed theeffective or the conduction mean free path. The corre-

Ns = Concentration of scatters

ℓ = u τ

Scatterer S

Scatterer S

u

Electron

Scattering

Fig. 2.1 Scattering of an electron from a scattering center.The electron travels a mean distance `D u� between col-lisions

sponding effective scattering cross section is

Seff D 1

NS`eff: (2.10)

The expressions for mobility and conductivity become

�D e��

m�

eD e�`

m�

e uD e`eff

m�

e uD e

m�

e uNSSeff(2.11)

and

� D e2��

m�

eD e2n�`

m�

e uD e2n`eff

m�

e uD e2n

m�

e uNSSeff:

(2.12)

Suppose that in a collision the electron is scattered atan angle to its original direction of travel as shown inFig. 2.2. It is convenient to introduce a quantity S� ./,called the differential scattering cross section, definedso that 2  sin S�d represents the probability of scat-tering at an angle between and Cd with respect tothe original direction. If the magnitude of the velocityis not changed then the fractional change in componentof the velocity along the original direction is 1� cos .The average number of collisions � required to random-ize the velocity is then

� D 1

h1� cos i ; (2.13)

where the average is given by

h1� cos i DR  

0 .1� cos / S� ./ sin dR  

0 S� ./ sin d:

(2.14)

The effective cross sectional area Seff is then

Seff D 2 

 Z

�m

.1� cos / S� ./ sin d : (2.15)

Scattering center

θ = Angle of scatteringθ

vx = 1

x

Fig. 2.2 An electron moving in the x-direction becomesscattered though an angle with respect to the originaldirection

PartA|2.2

22 Part A Fundamental Properties

As an example, consider conduction electrons scatter-ing from charged impurities in a nondegenerate semi-conductor where small angle deviations predominate.The differential cross section for coulombic scatter-ing from a charged impurity center with charge CZeis

S� ./D 16k2

u4� 1

4I k D Ze2

4 "0"rm�

e; (2.16)

where k D Ze2=4 "0"rm�

e and "r is the relative permit-tivity of the semiconductor. Let rm be the maximumeffective radius of action for the impurity at which theminimum scattering angle m occurs. Then the integralin (2.15) evaluates to

Seff D 2 k2

u4ln

�1C r2mu

4

k2

�: (2.17)

In a nondegenerate semiconductor, the equipartitiontheorem links velocity to temperature, m�

e u2=2 D

3kT=2, so that u / T1=2. Thus,

Seff D A

T2ln�1CBT2

�; (2.18)

where A and B are constants. The drift mobility due toscattering from ionized impurities becomes

�I D e

m�

e uSeffNI/ 1

T1=2

T2

A ln.1CBT2/

1

NI

� CT3=2

NI; (2.19)

where NI is the density of ionized impurities (it repre-sents NS), and C is a new constant. At low tempera-tures where lattice scattering is insignificant, we expect�I / T3=2=NI for nondegenerate semiconductors.

The above semiquatitative description is suffi-cient to understand the basic principles of conduction.A more rigorous approach involves solving the Boltz-mann charge transport equation and is addressed inSect. 2.6.

2.2 Matthiessen’s Rule

In general, the conduction electron whether in a metalor in a semiconductor can be scattered by a numberof mechanisms, such as lattice vibrations, impurities,lattice defects such as dislocations, grain boundaries,vacancies, surfaces, or any other deviation from a per-fectly periodic lattice. All these scattering processesincrease the overall resistivity of the substance by re-ducing the mean scattering time. The relation betweenthe types of scattering and the total scattering time canbe obtained by considering scattering from lattice vi-brations and impurities as shown in Fig. 2.3. We definetwo mean free times �L and �I:�L is the mean freetime considering only scattering from lattice vibrations(phonons) and �I is the mean free time considering onlycollisions with impurities. In a small unit of time dt,the total probability of scattering (dt=�) is simply thesum of the probability for phonon scattering (dt=�L)and the probability for impurity scattering (dt=�I), andthus

1

�D 1

�LC 1

�I: (2.20)

We have assumed that neither �L nor �I is affected by thepresence of the other scattering mechanism, that is eachtype of scattering is independent. The above expression

can be generalized to include all types of independentscattering mechanisms yielding

1

�DX

i

1

�i; (2.21)

where �i is the mean scattering time considering theith scattering process alone. Since the drift mobility �d

Impurity and its lattice deformation

τI

τL

Fig. 2.3 Scattering from lattice vibrations alone witha mean scattering time �L, and from impurities alone witha mean scattering time �I

Electrical Conduction in Metals and Semiconductors 2.3 Resistivity of Metals 23Part

A|2.3

is proportional to � , (2.21) can be written in terms ofthe drift mobilities determined by the various scatteringmechanisms. In other words,

1

�dDX

i

1

�i; (2.22)

where�i is the drift mobility limited just by the ith scat-tering process. Finally, since the resistivity is inverselyproportional to the drift mobility, the relation for resis-tivity is

�DX

�i ; (2.23)

where �i is the resistivity of the material if only theith scattering process were active. Equation (2.23) isknown as Matthiessen’s rule. For nearly perfect, purecrystals the resistivity is dominated by phonon scatter-ing �T. If impurities or defects are present, however,there are an additional resistivities �I from the scatter-ing off the impurities and �D from defect scattering, and�D �T C �I C �D.

Matthiessen’s rule is indispensable for predictingthe resistivities of many types of conductors. In somecases like thin films, the rule is obeyed only approx-imately, but it is nonetheless still useful for an initial(often quite good) estimate.

2.3 Resistivity of Metals

2.3.1 General Characteristics

The effective resistivity of a metal, by virtue ofMatthiessen’s rule, is normally written as

�D �T C �R ; (2.24)

where �R is called the residual resistivity and is due tothe scattering of electrons by impurities, dislocations,interstitial atoms, vacancies, grain boundaries and soon. The residual resistivity shows very little temperaturedependence whereas �T is nearly linear in absolute tem-perature. �T will be the main resistivity term for manyhigh-quality, nonmagnetic pure, crystalline metals. Inclassical terms, we can take the thermal vibrations ofa lattice atom with mass M as having a mean kineticenergy KE of .1=2/Ma2!2, where a and ! are theamplitude and frequency of the vibrations. This meanKE must be of the order of kT so that the ampli-tude a / T1=2. Thus the electron scattering cross sectionS D  a2 / T . Since the mean speed of conduction elec-tron in a metal is the Fermi speed and is temperatureinsensitive, �/ � / 1=S / T�1, and hence the resis-tivity � / T . Most nonmagnetic pure metals obey thisrelationship except at very low temperatures. Figure 2.4shows the resistivity of Cu as a function of temperaturewhere above � 100K, � / T .

The � / T behavior can also be derived more rigor-ously by noting that at high temperatures, the phononconcentration nph increases as T . The mean scatteringtime � is inversely proportional to nph so that � / T .Frequently, the resistivity versus temperature behav-ior of pure metals can be empirically represented bya power law of the form,

�D �0

�T

T0

�n

; (2.25)

where �0 is the resistivity at the reference tempera-ture, T0, and n is a characteristic index that best fitsthe data. For the nonmagnetic pure metals, n is closeto unity whereas it is close to 2 for the magnetic metalsFe and Ni [2.5]. Figure 2.5 shows � versus T for vari-ous metals. Table 2.1 summarizes the values �0 and nfor various metals.

100

10

1

0.1

0.01

0.001

0.0001

0.000011 10 100 1000 10000

Temperature (K)

3.5

3

2.5

2

1.5

1

0.5

00 20 40 60 80 100

T (K)

Resistivity (n m)

(n m)ρ

ρ

Tρ

T5ρ

= Rρ

T5ρ

Tρ

= Rρ ρ

Fig. 2.4 The resistivity of copper from low to high temperatures(near its melting temperature, 1358K) on a log–log plot. Aboveabout 100K, � / T , whereas at low temperatures, � / T5, and atthe lowest temperatures � approaches the residual resistivity �R.The inset shows the � versus T behavior below 100K on a linearplot. (�R is too small to see on this scale.) (After [2.5])

PartA|2.3

24 Part A Fundamental Properties

2000

1000

100

10100 500 1000 5000

Resistivity (nΩ m)

Temperature (K)

ρ ∝ T

Inconel-825

NiCr heating wire

Iron

Monel-400Tungsten

CopperSilverNickel

Platinum

AluminumGold

Fig. 2.5 The resistivities of various metals as a function oftemperature above 0 ıC. Tin melts at 505K, whereas nickeland iron go through a magnetic-to-nonmagnetic (Curie)transformation at about 627K and 1043 K, respectively.The theoretical behavior (� / T) is shown for reference.(After [2.5])

As apparent from Fig. 2.4, below � 100K, the � /T behavior fails, and � / T5. The reason is that, as thetemperature is lowered, the scattering by phonons be-comes less efficient, and it takes many more collisionsto fully randomize the initial velocity of the electron.The mean number of collisions � required to randomizethe velocity scales with T�2 [2.5], and at low tempera-tures, the concentration nphonon of phonons increases asT3. Thus,

� / �� / �

nphonon/ T�2

T3/ T�5 (2.26)

which explains the low-temperature ��T behavior inFig. 2.4. The low temperature � / T5 and high temper-ature � / T regimes are roughly separated by theDebyetemperature TD. For T > TD we expect � / T , and forT < TD we expect � / T5.

In the case of metals with impurities and for alloys,we need to include the �R contribution to the overall

Table 2.1 Typical resistivities at 273K (0 ıC) �0 and ther-mal coefficients of resistivity ˛0 at 0 ıC for various metalsabove 0 ıC and below melting temperature. The resistivityindex n in �D �0.T=T0/n is also shown. Note that n is fit-ted to resistivity data above 273K but below the meltingtemperature of the metal. Data selectively combined fromvarious sources, including [2.6, 7]

Metal �0 (n�m) n ˛0 �10�3 .K�1/

Aluminium, Al 24.2 1.20 4Antimony, Sb 302 1.27 4.7Beryllium, Be 30.2 1.73 8Bismuth, Bi 1070 – 4.5Cadmium, Cd 68 – 4.3Calcium, Ca 31.1 1.09 4.0Cerium, Ce 730 – 0.9Cesium, Cs 187 1.23 4.8Cromium, Cr 118 1.01 2.9Cobalt, Co 56 – 6.6Copper, Cu 15.4 1.16 4.3Gold, Au 20.5 1.13 4.0Hafnium, Ha 304 1.21 4.4Indium, In 80 1.31 5.0Iridium, Ir 47 – 4.5Iron, Fe 85.7 1.73 6.3Lead, Pb 192 1.14 4.2Magnesium, Mg 40.5 1.07 4.2Molybdenum, Mo 48.5 1.21 5.0Nickel, Ni 61.6 1.76 6.5Niobium, Nb 152 – 2.3Palladium, Pd 97.8 0.94 0.39Platinum, Pt 98.4 1.01 3.9Rhodium, Rh 43 – 4.4Ruthenium, Ru 71 – 4.1Silver, Ag 14.7 1.13 4.1Strontium, Sr 123 0.99 3.6Tantalum, Ta 122 0.93 3.6Tin, Sn 115 1.1 4.0Titanium, Ti 390 1.01 4.8Tungsten, W 48.2 1.24 4.8Vanadium, V 181 1.02 3.9Zinc, Zn 54.6 1.14 4.2Zirconium, Zr 388 1.00 4.2

resistivity. For T > TD, the overall resistivity is

�� AT C �R ; (2.27)

where A is a constant, and the AT term in (2.27) arisesfrom scattering from lattice vibrations. Normally, �Rhas very little temperature dependence, and hence veryroughly � versus T curves shift to higher values as �R isincreases due to the addition of impurities, alloying orcold working the sample (mechanical deformation thatgenerates dislocations) as illustrated for Cu–Ni alloysin Fig. 2.6.

Electrical Conduction in Metals and Semiconductors 2.3 Resistivity of Metals 25Part

A|2.3

Resistivity versus temperature behavior of nearly allmetals is characterized by the temperature coefficient ofresistivity (TCR) ˛0 which is defined as the fractionalchange in the resistivity per unit temperature increaseat the reference temperature T0, i. e.,

˛0 D 1

�0

�d�

dT

�

TDT0

; (2.28)

where �0 is the resistivity at the reference tempera-ture T0, usually at 273K (0 ıC) or 293K (20 ıC), d�D�� �0 is the change in the resistivity due to a small in-crease, dT D T �T0, in temperature. Assuming that ˛0is temperature independent over a small range from T0to T , we can integrate (2.28), which leads to the wellknown equation,

�D �0Œ1C˛0.T � T0/� : (2.29)

Equation (2.29) is actually only valid when ˛0 is con-stant over the temperature range of interest which re-quires (2.27) to hold. Over a limited temperature rangethis will usually be the case. Although it is not obviousfrom (2.28), we should, nonetheless, note that ˛0 de-pends on the reference temperature, T0 by virtue of �0depending on T0.

It is instructive to mention that if � � AT as we ex-pect for an ideal pure metal, then ˛0 D T�1

0 . If we takethe reference temperature T0 as 273K (0 ıC), then ˛0should ideally be 1/(273K) or 3:66�10�3 K�1. Exam-ination of a0 for various metals shows that � / T is nota bad approximation for some of the familiar pure met-als used as conductors, e.g., Cu, Al, Au, but fails badly

60

40

20

00 100 200 300

Resistivity (n m)Cu-3.32% Ni

Cu-2.16% NiCu-1.12% Ni(deformed)Cu-1.12% Ni

100% Cu(deformed)

100% Cu(annealed)

Temperature (K)

T

I

cw

Fig. 2.6 Resistivities of annealed and cold-worked (de-formed) copper containing various amounts of Ni (givenin atomic percentages) versus temperature

for others, such as indium, antimony and, in particular,the magnetic metals, e.g., iron and nickel.

Frequently we are given ˛0 at a temperature T0,and we wish to use some other reference temperature,say T 0

0, that is, we wish to use �0

0 and ˛0

0 for �0 and ˛0respectively in (2.29) by changing the reference fromT0 to T 0

0. Then we can find ˛1 from ˛0,

˛0

0 D ˛0

1C˛0.T 0

0 �T0/

and �D �0

0

�1C ˛0

0

�T � T 0

0

��: (2.30)

For example, for Cu ˛0 D 4:31�10�3 K�1 at T0 D0 ıC, but it is ˛0 D 3:96�10�3 K�1 at T0 D 20 ıC. Ta-ble 2.1 summarizes ˛0 for various metals. Note that ˛0in Table 2.1 is at 0 ıC and can be converted to ˛0

0 at20 ıC using (2.30).

2.3.2 Fermi Electrons

The electrical properties of metals depend on the behav-ior of the electrons at the Fermi surface. The electronstates at energies more than a few kT below EF arealmost fully occupied. The Pauli exclusion principle re-quires that an electron can only be scattered into anempty state, and thus scattering of deep electrons ishighly suppressed by the scarcity of empty states (scat-tering where the energy changes by more than a few kTis unlikely). Only the electrons near EF undergo scat-tering. Likewise, under the action of an external field,only the electron occupation near EF is altered. As a re-sult, the density of states (DOS) near the Fermi level ismost important for the metal electrical properties, andonly those electrons in a small range E around EF

actually contribute to electrical conduction. The den-sity of these electrons is approximately g.EF/ E whereg.EF/ is the DOS at the Fermi energy. From simplearguments, the overall conductivity can be shown tobe [2.5]

� D 1

3e2v 2

F�g.EF/ ; (2.31)

where vF is the Fermi speed and � is the scattering timeof these Fermi electrons. According to (2.31), what isimportant is the density of states at the Fermi energy,g.EF/. For example, Cu and Mg are metals with va-lencies I and II. Classically, Cu and Mg atoms eachcontribute 1 and 2 conduction electrons respectivelyinto the crystal. Thus, we would expect Mg to havehigher conductivity. However, the Fermi level in Mg iswhere the top tail of the 3p-band overlaps the bottomtail of the 3s band where the density of states is small.

PartA|2.4

26 Part A Fundamental Properties

In Cu, on the other hand, EF is nearly in the middle ofthe 4 s band where the density of states is high. Thus,Mg has a lower conductivity than Cu.

The scattering time � in (2.31) assumes that thescattered electrons at EF remain in the same energyband as, for example, in Cu, whose simplified energyband diagram around EF is shown in Fig. 2.7a. In cer-tain metals, there are two different energy bands thatoverlap at EF. For example, in Ni, 3d and 4s bandsoverlap at EF as shown in Fig. 2.7b. An electron canbe scattered from the 4s to the 3d band and vice versa.Electrons in the 3d band have very low drift mobili-ties and effectively do not contribute to conduction sothat only g.EF/ of the 4s band operates in (2.31). Since4s to 3d band is an additional scattering mechanism,by virtue of Matthiessen’s rule, the effective scatteringtime � for the 4s band electrons is shortened and hence� from (2.31) is smaller. Thus, Ni has poorer conduc-tivity than Cu.

Equation (2.31) does not assume a particular densityof states model. If we now apply the free electron modelfor g.EF/, and also relate EF to the total number of con-duction electrons per unit volume n, we would find thatthe conductivity is the same as that in the Drude model,

a)

b)

g(E)

4s

3d

Cu

EF Eg(E)

4s

3d

Ni

EF E

Fig. 2.7 Simplified energy band diagrams around EF for(a) copper and (b) nickel

that is

� D e2n�

me: (2.32)

2.4 Solid Solutions and Nordheim’s Rule

In an isomorphous alloy of two metals, i. e., a binaryalloy which forms a binary substitutional solid solu-tion, an additional mechanism of scattering appears, thescattering off solute phase atoms. This scattering con-tributes to lattice scattering, and therefore increases theoverall resistivity. An important semi-empirical equa-tion which can be used to predict the resistivity of analloy is Nordheim’s rule. It relates the impurity part ofthe resistivity �I to the atomic fraction X of solute atomsin a solid solution via

�I D CX.1�X/ ; (2.33)

where the constant C is termed the Nordheim coeffi-cient and represents the effectiveness of the solute atomin increasing the resistivity. Nordheim’s rule was orig-inally derived for crystals. Combining Nordheim’s rulewith Matthiessen’s rule (2.23), the resistivity of an alloyof composition X should follow

�D �matrix CCX.1�X/ ; (2.34)

where �matrix is the resistivity of the matrix due to scat-tering from thermal vibrations and from other defects,in the absence of alloying elements.

Nordheim’s rule assumes that the solid solution hasthe solute atoms randomly distributed in the lattice.For sufficiently small amounts of impurity, experimentsshow that the increase in the resistivity �I is nearlyalways simply proportional to the impurity concentra-tion X, that is, �I / X. For dilute solutions, Nordheim’srule predicts the same linear behavior, that is, �I D CXfor X � 1.

Originally the theoretical model for �I was de-veloped by Nordheim [2.8] by assuming that the so-lute atoms simply perturb the periodic potential andthereby increase the probability of scattering. Quantummechanical calculations for electron scattering withina single band, such as the s-band, at EF show that

�I / g.EF/V2scatterX.1�X/ ; (2.35)

where g.EF/ is the DOS at EF, and Vscatter is matrix ele-ment for scattering from one wavefunction to another atthe Fermi surface in the same band, which for an s-bandis

Vscatter D ˝ �

s j Vj s˛; (2.36)

where V is the difference between the potentials as-sociated with solvent and solute atoms, and s is the

Electrical Conduction in Metals and Semiconductors 2.4 Solid Solutions and Nordheim’s Rule 27Part

A|2.4

Table 2.2 Nordheim coefficients (at 20 ıC) for dilute al-loys obtained from �i D CX and X < 1 at.%. Note: Formany isomorphous alloys, C may be different at higherconcentrations; that is, it may depend on the compositionof the alloy [2.7, 9]. Maximum solubility data from [2.10]

Solute in solvent(element in matrix)

Nordheimcoefficient(n�m)

Maximum solubilityat 25 ıC(at.%)

Au in Cu matrix 5500 100Mn in Cu matrix 2900 24Ni in Cu matrix 1250 100Sn in Cu matrix 2900 0.6Zn in Cu matrix 300 30Cu in Au matrix 450 100Mn in Au matrix 2410 25Ni in Au matrix 790 100Sn in Au matrix 3360 5Zn in Au matrix 950 15

wavefunction of an electron in the s-band at EF. It isclear that C is only independent of X if g.EF/ and Vscatter

remain the same for various X which may not be true.For example, if the effective number of free electronsincreases with X, EF will be shifted higher, and C willnot be constant.

Table 2.2 lists some typical Nordheim coefficientsfor various additions to copper and gold. The value ofthe Nordheim coefficient depends on the type of soluteand the solvent. A solute atom that is drastically differ-ent in size to the solvent atom will result in a biggerincrease in �I and will therefore lead to a larger C. Animportant assumption in Nordheim’s rule in (2.33) isthat the alloying does not significantly vary the numberof conduction electrons per atom in the alloy. Althoughthis will be true for alloys with the same valency, thatis, from the same column in the Periodic Table (e.g.,Cu–Au, Ag–Au), it will not be true for alloys of differ-ent valency, such as Cu and Zn. In pure copper, there isjust one conduction electron per atom, whereas each Znatom can donate two conduction electrons. As the Zncontent in brass is increased, more conduction electronsbecome available per atom. Consequently, the resistiv-ity predicted by (2.34) at high Zn contents is greaterthan the actual value because C refers to dilute alloys.To get the correct resistivity from (2.34) we have tolower C, which is equivalent to using an effective Nord-heim coefficient Ceff that decreases as the Zn contentincreases. In other cases, for example, in Cu–Ni al-

160

140

120

100

80

60

40

20

00 10 20 30 40 50 60 70 80 90 100

Resistivity (nW m)

Composition (at.% Au)

Quenched

Annealed

Cu3Au CuAu

Fig. 2.8 Electrical resistivity versus composition at roomtemperature in Cu–Au alloys. The quenched sample(dashed curve) is obtained by quenching the liquid, andit has the Cu and Au atoms randomly mixed. The resistiv-ity obeys the Nordheim rule. On the other hand, when thequenched sample is annealed or the liquid slowly cooled(solid curve), certain compositions (Cu3Au and CuAu) re-sult in an ordered crystalline structure in which Cu and Auatoms are positioned in an ordered fashion in the crystaland the scattering effect is reduced

loys, we have to increase C at high Ni concentrations toaccount for additional electron scattering mechanismsthat develop with Ni addition. Nonetheless, the Nord-heim rule is still useful for predicting the resistivities ofdilute alloys, particularly in the low-concentration re-gion.

In some solid solutions, at some concentrations ofcertain binary alloys, such as 75% Cu–25% Au and50% Cu–50% Au, the annealed solid has an orderlystructure; that is, the Cu and Au atoms are not ran-domly mixed, but occupy regular sites. In fact, thesecompositions can be viewed as a pure compound – likethe solids Cu3Au and CuAu. The resistivities of Cu3Auand CuAu will therefore be less than the same composi-tion random alloy that has been quenched from the melt.As a consequence, the resistivity � versus compositionX curve does not follow the dashed parabolic curvethroughout; rather, it exhibits sharp falls at these specialcompositions, as illustrated in Fig. 2.8. The effectivemedia approximation may be used as an effective toolto estimate the resistivities of inhomogeneous media.

PartA|2.5

28 Part A Fundamental Properties

2.5 Carrier Scattering in Semiconductors

At low electric fields, ionized impurity scattering andphonon scattering predominate. Other types of scatter-ing include carrier-carrier scattering, inter-valley scat-tering, and neutral impurity scattering; these may gen-erally be ignored to a first approximation.

For phonon scattering, both polar and non-polarphonon scattering should be considered. In polar scat-tering, short wavelength oscillations of atoms on dif-ferent sub-lattices vibrating out of phase produce aneffective dipole moment proportional to the bond po-larity. Since such vibrational modes are optically active(since this dipole moment can interact with an incidentelectromagnetic field), this type of lattice scattering isusually referred to as polar optical phonon scattering.Since a sub-lattice is necessary for optical modes, thisscattering mechanism is not present in elemental semi-conductors such as Si, Ge, or diamond.

Non-polar phonon scattering comes from longwavelength oscillations in the crystal, involving smalldisplacements of tens to thousands of atoms. The wave-length depends on the material and its elastic properties.Such modes are very similar to sound vibrations andare thus referred to as acoustic modes. The associatedatomic displacements correspond to an effective built-instrain, with local change in the lattice potential, caus-ing carrier scattering known as deformation potentialacoustic phonon scattering. Since the change in po-tential is relatively small, the scattering efficiency isrelatively low as compared with polar optical phononscattering.

Each scattering process contributes to the drift mo-bility according toMatthiessen’s rule

1

�eDX

i

1

�i(2.37)

as discussed in Sect. 2.3 above. Figure 2.9 showsthe contributions of each scattering process for n-typeZnSe – a material that has applications in optoelec-tronics. At room temperature, polar optical phononscattering and ionized impurity scattering dominate.These processes depend on the carrier concentration.The curve for ionized impurity scattering decreasesmarkedly with increasing carrier concentration owingto the increasing concentration of ionized donors thatsupply these carriers. The ionized impurity scatteringmobility is roughly inversely proportional to the con-centration of ionized impurities. However, as the carrierconcentration increases, as in a degenerate semicon-ductor, the average energy per carrier also increases(i. e., carriers move faster on average) and thus car-riers are less susceptible to being scattered from theionized impurity centers. In contrast, the polar phonon

scattering rate is determined by the number of partici-pating phonons which depends on the thermal energyavailable to create a given quantized vibrational mode.

The temperature dependence of the mobility maybe estimated by considering the effect of temperatureon ionized impurity and phonon scattering and combin-ing these mechanisms using Matthiessen’s rule. Phonon

106

105

104

103

1014 1015 1016 1017 1018 1019

Carrier concentration n (cm–3)

Electron mobility μe(cm2/Vs)

Acoustic phonons

Impurity scattering

Piezoelectricphonons

Polar opticalphononsTotal

scattering

Temperature = 77K

Fig. 2.9 Dependence of electron mobility on carrier con-centration for ZnSe at 77 K. (After [2.11])

101 102 103

107

106

105

104

103

102

101

100

16 cm–3

Electron mobility μe(cm2/V s)

Acousticphonons

Impurityscattering (**)

Piezoelectric phonons

Temperature (K)

Polar-opticalphonons

Impurityscattering (*)

Inh. limit

Total mobility

16 cm–3

17 cm–3

17 cm–3

18 cm–3

18 cm–3

19 cm–3

NI

Fig. 2.10 Dependence of electron mobility on temperatureand doping for ZnSe. (After [2.11])

Electrical Conduction in Metals and Semiconductors 2.6 The Boltzmann Transport Equation 29Part

A|2.6

scattering increases strongly with increasing tempera-ture T due to the increase in the number of phononsresulting in a T�3=2 dependence for the polar phononmobility. For ionized impurity scattering, increasingthe temperature increases the average carrier velocityand hence increases the carrier mobility for a set con-centration of ionized impurities. Once a temperatureis reached such that impurity ionization is complete,the ionized impurity based carrier mobility can be

shown to increase with temperature T as approximately,TC3=2. At low temperatures, the mobility is basicallydetermined by ionized impurity scattering and at hightemperatures by phonon scattering leading to a peakedcurve. Invoking the previous discussions for the depen-dence of the total mobility on carrier concentration, it isclear that the peak mobility will depend on the dopinglevel, and the peak location will shift to higher temper-atures with increased doping as shown in Fig. 2.10.

2.6 The Boltzmann Transport Equation

A more rigorous treatment of charge transport requiresa discussion of the Boltzmann Transport Equation. Theelectronic system is described by a distribution func-tion f .k; r; t/ defined in such a way that the number ofelectrons in a six-dimensional volume element d3kd3rat time t is given by 1

4 �3f .k; r; t/d3kd3r. In equilib-

rium, f .k; r; t/ depends only on energy and reduces tothe Fermi distribution f0 where the probability of oc-cupation of states with momenta Ck equals that forstates with �k, and f0.k/ is symmetrical about k D0, giving no net charge transport. If an external fieldacts on the system (i. e., non-equilibrium), the occupa-tion function f .k/ will become asymmetric in k-space.If this non-equilibrium distribution function f .k/ iscompletely specified and appropriate boundary condi-tions supplied, the electronic transport properties canbe completely determined by solving the steady stateBoltzmann transport equation [2.12]

v � rrf C Pk � rkf D�@f

@t

�

c; (2.38)

where:

1. v � rrf represents diffusion through a volume ele-ment d3r about the point r in phase space due toa gradient rrf

2. Pk�rkf represents drift through a volume element d3kabout the point k in phase space due to a gradientrkf (for example, „ Pk D e

�EC 1

cv �B�in the pres-

ence of electric and magnetic fields)3. .@f =@t/c is the collision term and accounts for the

scattering of electrons from a point k (for example,this may be due to lattice or ionized impurity scat-tering).

Equation (2.38) may be simplified by using the re-laxation time approximation

�@f

@t

�

cD f

�D � f � f0

�(2.39)

which is based on the assumption that for small changesin f carriers return to equilibrium in a characteristic

time � , dependent on the dominant scattering mech-anisms. Further simplifications of (2.38) using (2.39)applicable for low electric fields lead to a simple equa-tion connecting the mobility� to the average scatteringtime h�i

�Š eh�im�

: (2.40)

The details of calculations may be found in various ad-vanced textbooks, for example Bube [2.13], Blatt [2.4].The average scattering time may be calculated assum-ing a Maxwell–Boltzmann distribution function anda parabolic band

h�i D 2

3kBT

R1

0 �.E/E3=2e�E=kBTdER

1

0 E3=2e�E=kBTdE: (2.41)

Quantum mechanical perturbation theory can be usedto calculate the carrier scattering rate for different pro-cesses i, giving,

�i.E/D aE�˛ ; (2.42)

where a and ˛ are constants and E is the electron en-ergy. Substituting (2.42) into (2.41) gives

< �i >D 4a� .5=2�˛/3 1=2.kBT/˛

(2.43)

in terms of the gamma function � . Using this approach,the expressions for the mobility for the case of latticeand impurity scattering may be easily found

�L / 4e

m�

p9 kB

T�32 ; (2.44)

�I / 8ek3=2B

NIm�

p TC

32 ; (2.45)

where NI is the concentration of ionized impurities.

PartA|2.7

30 Part A Fundamental Properties

2.7 Resistivity of Thin Polycrystalline Films

Two new dominant scattering mechanismsmust be con-sidered in polycrystalline thin films – scattering bygrain boundaries and scattering at the surface. Thescattering by grain boundaries is schematically shownin Fig. 2.11. As a first approximation, the conductionelectron may be considered free within a grain, but be-comes scattered at the grain boundary. Its mean freepath `grains is therefore roughly equal to the averagegrain size d. If �D `crystal is the mean free path of theconduction electrons in the single crystal, then accord-ing to Matthiessen’s rule

1

`D 1

`crystalC 1

`grainsD 1

�C 1

d: (2.46)

The resistivity is inversely proportional to the mean freepath which means the resistivity of the single crystal�crystal / 1=� and resistivity of the polycrystalline sam-ple � / 1=`. Thus,

�

�crystalD 1C

��

d

�: (2.47)

a)

b)

Grain 1

Grain 2

Grain boundary

Fig. 2.11 (a) Grain boundaries cause electron scatter-ing and therefore add to the resistivity according toMatthiessen’s rule. (b) For a very grainy solid, the elec-tron is scattered from grain boundary to grain boundary,and the mean free path is approximately equal to the meangrain diameter

Figure 2.12 clearly demonstrates that even simple con-siderations agree well with experimental data. However,in a more rigorous theory we have to consider a num-ber of effects. It may take more than one scattering ata grain boundary to totally randomize the velocity sothat we need to calculate the effective mean free paththat accounts for how many collisions are needed torandomize the velocity. There is a possibility that theelectron may be totally reflected back at a grain bound-ary (bounce back). Let � be the conductivity of thepolycrystalline (grainy) material and �crystal be the con-ductivity of the bulk single crystal. Suppose that theprobability of reflection at a grain boundary is R. If d isthe average grain size (diameter) then the two conduc-tivities may be related using the Mayadas and Shatzkesformula [2.15]

�

�crystalD 1� 3

2ˇC 3ˇ2 � 3ˇ3 ln

�1C 1

ˇ

�

where ˇ D �

d

�R

1�R

�;

(2.48)

The ˇ in (2.48) represents the �=d ratio adjusted forthe reflection to transmission ratio of the electron at thegrain boundary. When the grain size is large, ˇ is small

35

30

25

20

150 0.005 0.01 0.015 0.02 0.025

1/d (1/nm)

Resistivity (nΩ m) Grain size decreases

Fig. 2.12 Resistivity of Cu polycrystalline film versus re-ciprocal mean grain size (diameter), 1=d. Film thick-ness D D 250�900 nm does not affect the resistivity.The straight line �D 17:8 n�mC.595 n� nm/.1=d/. (Af-ter [2.14])

Electrical Conduction in Metals and Semiconductors 2.7 Resistivity of Thin Polycrystalline Films 31Part

A|2.7

and (2.48) simples to

�

�crystal� 1C 3

2ˇ : (2.49)

For highly polycrystalline films, the grain size would besmall and ˇ � 1

�

�crystal� 4

3ˇ : (2.50)

Equation (2.49) implies that the Matthiessen’s rule isreasonably well obeyed when the grains are largerthan the mean free path. For copper, typically R val-ues are 0:24�0:40, and R is somewhat smaller forAl. Equation (2.48) for a Cu film with R � 0:3 pre-dicts �=�crystal � 1:21 for roughly d � 3� or a grainsize d � 120 nm since in the bulk crystal �� 40 nm.Tellier et al. [2.16] have given extensive discussionsof grain boundary scattering limited resistivity of thinfilms.

Scattering from the film surfaces must also be in-cluded in any resistivity calculation. It is generallyassumed that the scattering from a surface is partiallyinelastic, that is the electron loses some of the velocitygained from the field. The inelastic scattering is alsocalled nonspecular. (If the electron is elastically re-flected from the surface just like a rubber ball bouncingfrom a wall, then there is no increase in the resistivity.)If the parameter p is the fraction of surface collisionswhich are specular (elastic) and if the thickness offilm D is greater than �, and �bulk D 1=�bulk, than inaccordance with Fuchs-Sondheimer equation [2.17, 18]the conductivity � of the film is

�

�bulkD 1� 3�

8D.1� p/ ; .�=D> 1/ : (2.51)

If D is much shorter than `,

�

�bulkD 3D

4�

�ln

��

D

�C 0:423

� .1C 2p/ ; .�=D � 1/ : (2.52)

For purely nonspecular (inelastic) scattering, an ap-proximate estimate can be obtained from

�

�bulk� 1C 3

8

��

D

�: (2.53)

Figure 2.13 shows the resistivity of Cu films as a func-tion of film thickness D. The Cu thin films in this caseare single crystal layers grown on the (001) surface ofa single crystal of MgO (which is the substrate). As

35

30

25

20

150 0.01 0.02 0.03 0.04 0.05

1/D (1/nm)

Resistivity (nΩ m) Thickness decreases

Fig. 2.13 The resisitivity �film of single crystal thin filmsof Cu against reciprocal film thickness 1=D at 25 ıC.The films are grown on the surface of a single crystalof MgO and the best straight line is �film D 17:0 n�m C.200 n�mnm/.1=D/. (Data extracted from [2.19])

the film is a single crystal, there is no grain boundaryscattering, and the observed increase in the resistivitywith decreasing film thickness is due to the scatteringof the electrons from the film surfaces. The experimentsin Fig. 2.13 can be explained by the simplified Fuchs–Sondheimer equation with an average p D 0:20.

For elastic or specular scattering p D 1 and thereis no change in the conductivity. The value of p de-pends on the film preparation method (e.g., sputtering,epitaxial growth etc.) and the substrate on which thefilm has been deposited. Table 2.3 summarizes the re-sistivities of thin Cu and Au gold films deposited byvarious preparation techniques. Notice the large differ-ence between the Au films deposited on a noncrystallineglass substrate and on a crystalline mica substrate.Such differences between films are typically attributedto different values of p. The p-value can also change(increase) when the film is annealed. Obviously, thepolycrystallinity of the film will also affect the resis-tivity as discussed above. Typically, most epitaxial thinfilms, unless very thin (D � `), deposited onto heatedcrystalline substrates exhibit highly specular scatteringwith p D 0:8� 1.

It is generally very difficult to separate the effectsof surface and grain boundary scattering in thin poly-crystalline films; the contribution from grain boundaryscattering is likely to exceed that from the surfaces. Inany event, both contributions, by Matthiessen’s generalrule, increase the overall resistivity. Figure 2.12 showsan example in which the resistivity �film of thin Cu poly-crystalline films is due to grain boundary scattering, andthickness has no effect (D was 250�900 nm and muchgreater than �). The resistivity �film is plotted against thereciprocal mean grain size 1=d, which then follows the

PartA|2.8

32 Part A Fundamental Properties

Table 2.3 Resistivities of some thin Cu and Au films at room temperature. PC: Polycrystalline film; RT is room tem-perature; D = film thickness; d = average grain size. FS and MS refer to Fuchs–Sondheimer and Mayadas–Shatzkesdescriptions of thin film resistivity. At RT for Cu, �D 38�40 nm, and for Au �D 36�38 nm. Data selectively combinedfrom various sources [2.14, 19–22, 24–27]

Film D (nm) d (nm) � (n�m) CommentCu filmsCu encapsulated inSiO2/Cu/SiO2 [2.20]

45.331.7

10141

28.035.5

DC sputtering. Cu film sandwiched in SiO2/Cu/SiO2, an-nealed at 150 ıC. MS with R D 0:50

Cu encapsulated inSiO2/Ta/Cu/Ta/SiO2 [2.21]

34.2 39.4 37.3 DC sputtering. Cu film sandwiched inSiO2/Ta/Cu/Ta/SiO2,annealed at 600 ıC. MS with R D 0:47

Cu on Ta/SiO2/Si(001) [2.21] 35 40 31 Sputtering and then annealing at 350 ıCCu (single crystal) onTiN/MgO (001) [2.22]

4013

11

2129.7

Cu single crystal epitaxial layer on TiN(100) on MgO surface(001). Ultra-high vacuum, DC sputtering. FS with p � 0:6 invacuum, p D 0 in air.

Cu on TiN, W and TiW [2.14] > 250 18644

2131

CVD (chemical vapor deposition). Substrate temperature200 ıC, � depends on d not D D 250�900 nm. MS.

Cu on crystalline Si (100)surface [2.23]

51.217.28.6

D=2:3 32.270.5126

Ion beam deposition with negative substrate bias. Resistivityfollows FS and MS equations combined; surface and grainboundary scattering. FS and MS, p D 0, R D 0:24, d D D=2:3

Au filmsAu epitaxial film on mica 30 25 Single crystal on mica. p � 0:8, highly specular scatteringAu PC film on mica 30 54 PC. Sputtered on mica. p is smallAu film on glass 30 70 PC. Evaporated onto glass. p is small, nonspecular scatteringAu on glass [2.24] 40

408.53.8

92189

PC. Sputtered films. R D 0:27�0:33

expected linear behavior in (2.49). On the other hand,Fig. 2.13 shows the resistivity of Cu films as a func-tion of film thickness D. In this case, the thin films aregrown epitaxially (as a single crystal) on MgO single

crystal surfaces and the grain boundary scattering is ab-sent, which is a clear cut case for surface scattering; thetransport is described by (2.51). Chapter 28 providesa more advanced treatment of conduction in thin films.

2.8 Inhomogeneous Media: Effective Media Approximation

The effective media approximation (EMA) attempts toestimate the properties of inhomogeneous mixture oftwo or more components using the known physicalproperties of each component. The general idea of anyEMA is to substitute for the original inhomogeneousmixture some imaginary homogeneous substance – theeffective medium (EM) – whose response to an ex-ternal excitation is the same as that of the originalmixture. The EMA is widely used for investigationsof non-uniform objects in a variety of applicationssuch as composite materials [2.28, 29], microcrystallineand amorphous semiconductors [2.30–33], light scatter-ing [2.34], conductivity of dispersed ionic semiconduc-tors [2.35] and many others.

Calculations of the conductivity and dielectricconstant of two component mixtures are reviewedby Reynolds and Hough [2.36] and summarized byRossiter [2.1]. For such a mixture we assume that thetwo components ˛ and ˇ are randomly distributed inspace with volume fractions of �˛ and �ˇ D 1��˛ .

The dielectric properties are described by an effectivepermittivity "eff given by the ratio

"eff D hDihEi ; (2.54)

where hEi is the average electric field and hDi is theaverage displacement field. The displacement field av-eraged over a large volume may be calculated from

hDi D 1

V

Z

V

Ddv D 1

V

0

B@Z

V˛

Ddv CZ

Vˇ

Ddv

1

CA

D �˛ hD˛i C�ˇ hDˇi ;(2.55)

where hD˛i and hDˇi are the average displacementfields inside regions of the respective components andV˛ and Vˇ are their volumes. Likewise the electric field

Electrical Conduction in Metals and Semiconductors 2.8 Inhomogeneous Media: Effective Media Approximation 33Part

A|2.8

is given by

hEi D �˛ hE˛i C�ˇ hEˇi : (2.56)

From (2.54) one gets

"eff D "ˇ C �"˛ � "ˇ

��˛f˛ (2.57)

or

."eff � "˛/ �˛f˛ C �"eff � "ˇ

��ˇfˇ D 0 (2.58)

where "˛ and "ˇ are the permittivities of the com-ponents and f˛ D hE˛i = hEi and fˇ D hEˇi = hEi areso-called field factors. The choice between (2.57) and(2.58) depends on particle geometry. Equation (2.57) isbetter when the particles of component are dispersed ina continuous medium ˇ. Equation (2.58) is preferredwhen the particle size of the two components is of thesame order of magnitude.

The field factors can be calculated analytically onlyfor phase regions with special specific geometries. Thefield factor for ellipsoids is given by (Stratton [2.48])

f˛ D3X

iD1

cos2 ˛i1CAi

�"˛"�

� 1� ; (2.59)

where ˛i are the angles between the ellipsoid axes andthe applied field and the constants Ai depend on the ax-

Table 2.4 Mixture rules for randomly oriented particles. (Compiled from Reynolds and Hough [2.36])

Particle shape Mixture rule Factors in (2.59) ReferencesA "�

Spheres"eff � "ˇ

"eff C 2"ˇD �˛

"˛ � "ˇ

"˛ C 2"ˇ13 "ˇ [2.37–41]

Spheres"eff � "ˇ

3"ˇD �˛

"˛ � "ˇ

"˛ C 2"ˇ13 "ˇ [2.42]

Spheres �˛"˛ � "eff

"˛ C 2"effC�ˇ

"ˇ � "eff

"ˇ C 2"effD 0 1

3 "eff [2.43]

Spheres"eff � "ˇ

3"effD �˛

"˛ � "ˇ

"˛ C 2"ˇ13 "eff [2.44]

Spheroids "eff D "ˇ C �˛

3 .1��˛/

3X

iD1

"˛ � "eff

1CAi

"˛"ˇ

� 1� A "ˇ [2.45]

Spheroids "eff D "ˇ C �˛

3

3X

iD1

"˛ � "ˇ

1CAi

"˛"eff

� 1� A "eff [2.46]

Lamellae "2eff D 2�"˛�˛ C "ˇ�ˇ

�� "eff"˛�˛

C "ˇ�ˇ

0 "eff [2.43]

Rods 5"3eff C5"0

p � 4"p�"2eff �

�˛"

2˛ C 4"˛"ˇ C�ˇ"

2ˇ

�� "˛"ˇ"p D 0

where1

"0

pD �˛

"ˇC �ˇ

"˛and

1

"pD �˛

"˛C �ˇ

"ˇ

12 [2.47]

ial ratios of the ellipsoids; they must satisfy

3X

iD1

Ai D 1 :

For a spheroid, A2 D A3 D A and A1 D 1� 2A. Fora random orientation of spheroids, cos2 ˛1 D cos2 ˛2 Dcos2 ˛3 D 1

3 . In the case of long particles with alignedaxes, cos2 ˛1 D cos2 ˛2 D 1

2 and cos2 ˛3 D 0. The val-ues of parameters entering (2.59) can be found inTable 2.4 which shows a set of mixture rules, i. e., a setof formulae allowing one to calculate "eff for some spe-cific cases (such as spheres, rods, lamellae, etc.). Thepresence of some degree of orientation somewhat sim-plifies the calculations as shown in the Table 2.5.

The same formulae can be used to calculate theconductivity of mixtures by substituting the appropri-ate conductivity � for ". For some special cases, themixture rules of Table 2.5 lead to very simple formulaewhich allows one to calculate the conductivity of inho-mogeneous alloys with those specific geometries. Thesemixture rules are summarized in Table 2.6.

The most general approach to calculating the effec-tive dielectric permittivity comes from

"eff D "ˇ

0

@1��˛1Z

0

G.L/

t� LdL

1

A ; (2.60)

PartA|2.8

34 Part A Fundamental Properties

Table 2.5 Mixture rules for partially oriented particles. (Compiled from Reynolds and Hough [2.36])

Particle shape Formula Factors in (2.59) ReferencesA "� cos˛1 D cos˛2 cos˛3

Parallel cylinders"eff � "ˇ

"eff C "ˇD �˛

"˛ � "ˇ

"˛ C "ˇ

12 "ˇ

12 0 [2.40, 41]

Parallel cylinders �˛"˛ � "eff

"˛ C "effC�ˇ

"ˇ � "eff"ˇ C "eff

12 "eff

12 0 [2.43]

Parallel lamellae(with two axes randomlyoriented)

"2eff D "˛�˛ C "ˇ�ˇ"˛�˛

C "ˇ�ˇ

0 "eff12 0 [2.43]

Lamellae with all axes aligned(current lines are perpendicularto lamellae planes)

1

"effD �˛

"˛C �ˇ

"ˇ0 "eff 0 1 [2.49]

Lamellae with all axes aligned(current lines are parallel tolamellae planes)

"eff D "˛�˛ C "ˇ�ˇ 0 "eff 1 0 [2.49, 50]

Spheroids with all axes aligned(current lines are parallel to oneof the axes)

"eff D "ˇ C �˛�"˛ � "ˇ

�

1CA"˛"ˇ

� 1� A "ˇ 0 1 [2.51]

Spheroids with all axes aligned(current lines are parallel to oneof the axes)

"eff

"ˇD 1C �˛

"˛"ˇ

� 1�

�1 CA�˛

A "ˇ 0 1 [2.52]

Table 2.6 Conductivity/resistivity mixture rules. (After [2.5])

Particle shape Formula CommentaryLamellae with all axes aligned(current lines are perpendicular to lamellaeplanes)

�eff D �˛�˛ C�ˇ�ˇ Resistivity mixture rule: �˛ and �ˇ are the resistivitiesof two phases and �eff is the effective resistivity ofmixture

Lamellae with all axes aligned(current lines are parallel to lamellaeplanes)

�eff D �˛�˛ C�ˇ�ˇ Conductivity mixture rule: �˛ and �ˇ are the con-ductivities of two phases and �eff is the effectiveconductivity of mixture

Small spheroids (˛-phase) in medium(ˇ-phase)

�eff D �ˇ

�1C 1

2�˛�

.1��˛/�˛ > 10�ˇ

Small spheroids (˛-phase) in medium(ˇ-phase)

�eff D �ˇ.1��˛/

.1C 2�˛/�˛ < 0:1�ˇ

Table 2.7 Mixture rules and corresponding spectral functions G.L/

Mixture rule by Bruggemann [2.43]: �˛"˛ � "eff

"˛ C 2"effC�ˇ

"ˇ � "eff

"ˇ C 2"effD 0

G.L/D 3�˛ � 1

2�˛ı.L/�.3�˛ � 1/C 3

4 �˛L

q.L� L�/

�LC � L

��.L� L�/�.LC � L/

where LC=� D 1

3

�1C�˛ ˙ 2

q2�˛ � 2�2˛

�

Mixture rule by Maxwell-Garnett [2.53]:"eff � "ˇ

"eff C 2"ˇD �˛

"˛ � "ˇ

"˛ C 2"ˇ

G.L/D ı

�L� 1��˛

3

�

Electrical Conduction in Metals and Semiconductors 2.9 The Hall Effect 35Part

A|2.9

Table 2.7 (continued)

Mixture rule by Looyenga [2.54]: "1=3eff D �˛"1=3˛ C�ˇ"

1=3ˇ

G.L/D �2˛ı.L/C 3p3

2 

�2ˇ

ˇˇ̌ˇL� 1

L

ˇˇ̌ˇ

1=3

C�˛�ˇ

ˇˇ̌ˇL� 1

L

ˇˇ̌ˇ

2=3!

Mixture rule by Monecke [2.55]: "eff D 2��˛"˛ C�ˇ"ˇ

�2 C "˛"ˇ

.1C�˛/ "˛ C .2��˛/ "ˇ

G.L/D 2�˛1C�˛

ı.L/C 1��˛

1C�˛ı

�L� 1C�˛

3

�

Mixture rule for hollow sphere equivalent by Bohren and Huffman [2.56]:

"eff D "˛.3� 2f / "ˇ C 2f "˛f "ˇ C .3� f / "˛

G.L/D 2

3� fı.L/C 1� f

3� fı

�L� 3� f

3

�

where f D 1� r3ir3o

and ri=o is the inner/outer radius of the sphere

where t D "ˇ=."ˇ � "˛/ and G.L/ is the spectral func-tion which describes the geometry of particles in termsof the depolarization factor L. Equation (2.60) is com-monly known as the Bergman representation [2.57] andrepresents a general approach to modeling a heteroge-neous mixture in terms of effective properties [2.58].The advantage of the spectral representation is thatit distinguishes between the influence of geometri-

cal quantities and that of the dielectric properties ofthe components on the effective behavior of the sys-tem. Although the spectral function G.L/ is gener-ally unknown for an arbitrary two-phase composite,it’s analytically known or can be numerically derivedfor any existing mixture rule. Examples of spectralfunctions and corresponding solutions are shown inTable 2.7.

2.9 The Hall Effect

The Hall effect is closely related to the phenomenon ofconductivity and is observed as the occurrence of a volt-age appearing across a conductor carrying an electriccurrent in a magnetic field. The schematic of the ex-periment is shown in the Fig. 2.14. The effect is oftencharacterized by the Hall coefficient

RH D Ey

JxBz; (2.61)

where Ey is the Hall effect electric field in the y-direction, Jx is the current density in the x-direction andBz is magnetic field in the z-direction.

The Hall effect for ambipolar conduction in a sam-ple where there are both negative and positive chargecarriers, e.g., electrons and holes in a semiconduc-tor, involves not only the concentrations of electronsand holes, n and p respectively, but also the elec-tron and hole drift mobilities, �e and �h. In the firstapproximation, the Hall coefficient can be calculatedin the following way. Both charge carriers experiencea Lorentz force in the same direction since they are

drifting in the opposite directions but of course haveopposite signs as illustrated in Fig. 2.14. Thus, both

– –

+ + ––

Jy = 0

JxJx

Bz

Bz

y

xzeEy

Ey

Ex

eEy evexBzevhxBz

A

V

vexvhx

+ +

+ +– –

+ –

Fig. 2.14 Hall effect for ambipolar conduction. The mag-netic field Bz is out of the plane of the paper. Both electronsand holes are deflected toward the bottom surface of theconductor and so the Hall voltage depends on the relativemobilities and concentrations of electrons and holes

PartA|2.9

36 Part A Fundamental Properties

holes and electrons accumulate near the bottom surface.The magnitude of the Lorentz force, however, will bedifferent since the drift mobilities and hence drift ve-locities will be different. Once equilibrium is reached,there should be no current flowing in the y-direction aswe have an open circuit. The latter physical argumentslead to the following Hall coefficient [2.5],

RH D p�2h � n�2

e

e .p�h C n�e/2

or RH D p� nb2

e .pC nb/2; (2.62)

where b D �e=�h.It is clear that the Hall coefficient depends on both

the drift mobility ratio and the concentrations of holesand electrons. For p > nb2, RH will be positive and forp< nb2, it will be negative. Note that the carrier con-centration is not zero when the Hall coefficient is zerobut rather n=p D .�h=�e/

2. As an example, Fig. 2.15shows the dependence of Hall coefficient versus elec-tron concentration for a single crystal silicon. The cal-culations are based on (2.62) and the law of mass action

np D n2i ; (2.63)

where ni is the electron concentration in the intrinsicsemiconductor.

In the case of monopolar conduction, e.g., con-duction in metals or in doped semiconductors, (2.62)reduces to

RH D � 1

en; .for n � p/ (2.64)

or

RH D 1

e p; .for p � n/ : (2.65)

Therefore, (considering as an example a n-type semi-conductor where � D ne�n) one can write

�H D �

neD ��RH (2.66)

which provides a simple expression for determining theelectron mobility known as the Hall mobility. Note,however, that the Hall mobility may differ from the driftmobility discussed in the previous sections. The differ-ence arises from the carriers in a semiconductor having

0.2

0.1

0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.60.01 0.1 0

(eni)RH

n /ni

1.14

1/3

0.17

1 1

Fig. 2.15 Normalized Hall coefficient versus normalizedelectron concentration plot for single-crystal silicon. Thevalues 0:17, 1:14 and 0:33 shown are the n=ni valueswhen the magnitude R reaches maxima and zero respec-tively. In single-crystal silicon, ni D 1:5�1010 cm�3,�e D1350 cm2V�1s�1 and �h D 450 cm2V�1s�1

a distribution of energies. An average is used to describethe effect of carriers occupying different allowed states.(This is distinct from the earlier discussions of mobilitywhere it was assumed that all the carriers have the samemean free time between collisions.) To include this, it isnecessary to use a formal analysis based on the Boltz-mann transport equation, as discussed in Sect. 2.6. Ifwe express the averaging of the carriers with energy Eand distribution function f .E/, we can write the energy-averaged �.E/ (i. e., h�i) as

h�i DR�.E/f .E/dERf .E/dE

(2.67)

and the energy-averaged �2 (i. e., h�2i) as

h�i DR�2.E/f .E/dERf .E/dE

: (2.68)

The rigorous analysis [2.12] shows that the Hall mobil-ity �H in terms of the drift mobility �d is

�H D rH ��d ; (2.69)

where rH is the Hall factor given by the ratio

rH D h�2ih�i2 : (2.70)

The magnitude of the Hall factor will depend on themagnitude of the scattering mechanisms that contributeto � , but usually at low magnetic fields, rH � 1 and thetwo mobilities are identical.

Electrical Conduction in Metals and Semiconductors 2.10 High Electric Field Transport 37Part

A|2.10

2.10 High Electric Field Transport

The previous sections focused on carrier transport inweak electric fields where the energy gained by carri-ers from the field is lost to the lattice through collisionswith phonons or ionized impurities. However at higherfields, the efficiency of collision mechanisms dimin-ishes and the carrier system contains more energy thanthe lattice. The carriers are then called hot with effec-tive temperatures Te for electrons and Th for holes. Inthis case, the drift velocity no longer obeys Ohm’s law,and becomes non-linear in the applied field with a cleartendency to saturation due to the appearance of a newdissipation mechanism involving optical phonon gen-eration. Figure 2.16 shows the drift velocity saturationfor both electrons and holes in Si and electrons in GaAs;GaAs shows a region of electron velocity overshoot andthen negative differential resistivity due to inter-valleyscattering, i. e., the transfer of electrons from the �minimum to the L conduction band minimum.

Solving the Boltzmann transport equation by anal-ogy with (2.6) and (2.7) the electron (or hole) mobilitymay be defined as

�D e h�.Te/im�

; (2.71)

where e is elementary charge and m� is the electron (orhole) effective mass and h�.Te/i is the mean free time

108

107

106

105

102 103 104 105 106

Drift velocity (cm/s)

Electric field (V/cm)

GaAs (Electrons)

Si(Electrons)

Si (Holes)

GaAs (Holes)

T = 300K

Fig. 2.16 Dependence of carrier drift velocity on electricfield for GaAs and Si. (After [2.59])

which now strongly depends on Te. Therefore, the high-field mobility is related to the low field mobility �0 by

�D �0

�TeT

�ˇ

; (2.72)

where ˇ depends on electric field and scattering mech-anisms. Thus, in order to determine the dependence ofthe mobility on electric field, the dependence of theeffective carrier temperature on field is required. Thismay be found by using the time-dependent Boltzmanntransport equation. Suppose that F is the field, then fornon-degenerate conditions with Te � T

TeT

/ F�2

2ˇ�1 (2.73)

and hence

�.E// �0F�

22ˇ�1 : (2.74)

For acoustic phonon scattering, ˇ D �3=2 and the driftmobility shows no saturation, increasing with fieldas F1=4. Saturation in the drift velocity may be achievedonly when ˇ ! 1 due to optical phonon scatteringwhere large energy changes are involved. The satura-tion velocity vs (related to the saturation mobility byvs D F�s) may be calculated using the energy and mo-mentum rate equations for optical phonon scattering

d hEidt

D eFvs � Eop

�e(2.75)

d hm�vsidt

D eF � m�vs�m

(2.76)

where Eop is the optical phonon energy, �e and �m arethe energy and momentum relaxation times, respec-tively. At steady state and for not extremely high fields,one may assume that �e � �m. (It is worth noting that atthe highest electric fields �e > �m and may lead to theappearance of avalanche, as discussed in Sect. 2.11).Therefore, the solution of (2.75) and (2.76) is

vs D�Eop

m�

�1=2

: (2.77)

in agreement with the experimental values shownin Fig. 2.16.

PartA|2.11

38 Part A Fundamental Properties

2.11 Impact Ionization

At very high electric field (in the range 2�105 V=cm orlarger) a new possibility appears: a carrier may have ki-netic energy in excess of the binding energy of a valenceelectron to its parent atom. In colliding with an atom,such a carrier can rupture the covalent bond and pro-duce an electron-hole pair. This process is called impactionization and is characterized by the impact ionizationcoefficient ˛ (˛e for electrons and ˛h for holes). Thereleased electrons and holes may, in turn, impact ion-ize more atoms producing new electron-hole pairs. Thisprocess is called avalanche and is characterized by themultiplication factor, M, which is the ratio of numberof collected carriers to the number of initially injectedcarriers.

The field dependence of the impact ionization coef-ficient, at least over the limited fields where avalancheis observed, is usually modeled by experimentalists byusing

˛ D A exp

���B

F

�n; (2.78)

where F is the field A, B, n are constants that depend onthe semiconductor material properties such as the E-kelectronic structure, phonon energies and spectra, scat-tering mechanisms and so on. The constant B has beensemiquantitatively argued to depend on Eg=� where Eg

is the bandgap and 1=� is the phonon scattering rate;higher bandgap semiconductors tend to have steeperslopes on log ˛ versus 1=F plots and the log ˛ versus1=F curve tends to shift to higher fields. Typically, it hasbeen found that n � 1 at low fields and n � 2 for highfields. Figure 2.17 shows experimental data for a varietyof materials over a wide range of electric fields.

Avalanche multiplication in crystalline semicon-ductors has been well studied over several decades andis now well developed and used in various commercialapplications, such as avalanche photodiodes in opticalcommunications, high sensitivity radiation detectors,and x-ray and nuclear detectors [2.69, 70]. In the caseof amorphous semiconductors, the present work pointsto only a-Se exhibiting avalanche multiplication in thebulk for reasons explained in reference [2.71], with re-ports dating back to the 1980s [2.61, 62, 72]. Avalanchein a-Se has also been commercialized in ultrasensitivevideo tubes [2.73], with potential application in medicalimaging [2.74].

The origin of (2.78) in its simple n D 1 form liesin Shockley’s [2.75] lucky electron model. When a car-rier moves a distance z down-stream (along the field)without being scattered, it gains an energy eFz. Anunlucky carrier is scattered so frequently that its eFz

5.5

4.5

3.5

2.5

1.5

0.50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Log (ionization coefficient)

1/F (cm/MV)

H: HolesE: Electrons

Si (1.1 eV)

Si (1.1 eV)

E

HH

InP (1.42 eV)

InP (1.42 eV)

EH

GaInP (2.0 eV)

a-Se (2.1 eV)

a-Se (2.1 eV)

H

E

EGaP

(2.27 eV)

AlGaAs (2.0 eV)

E

EGaN

(3.4 eV)

Fig. 2.17 Semilogarithmic plot of the dependence of theimpact ionization coefficient on the reciprocal field forvarious crystalline and amorphous semiconductors forcomparison; H indicates holes and E electrons. The y-axis is a base-10 logarithm of ˛ in which ˛ is in 1=cm.Data for a-Se electrons and holes from [2.60–62]; Datafor crystalline semiconductors are for GaP [2.63]; GaInP(Ga0:52In0:48P) [2.64]; Al0:60Ga0:40As [2.65]; InP [2.66];Si, [2.67]; GaN (calculation only) [2.68]

never reaches the threshold ionization energy EI forimpact ionization. On the other hand, a lucky electronis a ballistic electron that avoids scattering for a sub-stantial distance, and hence is able to build its eFz toreach EI and thereby cause impact ionization. If � isthe mean free path of collisions, then Shockley’s modelgives

˛ D eF

EIexp

�� EI

e�F

�: (2.79)

The main problemwith the Shockley model is that thereare just not enough ballistic electrons to cause sufficientimpact ionizations to explain the experiments. A bettermodel was developed by Baraff [2.76] who numericallysolved the Boltzmann transport equation for a simpleparabolic band and energy independent mean free pathto provide a relationship between ˛ and F in terms offour parameters, that is, threshold energy for impactionization, mean free path associated with ionizations,optical phonon energy, and mean free path for opticalphonon scattering. Baraff’s theory served experimental-

Electrical Conduction in Metals and Semiconductors 2.11 Impact Ionization 39Part

A|2.11

ists quite well in terms of comparing their results eventhough the model was not intuitive and was limited interms of its assumptions and applicability to real semi-conductors.

Impact ionization theory in crystalline solids onlyreached an acceptable level of confidence and under-standing in the 1980s and 1990s with the developmentof the lucky-drift model by Ridley [2.77] and its exten-sion by Burt [2.78], andMackenzie and Burt [2.79]. Thelatter major advancement in the theory appeared as thelucky drift (LD) model, and it was based on the re-alization that at high fields, hot electrons do not relaxmomentum and energy at the same rates. Momentumrelaxation rate is much faster than the energy relaxationrate. An electron can drift, being scattered by phonons,and have its momentum relaxed, which controls thedrift velocity, but it can still gain energy during thisdrift. Stated differently, the mean free path �E for en-ergy relaxation is much longer than the mean free path� for momentum relaxation.

In the Mackenzie and Burt [2.79] version of theLD model, the probability P.E/ that a carrier attains anenergy E is given by,

P.E/D exp

0

@�EZ

0

dE0

eF�.E0/

1

ACEZ

0

dE1

eF�.E1/

� exp

0

@�E1Z

0

dE0

eF�.E0/

1

A

� exp

0

@�EZ

E1

dE0

eF�E.E0/

1

A ;

(2.80)

where as mentioned above � is the mean free path asso-ciated with momentum relaxing collisions and �E is themean energy relaxation length associated with the en-ergy relaxing collisions. The first term is the Shockleylucky electron probability, i. e., the electron moves bal-listically to energy E. The second term is the lucky driftprobability term which is composed of the following:the electron first moves ballistically to some interme-diate energy E1 (0< E1 < E) from where it begins itslucky drift to energy EI; hence the integration over all

possible E1. The impact ionization coefficient can thenreadily be evaluated from

˛ D eFP.EI/R EI0 P.E/dE

: (2.81)

The model above is based on a hard threshold ionizationenergy EI, that is, when a carrier attains the energy EI,ionization ensues. The model has been further refinedby the inclusion of soft threshold energies which repre-sent the fact the ionization does not occur immediatelywhen the carrier attains the energy EI, and the carrierdrifts further to gain more energy than EI before impactionization [2.80–84].

Assuming � and �E are energy independent, whichwould be the case for a single parabolic band in thecrystalline state, (2.80) and (2.81) can be solved ana-lytically to obtain

˛ D 1

��

��E

exp

�EIeF�E

�C

��E

�2exp

��EIeF�

�

1� exp

�EIeF�E

��

��E

�2 �1� exp

��EIeF�

�� :

(2.82)

For �E > �, and in the low field region, where typi-cally (˛�) < 10�1, or x D EI=eF� > 10, (2.82) leads toa simple expression for ˛,

˛ D�

1

�E

�exp

�� EI

eF�E

�: (2.83)

For crystalline semiconductors, one typically also as-sumes that �E depends on the field F, � and the opticalphonon energy „! as

�E D eF�2

2„! coth

� „!2kT

�: (2.84)

As the field increases, �E eventually exceeds �, and al-lows lucky drift to operate and the LD carriers to reachthe ionization energy.

It is worth noting that the model of lucky drift issuccessfully used not only for crystalline semiconduc-tors but to amorphous semiconductors [2.83].

PartA|2.12

40 Part A Fundamental Properties

2.12 Two-Dimensional Electron Gas

Heterostructures offer the ability to spatially engineerthe potential in which carriers move. In such struc-tures having layers deposited in the z-direction, whenthe width of a region with confining potential tz < �dB,the de Broglie electron wavelength, electron states be-come stationary states in that direction, retaining Blochwave character in the other two directions (i. e., x-and y-directions), and is hence termed a 2-D electrongas (2DEG). These structures are notable for their ex-tremely high carrier mobility.

High mobility structures are formed by selectivelydoping the wide bandgap material behind an ini-tially undoped spacer region of width d as shown inFig. 2.18a. Ionization and charge transfer leads to car-

Nir

Nis

Nib

V0

0–L –d 0

E1

E0

EF

a) Impurity concentration

Distance zb) Energy

Distance z

S1S2

Fig. 2.18 (a) Doping profile for a selectively doped 2DEGheterostructure. The S1 is a narrow bandgap material ad-jacent to the heterointerface. The S2 is selectively dopedwide-bandgap material including an initially undopedspacer region of width d. (b) Conduction energy bandstructure for a selectively doped 2DEG heterostructure.The equilibrium band bending (through Poisson’s equa-tion) in the well region results from the equalization of theionized donor concentration in the wide-bandgap materialand the 2DEG concentration adjacent to the heterointer-face. When the associated interfacial field is sufficientlystrong, carriers are confined within �dB and electron statesare quantized into the sub-bands E0 and E1. (After [2.84])

rier build-up in the low potential region of narrowbandgap material adjacent to the hetero-interface. Theequilibrium band bending (i. e., through Poisson’s equa-tion) in the well region, as shown in Fig. 2.18b, resultsfrom the equalization of ionized donor concentrationin the wide bandgap material and 2DEG concentrationadjacent to the heterointerface. When the associated in-terfacial field is sufficiently strong, carriers are confinedwithin �dB and electron states are quantized into sub-bands (i. e., E0 and E1/, as shown in Fig. 2.18b.

Figure 2.19 shows the contributions of componentscattering mechanisms to the low temperature mobilityof a 2DEG formed at a Ga0:70Al0:3As-GaAs heteroint-erface, as a function of the electron gas density. As forbulk samples, the most important mechanism limitingthe low temperature mobility is ionized impurity scat-

109

108

107

106

105

1011 1012

cm2

V sMobility μe

Electron-gas density NS (cm–2)

Total (5K)

Total (77K)

GaAs–Ga1–xAlxAsx = 0.3Spacer width = 150Å

Alloy disorder

Deformation potential (5K)

Piezoelectric (5K) Absolute limit

Inherent limit (5K)Background impurity (5K)

Remote impurity

μe

107

NS

Fig. 2.19 Dependence of 2DEG electron mobility on car-rier concentration for a Ga0:70Al0:3As-GaAs heterostruc-ture. (After [2.84])

Electrical Conduction in Metals and Semiconductors 2.12 Two-Dimensional Electron Gas 41Part

A|2.12

107

106

105

104

1 10010

cm2

V sMobility μe

Temperature (K)

Ga0.7Al0.3As–GaAs

d = 200 Å

Background impurity

Opticalphonon

Deformationpotential + piezoelectric

Remote impurity

Inherent limit

Total

Absolute limit

Fig. 2.20 Dependence of 2DEG electron mobility on tem-perature for a Ga0:70Al0:3As-GaAs heterostructure. (Af-ter [2.84])

tering, except at high electron densities, where so-calledalloy disorder scattering is significant. Ionized impurityscattering may be further broken down into scatteringfrom ionized impurities that are with the GaAs quantumwell, known as background impurities, those beyondthe spacer region, termed remote impurity scattering.For high purity growth, the unintentional backgroundimpurity concentration can be kept to very low limitsand impurity scattering based mobility values are thendictated by remote impurity scattering. Since carriersin the well are only weakly scattered by the tail fieldof these remote Coulomb centers, the mobility of such2DEG systems can be orders of magnitude higher thanbulk samples. The temperature dependence of the elec-tron mobility for such a system is shown in Fig. 2.20.Notice how, similar to bulk sample, increasing tem-perature increases the phonon population such that for

108

107

106

105

0 200100 300

cm2

V sMobility μe

Spacer width (Å)

Backgroundimpurity

Remote impurity

Inherent limit

Total

Absolute limit

Phonons

Alloy disorder

Fig. 2.21 Dependence of 2DEG electron mobility onspacer width for a Ga0:70Al0:3As-GaAs heterostructure.(After [2.84])

the example shown, above about 100K, polar opticalphonon scattering controls the mobility.

Figure 2.21 shows the dependence of the 2DEGmobility on the spacer width. Two competing factorsare active – for narrow spacer widths, the transfer ef-ficiency of carriers to the GaAs well is high and soa lower remote doping concentration is sufficient to pro-vide for a given constant 2DEG concentration, but sincethe Coulomb scatters are so close to the 2DEG, theyscatter very efficiently and limit the mobility. On theother hand, for large spacer widths, carrier transfer ef-ficiency is quite poor requiring higher remote dopingto supply the given 2DEG concentration; however, be-ing relatively far away each of the scattering centers areless effective at lowering the mobility, but given theirhigh concentration, the net effect is still a decrease inthe mobility at large spacer widths as seen in the figure.

PartA|2.13

42 Part A Fundamental Properties

2.13 One-Dimensional Conductance

In the case where carriers are confined within regionsof width Lx, Ly < �dB in two directions x and y, re-spectively, the electron energy in those two directionsbecome quantized with quantum numbers nx and ny, re-spectively. In the third direction z, electrons travel asBloch waves with energy that may be approximated by„2k2z=2m

� giving an expression for the total energy Eof the so-called 1-D electron system or quantum wire as

E.nx; ny; nz/D „2 2

2m�

n2xL2x

C n2yL2y

!

C „2k2z2m�

: (2.85)

The associated electron wavefunctions are

�.x; y; z/D 1

2pLxLyLz

sin�nx x

Lx

�

� sin�ny y

Ly

�eikzz : (2.86)

Using these equations, one can readily derive an expres-sion for the density of states per unit energy range

DOS D 2� 2�Lz2 

� �rkzE�

�1

D 2Lzh

sm�

2.E�Enx;ny/: (2.87)

In order to evaluate the conductance of this quantumwire, consider the influence of a weak applied po-tential V. Similar to the case for bulk transport theapplied field displaces the Fermi surface and results ina change in the electron wave-vector from k0 (i. e., withno applied potential) to kV (i. e., when the potential isapplied). When V is small compared with the electronenergy

k0 Ds

2m�.E�Enx;ny/

„2; (2.88)

kV D k0

s

1C eV

E�Enx;ny

� k0

1C 1

2

eV

E�Enx;ny

!

: (2.89)

This leads to establishing a current density J in the wire

J D 2e2 .DOS/p.EF �Enx;ny/p

2m�

: (2.90)

Which may be simplified to the following expressionsfor J and the current flowing in the wire for a givenquantum state Efnx; nyg, I

Jnx;ny D 2e2VLzh

and Inx;ny D 2e2V

h: (2.91)

The expression for the conductance through one chan-nel corresponding to a given quantum state fnx; nyg isthen given by

Gnx;ny D Inx;nyV

D 2e2

h: (2.92)

Notice how the conductance is quantized in units ofe2=h with each populated channel contributing equallyto the conductance – moreover, this is a fundamentalresult, being independent of the material considered.In practice, deviations from this equation can occur(although generally less than 1%) owing to the finite-ness of real nanowires and impurities in or near thechannel, influencing the conductivity and even result-ing in weak localization. Generally, unlike both bulkand 2DEG systems, ionized impurity scattering is sup-pressed in nanowires. The main reason for this is thatan incident electron in a quantum state fnx; nyg travel-ing along the wire with wave-vector kzfnx; nyg, can notbe elastically scattered into any states except those ina small region of k-space in the vicinity of �kzfnx; nyg.Such a scattering event involves a large change in mo-mentum of � 2kzfnx; nyg and thus, the probability ofsuch events is very small. As a result, the mean freepath and mobility of carriers in such quantum wires aresubstantially increased.

The nature of carrier transport in quantum wires de-pends on the wire dimensions (i. e., length LWire anddiameter dWire) as compared with the carrier mean freepath, lCarrier. When lCarrier � LWire, dWire the only po-tential seen by the carriers is that associated with thewire walls, and carriers exhibit wavelike behavior, be-ing guided through the wire as if it were a waveguidewithout any internal scattering. Conversely, if dWire ��DeBroglie, only a few energy states in the wire are ac-tive, and in the limit of an extremely small waveguide,only one state or channel is active, analogous to a singlemode waveguide cavity – this case is termed quantumballistic transport. In the limit, lCarrier � LWire, dWire,scattering dominates transport throughout the wire –with numerous scattering events occurring before a car-rier can traverse the wire or move far along its length.In such a case the transport is said to be diffusive.

Electrical Conduction in Metals and Semiconductors 2.14 The Quantum Hall Effect 43Part

A|2.14

As discussed previously, ionized impurity and latticescattering contribute to lCarrier, with lCarrier decreasingwith increasing temperature due to phonon scattering.For strong impurity scattering, this may not occur un-

til relatively high temperatures. In the intermediate caseof LWire � lCarrier � dWire and where dWire � �DeBrogliescattering is termed mixed mode and is often calledquasi-ballistic.

2.14 The Quantum Hall Effect

The observation of, and first explanation for the Hall Ef-fect in a 2DEG by von Klitzing et al. [2.85], won thema Nobel Prize. As shown in Fig. 2.22 the Hall resis-tivity exhibits plateaus for integer values of h=e2, in-dependent of any material dependent parameters. Thisdiscovery was later shown to be correct to a preci-sion of at least one part in 107, enabling extremelyaccurate determinations of the fine structure constant(i. e., ˛ D .�0ce2=2h/� 1=137) and a fundamental re-sistance standard to be established.

For the six point Hall geometry used (as shownin the insert to Fig. 2.18), one can define the Hallresistivity �xy D g1VH;y=Ix and longitudinal resistivity�xx D g2VL;x=Ix where g1 and g2 are geometric con-stants related to the sample geometry. These resistivitiesare related to corresponding conductivities through the

25

20

15

10

5

00 5 10 15 20 25

VH (mV)

Vpp (mV)

Vg (V)

Vpp

VH

n = 0 n = 1 n = 2

p-substrate Hall probe

Source

Potential probes

Drain

Gates

Surface channel n*

Fig. 2.22 Hall voltage VH and voltage drop across elec-trodes VPP as a function of gate voltage Vg at 1:5K, whenB D 18T. Source-drain current is 1�A. Insert shows planview of a device with length 400�m, width 50�m and aninterprobe separation of 130�m. (After [2.85])

conductivity and resistivity tensors

�xy D �xy

�2xx C �2

xy

;

�xx D �xx

�2xx C �2

xy

;

�yx D � �xy ;�yy D �xx : (2.93)

Starting from a classical equation of motion for elec-trons in an electric field Ex, magnetic field Bz, anddefining the cyclotron frequency !c D eBz=m�, the ve-locities perpendicular to the applied magnetic field canbe deduced as

vx D Ex

Bxsin!ct and vy D Ex

Bx.cos!ct� 1/

(2.94)

with the time averaged velocities

hvxi D Ex

Bxand

�vy�D Ex

Bx(2.95)

leading to Hall and longitudinal resistivities of

�xx D �yy D 0 and �xy D ��yx D nse

Bz; (2.96)

where ns is the areal electron concentration. Below,a quantum approach is used to establish a relation-ship for the electron concentration ns. Note that themotion of the electrons in the crossed fields arequantized with allowed levels termed Landau levelsEn

En D�nC 1

2

�„!c C g��BBz C "z ; (2.97)

where n is the quantum number describing the partic-ular Landau Level, g� is the Landé factor, �B is theBohr magneton and g��BBz is the spin magnetic en-ergy, and Ez is the energy associated with the z-motion

PartA|2

44 Part A Fundamental Properties

of the carriers. xy-plane carrier motion is characterizedby the cyclotron energy term Exy,

Exy D „2k2xy2m�

e

D�nC 1

2

�„!c : (2.98)

Following this description and noting that the motionof electrons in the xy-plane may be expressed in termsof wavefunctions of the harmonic oscillator using theLandau gauge of the vector potential Œ0; xBz; 0�, wemay write the density of states per unit area, DOSA as

DOSA D m�

e !cLxLy2 „ : (2.99)

Since the degeneracy of each Landau level is one (i. e.,since they are single spin states), this enables one tofind ns assuming Landau state filling up to the pth level(for integer p)

ns D m�

e !cp

2 „ : (2.100)

Using the definition of the cyclotron frequency gives the

final form

ns D peBz

„ (2.101)

which may be used to rewrite the previous expressionsfor Hall and longitudinal resistivity

�xx D �yy D 0 and �xy D ��yx D h

pe2: (2.102)

Equation (2.102) shows that Hall resistivity is quan-tized in units of h=pe2 whenever the Fermi energy liesbetween filled Landau levels. Consistent with obser-vation, the result is independent of the semiconduc-tor being studied. Although this model provides anexcellent basis for understanding experiments, under-standing the details of the results (i. e., in particularthe existence of a finite width for the Hall effectplateaus and zero longitudinal resistance dips) requiresa more complete treatment involving so-called localizedstates.

Acknowledgments. The authors thank Dr. Robert Jo-hanson for helpful discussions on the subject.

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