Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Problems 563

6. Scaling theory of localization II: Let T be the transfer matrix for a tight-

binding Hamiltonian on an L x L square lattice,

IK/

). (18.160)

The subscript / is needed because the matrices Ä/ have random diagonal ele-

ments, chosen with equal probability to lie within [-W/2, W/2]. Let

Q=n

161

)

i=\

Then according to Pichard and André (1986), the conductance G of an L x

Lx L cube, in units of e

/h, is

i =

(18.162)

LQQ*

+ (QQ*)-'+2j

(a) Rewrite the expression for G in terms of the eigenvalues of the matrix QQ*.

(b) Calculate the resistance of a 3 x 3 x 3 block at W = 0 and £ = 0 and find

In R= 1.4818.

—

10 and £ = 0.

(d) The resistance found in the previous part will depend greatly upon the partic-

ular values of the random disorder. To obtain a more meaningful result, carry

out averages over many realizations of the randomness. That is, compute In R

for one 3x3x3 block, obtaining In R\. Without initializing the random num-

ber generator, find the resistance of a second 3x3x3 block, obtaining In Rj.

Continue in this way, generating the series of resistances In /?/. The fluctua-

tions in resistance are so large that R is not well defined, but In R will average

well. Let In /?/ be the average of In R after / computations. Keep computing

until the fluctuations in In /?/ settle down to within around 4%.

Scaling theory of localization III:

Use the numerical routine of problem 6 to compute the conductance G, de-

fined in Eq. (18.162), and therefore find the scaling function for localization

in three dimensions. That is, reproduce Figure 18.9. This figure was produced

by using tens of thousands of cubes of size up to 7 x 7 x 7, but if systems this

size are too time-consuming, an adequate figure can be produced with cubes

of size up to 5 x 5 x 5.

8. Weak localization: The goal of this problem is to carry out the calculations

leading from Eq. (18.101) to Eq. (18.103). Specialize to the center of the

band, £ = 0, and write 3(g, £ = 0) = 5(g).

(a) Show that

]

= -J dg?(g) In \g\. (18.163)

564 Chapter 18. Microscopic Theories of Conduction

(b) Show that if

(f/)

is very narrow and centered on zero, then to leading order

W/i

one obtains

?(g) =

\?(-l/g).

(18.164)

3"=

(18.165)

where C is a constant.

(d) Show that to normalize 3

,C = 0.2696

....

(e) Equation (18.165) is completely independent of the probability distribution

From Eq. (18.101), an approximate form for

that involves

r . ,„, t . „ C

f(s)

</,v

(,8,66,

Iterating Eq. (18.101) further would produce even better approximations for

U, but to leading order in

W/i

it is not necessary.

(f) Inserting

Eqs.

(18.166) and (18.81) into Eq. (18.163), and working to leading

order in W/i, verify Eq. (18.103).

9. Luttinger Liquids:

(a) Verify that

[pti-q),

*] = - £ ^re-V'VM-q),

p,(</)]

(18.167)

-e

iqx

Ü.

(18.168)

(b) Verify Eq. (18.145). Use Eq. (13.119) to treat averages of exponentials of

Bose operators. Keep only terms that survive the expectation value

the

ground state.

References

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan (1979), Scaling theory

of localization: Absence of quantum diffusion in two dimensions, Physical Review Letters, 42,

673-676.

M. Ahlskog, R. Menon, A. J. Heeger, T. Noguchi, and T. Ohnishi (1997), Metal-insulator transition

in oriented poly(p-phenylenevinylene), Physical Review B, 55, 6777-6787.

L. Al'tshuler and P. A. Lee (1988), Disordered electronic systems, Physics

Today,

41(12), 36-44.

W. Anderson (1958), Absence of diffusion incertain random lattices, Physical Review, 109, 1492-

1505.

References

565

W. Ashcroft and L. J. Guild (1965), The resistivity of liquid aluminium, Physics Letters, 14,

23-24.

W. Ashcroft and D. Stroud (1978), Theory of the thermodynamics of simple liquid metals, Solid

State Physics: Advances in Research and Applications, 33, 1-81.

J. Bass, W. P. Pratt, and

A. Schroeder (1990), The temperature-dependent electrical resistivities of

the alkali metals, Reviews of Modern Physics, 62, 645-744.

Belitz and T. R. Kirkpatrick (1994), The Anderson-Mott transition, Reviews of Modern Physics,

66,

261-380.

A. B. Bhatia and K. S. Rrishnan (1948), "Diffuse scattering" of the Fermi electrons in monovalent

metals in relation to their electrical resistivities, Proceedings of

the

Royal Society of London, A

194,

185-205.

M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and

L. McEuen (1999),

Luttinger-liquid behaviour in carbon nanotubes, Nature, 397,

598-601.

A. R. Bulsara and L. Gammaitoni (1996), Tuning in to noise, Physics Today, 49(3), 39^45.

Chattopadhyay and H. J. Queisser (1981), Electron scattering by ionized impurities in semicon-

ductors, Reviews of Modern Physics, 53, 745-768.

E. Cusack (1987), The Physics of Structurally Disordered Matter, Adam Hilger, Bristol.

Dean (1972), The vibrational properties of disordered systems: Numerical studies, Reviews of

Modern Physics, 44, 127-168.

Dutta and P. M. Horn (1981), Low-frequency fluctuations in solids: 1/f noise, Reviews of Modern

Physics, 53, 497-516.

E. N. Economou (1983), Green's Functions in Quantum Physics, Springer-Verlag, Berlin.

P. Edwards and M. J. Sienko (1982), The transition to the metallic state, Accounts of Chemical

Research, 15,

87-93.

R. J. Elliott, J. A. Krumhansl, and P. L. Leath (1974), The theory and properties of randomly disor-

dered crystals and related physical systems, Reviews of Modern Physics, 46, 465-543.

H. Fritzsche (1984), Noncrystalline semiconductors, Physics

Today,

37(10),

34^41.

T. Giamarchi (2003), Quantum Physics in One Dimension, Oxford University Press, Oxford.

I. S. Grigoriev and E. Z. Meilkhov, eds. (1997), Handbook of Physical Quantities, CRC Press, Boca

Raton, FL.

D. M. Haldane (1981), 'luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of

the Luttinger model and their extension to the general ID interacting spinless Fermi gas, Journal

of Physics C: Solid State Physics, 14(19), 2585-2609.

S. John (1991), Localization of light (in dielectric microstructures), Physics Today, 44(5), 32^10.

J. B. Johnson (1927), Thermal agitation of electricity in conductors, Physical Review, 29, 367-

368.

The complete text of the article reads: Ordinary electric conductors are sources of random

voltage fluctuations, as a result of thermal agitation of the electric charges in the conductor. The

average effect of the fluctuations has been measured by means of a vacuum tube amplifier, where

it manifests itself as a component of the phenomenon commonly called "tube noise." A part of

the "tube noise" arises in the first tube and other elements of the apparatus; the remainder in

the input resistance, with a mean square voltage fluctuation (V )

which is proportional to the

resistance R ofthat conductor. The ratio (V )

/R, of the order of 10 watt at room temperature,

is independent of the material and shape of the conductor, but is proportional to its absolute

temperature. In the range of audio frequencies, at least, the noise contains all frequencies at equal

amplitudes. The noise of an input resistance of only 5000 ohms may exceed that of the rest of the

circuit, so that the limit of useful amplification is at times set by the thermal agitation of charges

in the input resistance of the amplifier.

C. Kane, L. Balents, and M. P. A. Fisher (1997), Coulomb interactions and mesoscopic effects in

carbon nanotubes, Physical Review Letters, 79, 5086-5089.

A. Kawabata (2007), Electron conduction in one-dimension, Reports on Progress in Physics, 70,

566 Chapter 18. Microscopic Theories of

Conduction

219-254.

W. Kohn (1957), Shallow impurity states in silicon and germanium, Solid State Physics: Advances

in Research and Applications, 5, 257-320.

Kramer and A. MacKinnon (1993), Localization: Theory and experiment, Reports on Progress in

Physics, 56, 1469-1564.

L. D. Landau and E. M. Lifshitz (1977), Quantum Mechanics (Non-relativistic Theory), 3rd ed.,

Pergamon Press, Oxford.

A. Luther and I. Peschel (1974), Single-particle states, Kohn anomaly, and pairing fluctuations in

one dimension, Physical Review B, 9, 2911-2919.

J. M. Luttinger (1963), An exactly soluble model of a many-fermion system, Journal of Mathemati-

cal Physics, 4, 1154-1162

March (1990), Liquid Metals: Concepts and Theory, Cambridge University Press, Cambridge.

C. Mattis and E. H. Lieb (1965), Exact solution of a many-Fermion system and its associated

boson field, Journal of Mathematical Physics, 6, 304-312

R. Menon, C. O. Yoon, D. Moses, A. J. Heeger, and

Cao (1993), Transport in polyaniline near the

critical regime of the metal-insulator transition, Physical Review B, 48,

685-17 694.

A. Messiah (1999), Quantum Mechanics, Dover, New York.

F. Mott (1978), Metal-insulator transitions, Physics

Today,

31(11), 42-47.

F. Mott (1990), Metal-Insulator Transitions, 2nd ed., Taylor and Francis, London.

F. Mott and E. A. Davis (1979), Electronic Processes in Non-Crystalline Materials, 2nd ed.,

Clarendon Press, Oxford.

H. Nyquist (1928), Thermal agitation of electric charge in conductors, Physical Review, 32, 110-113.

S. T. Pantelides (1978), The electronic structure of impurities and other point defects in semiconduc-

tors,

Reviews of Modern Physics, 50, 797-858.

R. E. Peierls (1929), On the kinetic theory of heat transport in crystals, Annalen der

Physik,

3, 1055-

1101.

In German.

J. L. Pichard and G. André (1986), Many-channel transmission: Large volume limit of the distribu-

tion of localization lengths and one-parameter scaling, Europhysics Letters, 2, 477^86.

T. F. Rosenbaum (1985), The disordered insulator: Electron glasses and crystals, in Localization and

Metal-Insulator Transitions, H. Fritzsche and D. Adler, eds., pp. 1-8, Plenum, New York.

M. V. Sadovskii (1981), Electron localization in disordered systems: Critical behavior and macro-

scopic manifestations, Soviet Physics Uspekhi, 24, 96-115.

Sheng (1995), Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Aca-

demic Press, San Diego.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret (2006), Observation of the critical regime near

Anderson localization of light, Physical Review Letters, 96(6), 063 904.

A. P. Sutton (1993), Electronic Structure of Materials, Clarendon Press, Oxford.

S. Tomonaga (1950), Remarks on Bloch's method of sound waves applied to many-fermion prob-

lems,

Progress of Theoretical Physics, 5, 544-569.

R. A. Webb and S. Washburn (1988), Quantum interference fluctuations in disordered metals, Physics

Today,

41(12),

46-53.

J. M. Ziman (1961), A theory of the electrical properties of liquid metals. I The monovalent metals,

Philosophical Magazine, 6, 1013-1034.

J. M. Ziman (1979), Models of Disorder : The Theoretical Physics of Homogeneously Disordered

Systems, Cambridge University Press, Cambridge.

A. Zunger (1986), Electronic structure of 3d transition-atom impurities in semiconductors, Solid

State Physics: Advances in Research and Applications, 39, 275-464.

19.

Electronics

19.1 Introduction

Electronics is the art of controlling the flow of electrons. It began with the dis-

covery of the ancient Greeks that a piece of amber (rjXexxpov) could attract and

hold small objects after being rubbed. The path leading to electrons' controlled

removal from matter included the observation early in the eighteenth century that

the electrical conductivity of air increases in the vicinity of a hot poker, continued

with Franklin's kite experiment relating lightning to static electricity, and culmi-

nated in the middle of the nineteenth century with experiments of Crookes who

passed electricity between high-voltage plates in evacuated tubes. Edison noticed

in 1883 that if he placed a metal plate inside a light bulb, current would flow to the

plate if it was at a positive voltage with respect to the filament, but not otherwise.

Edison did not think this observation of

rectification

particularly significant, but it

has turned out to have as many consequences as his electric lights.

Credit for the discovery of the electron is given to J. J. Thomson, whose exper-

iments on the flow of electricity from heated filaments in evacuated tubes isolated

it as a particle with a definite ratio of charge to mass. The name "electron" was

proposed by G. J. Stoney in 1894 for the unit of charge equal to 10~

coulombs,

and this term gradually superseded Thomson's term "corpuscle" for the new par-

ticle.

The first practical electronic device was built by J. A. Fleming, who built

upon the work of Thomson and Edison to create a cathode ray tube with a heated

filament capable of rectifying oscillating currents. He called it a "valve," but it is

now better known as the diode, and is depicted in Figure 19.1. Commercial radio

heated cathode

Forward Bias

.anode I |l|l

Reverse Bias

J—

PII—|

I——l|l|——"I

cathode anode cathode anode

Figure 19.1. The essential action of

diode is to send current in one direction in response

to an applied voltage, and not the other. The origin of

this

asymmetry is the fact that met-

als at elevated temperatures emit electrons long before they emit positively charged ions.

A heated cathode therefore sends off an appreciable current toward a positively charged

anode, but almost none toward a negatively charged anode.

567

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

568 Chapter 19. Electronics

Figure 19.2. The triode serves not only to rectify current, but also to amplify small signals.

It accomplishes this task through the interposition of a grid between cathode and anode.

The potential between cathode and grid determines whether electrons begin the journey

between cathode and grid at all. However, once electrons arrive at the grid, they discover

that a much larger positive potential awaits them at the anode and accelerate toward it.

Because only a small proportion of the electrons enters the grid, small grid currents control

large cathode—+ anode currents.

transmission became feasible soon after with the invention by L. De Forest of the

triode, shown in Figure 19.2. Rectification is essential to practical radio transmis-

sion because the time average electrical field of a propagating radio wave is zero,

and even if the amplitude of such a wave is modulated to encode sound, it cannot

directly drive a speaker. Once the signal is passed through a rectifier, time averages

no longer vanish, and the signal can easily be decoded. The triode was essential not

only because it allowed amplification of weak signals, but also because by feeding

a portion of the output back in to the control grid, it could be made into a powerful

and stable source of radio-frequency oscillations.

Up through the 1970s much of electronics consisted in the study of cathode ray

tubes.

They have now almost entirely been superseded by semiconductor devices,

which are much more reliable, and have slowly managed to capture even high-

power and high-frequency applications that at first seemed out of reach. However,

the basic concepts of controlling current, rectification, amplification, and switching

all first developed in the context of cathode ray tubes and were then taken over and

further developed by semiconductor descendants. Even the basic physics of the

various devices has many points of similarity. For this reason, it is advisable to

begin the study of electronics with the physics that made the cathode ray tubes

possible.

19.2 Metal Interfaces

As sketched in Figure 19.1, the cathode ray tube diode relies upon the fact that a

heated piece of metal emits electrons, but not positively charged ions. This effect

becomes most clearly visible when air is evacuated from the region in which the

Metal Interfaces 569

electrons are to travel, and for this reason cathode ray tubes are also known as

vacuum tubes or electron tubes. The physical question that needs to be answered

is how a metal surface in contact with vacuum emits electrons as a function of

temperature and electrical potential.

19.2.1 Work Functions

All the calculations of electronic energy levels up until now have been carried out

relative to one another. For example, the Fermi level was sometimes calculated

relative to the energy of the lowest single-particle electronic level, and in the band

structure diagrams of Section 10.4, the Fermi level was defined to be zero. None of

these calculations answers the question of the amount of energy needed to remove

an electron from an electrically neutral solid in vacuum. This energy is defined to

be the work function and is often denoted by

cf>.

It can be measured by optical meth-

ods to be discussed in Section

23.6.1,

or by properties of thermal emission to be

discussed immediately below. To calculate it requires understanding what happens

when an electron is dragged through the interface between metal and vacuum.

The work function must be contrasted with the chemical potential, which is

the energy required to take an electron from the bulk and remove it to infinity.

Inspection of Table 23.2 shows that the experimentally measured energy required

to pass an electron through one crystal surface typically differs by 10-20% from

the energy required to move it through an inequi valent one. As the energy required

for transit from bulk to infinity cannot depend upon path, it is necessary to establish

carefully what the experiments actually determine.

metal

Figure 19.3. An electron at a distance x from

a metal surface is attracted to the surface by

an image charge of opposite sign which guar-

antees that electric field lines will be normal

to the surface.

The electrostatic potentials contributing to the work function operate on three

separate length scales:

Atomic. In passing through the surface of a crystal, an electron passes through a

highly inhomogeneous environment where the crystal terminates. Although

the surface is almost completely electrically neutral, there is always a strong

dipole layer. The electron is buffeted by strong forces as it passes through this

layer, but the range over which these forces operate is only on the order of a

few lattice spacings, due to the effectiveness of screening in a metal.

570

Chapter 19. Electronics

Micron. After the electron exits the metal, it interacts with an image charge,

whose presence enforces the boundary condition that the tangential electric

field vanish on the surface, as shown in Figure 19.3. The force F at distance x

from the surface is

which implies that the electron has a potential energy

U(x) = = —3.6-10~VmeV. (19.2)

4x x

Work functions are on the order of several electron volts, and therefore further

changes due to the image charge energy become negligible by the time an

electron has traveled a few microns from the surface.

Macroscopic. Because different crystal faces have different dipole layers, the re-

gions outside them must be at different electrical potentials, produced by

minute shifts in electron density near the crystal surface. The existence of

this potential is absolutely necessary, because there is no other way to bring

an electron out of the bulk through one surface, return it through another sur-

face,

and have it return to the original bulk energy. The spatial scale for the

variation of this potential is the size of the crystal

itself.

Figure 19.4. The work function is denned as

the energy needed

remove

electron from

the bulk of

metal, and bring it within about

a micron of

particular surface.

Therefore, as indicated in Figure 19.4 the work function is defined to be the

energy of electrons brought out to distances on the order of microns from crys-

tals whose dimensions are larger than microns. Hölzl and Schulte (1979) describe

many additional complications that can arise in attempting to calculate or to mea-

sure work functions, such as what happens when surfaces are rough, or contain a

layer of adsorbate atoms.

19.2.2 Schottky Barrier

Equation (19.2) fails when an electron is too close to the crystal surface, because

the potential energy U diverges, and more realistic calculations require explicit

description of the electronic surface states of a metal. However, it is adequate for

the purpose of estimating the effect of an externally applied electrical potential on

the work function. Suppose that a positively charged metal plate is placed at a large

Metal Interfaces 571

distance to the right of the metal surface, creating a linear electric field of strength

—E. Now the potential energy U(x) of the electron is

U(x) = -—-e\E\x, (19.3)

which creates a barrier, shown in Figure 19.5, at distance

=J—^>U(x

)

-e^e\E\.

(19.4)

Therefore an externally applied electric field changes the barrier restraining an

Positive potential

electron within the metal by

e\Je\E\.

Ground

^______ No electric field

/^~~

Small electric field

Larger electric field

Figure 19.5. An externally applied electric field, created by placing at a distance a large

plate at an elevated voltage, lowers the barrier an electron must surpass in order to exit a

metal.

Having first found the effect of applied electric fields, the next goal is to exam-

ine the thermionic emission of electrons that results from heating the metal. This

task is accomplished by considering the electrons in the metal to be in equilibrium

with a dilute gas of electrons hovering outside it. For simplicity the electrons are

treated within the semiclassical approximation, which makes

possible to speak

of the probability for an electron to have wave number

at position

outside the

metal

f-. = : .

Compare with Eq. (17.27). Here

is more (19.5)

•** ß(£S+U(x)—n)

convenient

keep

constant and describe

"i"

the spatial change in potential through

(?).

The probability of finding electrons does not vanish as x travels far from the metal.

Finding the vacuum full of electrons may seem unacceptable, but is an inevitable

consequence of

the

fact that no solid or liquid can ever be in equilibrium with a vac-

uum at nonzero temperature. Entropy always favors total evaporation. However,

solid can exist in equilibrium with a dilute vapor of a particular concentration, and

that is what Eq. (19.5) implies for the electrons in

metal. The properties of the

electron vapor are fixed by the observation that because

is in equilibrium with

the metal, the two must have the same chemical potential. Taking "far from the

572

Chapter 19. Electronics

metal" to denote distances on the order of a micron, the chemical potential must

be replaced by the negative of the work function,

4> —

> 0- Work functions

are typically on the order of several electron volts, as shown in Table 23.2, so for

temperatures much less than 10000 K, Eq. (19.5) can be replaced by

/^««-^

+t/W+0)

(19.6)

In order to find the current drawn from the metal

the presence

an applied

electric field, one should write down the Boltzmann equation and calculate the

nonequilibrium function g ^. However, it is adequate to consider a simple approx-

imation, which is to assume that the electron gas is in equilibrium at all points to

the left of

and that all electrons that reach

and are traveling to the right escape

over the barrier and run off as a current. This idea predicts a current

Eq. (6.15).

j=-eexp{-ß[<j> + U(xo)]} f[dk] ^9(k

)e~^

2k2

For [dit],

see (19.7)

J m

Eq. (6.15).

exp l-ß 4>-e^fë\Ë\ \ ,

From

Eq.

(19.4).

(19.8)

where

A=^—rk

= 120.2 Acm"

. (19.9)

o il

jj-

nn i A

„™—2- v—2.

2n^

Equation (19.8)

called the Richardson-Dushman equation when used

for

E = 0, while the reduction of the work function by the square root of an applied

field is called the Schottky effect. The current does not vanish when the electric

field E goes away, which means that

one places a cold grounded plate at some

distance from the heated metal and provides a path for the electrons departing from

the metal to return to it, current will flow through the vacuum even in the absence

of a voltage difference. The factors outside the exponential in Eq. (19.8) are not

particularly to be trusted, but the exponential scaling with temperature and electric

field can be verified experimentally and can be used to measure the work function

</>. Equation (19.9) was derived by Schottky (1938), and

provided theoretical

underpinning for the development of cathode-ray tube electronics.

19.2.3 Contact Potentials

Whenever two dissimilar materials are brought together, charge moves between

them. The reason is that they have in general different work functions, and elec-

trons from the material with the smaller work function rush into the material with

the larger one. As this process occurs, charge builds up in the second material, and

at some point Coulomb repulsion brings the charge transfer to a halt. The effects

of Coulomb repulsion can, however, be minimized if the electrons that flow to the

second material are located as close as possible to the (positive) holes flowing to the

first material. For this reason, the electrons and holes arrange themselves as surface

charges along the interface between the two materials, the electrons on the side of

the second material, the holes on the side of the first. Variations on this basic sce-

nario follow mainly from the widths of the regions with surface charge. When two