Marder M.P. Condensed Matter Physics

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19.5.3 Quantum Dot

Figure 19.22. The quantum dot is a puddle of charge trapped between two quantum point

contacts. Because of the micron-scale dimensions of the trapping region, the number of

trapped electrons can be very small. One side of the trap is a plunger whose voltage can

be raised and lowered to alter the electrostatic properties of

the

dot. The current J flowing

from source to drain is the main experimental observable.

The quantum dot, as shown in Figure 19.22, is a structure one level more com-

plex than the quantum point contact. It mainly consists of two quantum point

contacts in series, but there is an additional interesting twist. The region between

the point contacts is rather small, an area on the order of 0.5/i mx0.5 /jm. In rough

analogy with the gate region of a transistor, there is also a metallic contact called

the plunger whose voltage can be raised and lowered in order to affect the number

of electrons in the central region.

Although named the quantum dot, the basic operation of this device is in large

part curiously classical. The kinetic energies of electrons, so decisive in metals,

are relatively small in this case. Consider, for example, placing N electrons into a

quantum dot with area d

. The single-electron quantum states would have energies

approximately

^r =

1510

rfvüH

(19

91)

This energy should be compared with the Coulomb repulsion of two electrons at

distance d, which is

, eV

- =

1.4-10-

—

(19.92)

d dj l/tmj

The difference in scale between the two energies is overstated by Eq. (19.92), be-

cause the Coulomb repulsion is diminished by screening effects to be discussed

next, but it is still correct to start with Coulomb repulsion as the main physical

effect and then add kinetic energy later as a small perturbation. The energy of

electrons in the quantum dot can be treated as a purely classical problem of adding

particles to a box, because no matter what the shape of their wave functions, only

the Coulomb integral has much importance.

604 Chapter 19. Electronics

Screening and Capacitance. To begin, suppose that the quantum point contacts

to the left and right of the dot are impenetrable barriers, and investigate how the

energy of the dot would vary as a function of the number of electrons placed in it.

The electrons

the dot cannot be taken independent

the rest

the universe.

They are

close proximity

the various metal gates, and charge must flow

and out of these gates so as to maintain them at externally applied voltages when-

ever electrons enter or leave the dot. Classical electrostatics handles this screening

problem by defining a capacitance matrix C

ß which posits that the charge Q

ß on

any of the gates, or in the dot, is a linear function of the electrostatic potentials V

of the gates and the dot. To make things simple, suppose that the charge on the

dot ßd depends only upon the potential within the dot,

V^,

and the potential of the

plunger, V

. Write the charge on the dot and plunger as

öd

QVd

—

CdpVp, The minus sign in front ofCdp is conventional, (19.93)

and it ensures that Q

will be positive.

ßp

-Cp

Vp. (19.94)

Because the possibility of electron motion through

the

junctions is being neglected

for the moment, the only feature of the outside world with which electrons in the

dot interact is the plunger. Therefore the charge on the dot must be

function of

Va —

Vp,

which means that

The capacitance matrix must

symmetric,

be-

(~>

cause

is given by second derivatives

the en-

/in

QC\

<-dp

<-pd-

ergy

(/ in

Eq. (19.96) with respect

potential,

yiy.yj)

and therefore C

The plunger is not similarly isolated. It is connected to a large reservoir of electrons

at potential

that enables

to remain at potential V

no matter what happens on

the dot. The electrostatic energy of the system is therefore

£/electrostat,c = ^ [ôd^d + ôpVp] + [ôreservoir - ßp]

Vp.

(19.96)

ßj

The remaining terms depend only upon

— T~^

Vptid

•

• • •

are independent of

Qd,

and so can be dropped, \iy-yl)

^A Make use of

Eqs.

(19.93),

(

19.94),

and

(

19.95).

The number of electrons preferred on the dot in equilibrium is given by minimizing

Eq. (19.97) with respect to ßd and is

^^öd

QV^

(]99g)

—e

were a capacitance on the order of

farad, this equilibrium number would be

immense. The point of the quantum dot is to generate capacitances

so small that

the equilibrium occupation is of order unity—that is, capacitances on the order of

aF= 10~

F (attofarad). In terms of this unit, Eq. (19.98) can be rewritten as

"'^iiiiFv

<l9

99)

Problems

605

showing that voltages on the order of millivolts applied to the plunger should pro-

duce changes of order unity in the number of electrons sitting on the dot. For such

small numbers of electrons, one must take into account the fact that N is an inte-

ger. What Eqs. (19.98) and (19.99) in fact predict is that the number of electrons

N increases in steps, with a transition occurring whenever states with N and N + 1

electrons have the same energy, at a voltage

= \N+ 1/21— Set Eq. (19.97) equal to itself evaluated at (19 100)

Q -Qd/e = N and N+l, using Eq. (19.98),

and solve for V

Having established the energetics of the problem in a simple classical fashion,

quantum features begin to creep into the interpretation of the results. The theory

based upon Eq. (19.98) is the theory of the Coulomb blockade, and it makes three

main predictions.

very small voltage is applied across the quantum dot, from source to drain,

the current from source to drain should show sharp narrow peaks as a function

the

plunger voltage V

, with the peaks spaced in voltage by a distance e/Q.

For fixed plunger voltage V

, current from source to drain should be relatively

tiny until the voltage from source to drain exceeds a critical threshold, either

positive or negative. The gap in voltage between the negative and positive

thresholds is e/C^.

The characteristic energy scale on which temperature fluctuations should de-

stroy these effects is kßT ~

/2Cd,

which works out to be a few kelvin.

The logic behind these predictions has to do with imagining physically how

electrons will manage to traverse the quantum dot in the presence of a voltage

between source and drain. In order to do so, an electron must tunnel across the first

quantum point contact, dwell for some time on the dot, and then tunnel across the

second quantum point contact. If according to Eq. (19.98) the electrostatic energy

of the dot goes up when an extra electron hops on, tunneling will be made difficult.

There are three ways around. First, whenever the plunger voltage sits at one of

the values indicated by Eq. (19.100), the energies of having N and N +1 electrons

on the dot are degenerate. There is no energy penalty preventing an electron from

flowing in and out of

the

dot, so for these special plunger voltages the dot has a high

conductivity, leading to prediction 1. Second, for an arbitrary plunger voltage, the

voltage between source and drain can be made large enough that it supplies the

energy needed to hop on and off the dot. Hence a prediction that current through

the dot will rapidly increase after a critical threshold no matter what the plunger

voltage. Third, thermal fluctuations may be large enough to supply the missing

energy, leading to the final prediction of a temperature scale on which the quantum

effects disappear.

All three of these predictions are beautifully verified by experiment. Figure

19.23 shows both (A) the periodic current peaks as a function of plunger voltage,

and (B) the very nonlinear relation between current and source-drain voltage.

606

Chapter 19. Electronics

-s

•a

0.020 -

0.015 -

0.010 -

0.005 -

o.ooo

I—ä—ii-Li

0.122 0.124 0.126 0.128 -.0.002 -0.001 0.000 0.001 0.002

(A) Plunger voltage

(V) (B) Drain-source voltage V

(V)

Figure 19.23. (A) Conductance as a function of plunger voltage displaying sharp, equally

spaced peaks. (B) Current across a quantum dot plotter versus the voltage

across the

dot. [Source: Meirav and Foxman (1996),

257.]

Problems

Chemical potential in intrinsic semiconductor: Consider a crystal of sil-

icon, with a very low level of doping, but a slight excess of acceptors over

donors.

(a) At sufficiently low temperatures, the chemical potential moves far away from

the center of the gap. At what temperature does this happen, and why?

(b) Next, consider the same situation, but with a slight excess of donors over

acceptors, and answer the same questions.

Ohmic junction: Diagrams of electronic circuits frequently show wires con-

nected to portions of semiconductors. These connections are supposed to be

ohmic, to conduct current with equal ease in either direction, and in linear

proportion to applied voltage. Because metal-semiconductor junctions have

intrinsic rectifying properties, ohmic response cannot be taken for granted.

(a) To have a sense of ways to obtain such junctions, copy the three parts of

Figure

19.11,

but assuming that the work function of the semiconductor is

greater than the work function of the metal.

(b) Argue from the graphical construction that the rectifying powers of

the

junc-

tion are plausibly diminished, by considering as in Figure 19.12 how the band

bending is affected by an applied voltage.

Thermopower of semiconductors:

(a) Consider an «-doped semiconductor, where transport is dominated by elec-

trons in the conduction band. Measure all energies £ relative to the bottom of

the conduction band, so £

= 0. Assume that £ = m*v

/2, that D(£) oc \/£,

that r

= a£~

, and that the matrices of

Eq.

(17.73) are all diagonal. Equation

References 607

(17.74) is still valid, but for semiconductors the approximation in Eq. (17.77)

cannot be used. Show that the thermopower a is given by

-s-

The identity T{\+x) =xT(x) is helpful. (19.101)

(b) Now consider a p-doped semiconductor, where transport is dominated by

holes.

How does Eq. (19.101) change in this case?

Carriers in depletion region: Using

Eqs.

(19.67), verify

Eqs.

(19.71). Notice

that because (19.71) is obtained from expressions for the current, it cannot be

used directly to predict it.

Ebers-Moll equations:

(a) Write down the three equations analogous to Eqs. (19.74) for minority carri-

ers in the three quasi-neutral regions of the transistor.

(b) Write down the solutions of these equations; in the collector and emitter

regions, the solutions are immediately determined up to an overall constant,

while in the base region, there are two unknown constants to calculate.

show that the currents described by (19.80) can be put in the form (19.81).

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PartV

OPTICAL PROPERTIES

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

20.

Phenomenological Theory

20.1 Introduction

Experiments with light provide unexpected ways to illuminate the inner workings

of solids. Oscillating electrical and magnetic fields can explore the immense range

of length and time scales sketched out in Figure 20.1, a range so large that the

interaction of light with matter is impossible to treat thoroughly inside any single

framework. The theoretical starting point that is appropriate for infrared radiation

falling on an ionic insulator has nothing to say about soft X-rays knocking core

electrons out of a metal. The underlying equations, Schrödinger's equation for

the solid coupled to the quantized radiation field, are too complicated to be of

much use. However, there is a collection of simple ideas that makes it possible

to understand most experiments. The main effect of incoming light is to make

electrons oscillate. Their response depends upon interactions with the atomic solid

in which they live and with each other. Fortunately, the interaction of an electron

with a lattice can often be modeled by a simple idea—the effective mass—and the

interaction of electrons with each other can often be explained with the classical

idea of a dielectric function.

Drude's model, Section

16.1.1,

provides a simple initial description for the

optical response of

metals.

Rewriting Eq. (16.1) with B = 0 as

A - v

mv = — eE

—

m—,

(20.1)

and supposing that E and v have time dependence of the form exp[—iuit] gives

• p ^

—loomv

= — et

—

m—

_ ne

T/m

=4>

j = —nev = E

— iuiT

. , ne

T/m

=>^M =

i——

1 — ILÜT

Therefore for frequencies much smaller than the relaxation rate

2TT/T,

current is

in phase with applied electric fields, and the conductivity is given by the low-

frequency form, Eq. (16.6). At higher frequencies, current and field fall out of

phase, and the low-frequency relation no longer applies. Many features of the

optical response of metals can be explained in this way.

However, there are types of optical response that Eq. (20.4) does not capture.

The photoelectric effect was discovered by Hertz (1887) and Thomson (1897).

611

(20.2)

(20.3)

(20.4)

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

612 Chapter 20. Phenomenological Theory

Light impinges upon a piece of metal, and a current of electrons emerges in ex-

change. The intensity of the current is proportional to the intensity of the incoming

light, but there is a cutoff frequency below which no electrons emerge, no matter

what the intensity. Einstein (1905) explained this experiment by supposing that

light could be viewed as a stream of particles carrying quanta of energy, an idea

that marks the origin of quantum mechanics.

Two sets of ideas come together to make an enlightened study of matter possi-

ble.

The first is the study of how electrons in a solid respond to external electrical

and magnetic fields, dealt with in Chapter 16. The second is the phenomenologi-

cal description of matter immersed in electromagnetic fields known as Maxwell's

equations.

_10

Î 10

12|/

Wavelength A (cm)

10"

1CT

10"

7U»V

Core-level

photoemission

Plasma frequency

in metals

Insulator band gaps

(

Electron scattering

rates 2-K/T

Semiconductor band gapsQ 1

10"

Characteristic transverse optical phonon

-Cyclotron resonance

Brillouin scattering

Ionic crystal excita;

^L_

io-

110"

no-

10"

1 10

1Ö

Wave number 1/A (cm

-1

)

Figure 20.1. Overview of length and energy scales important for the interaction of light

with condensed matter.