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Effects of the Electron–Phonon Interaction in Semiconductor Quantum Dots 369

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73. The states with L

 0 can be split into symmetrical and antisymmetrical states under the

transformation of the polar angles θ

e,h

→  θ

e,h

and φ

→  φ

of the vectors r

ជ

e,h

and q

ជ

respectively.

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370 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

79. B.A. Carpenter , E.A. Zibik , M.L. Sadowski , L.R. Wilson , D.M. Whittaker , J.W. Cockburn ,

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84. F. Koberling , A. Mews , and T. Basché , Phys. Rev. B 60 , 1921 ( 1999 ) .

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86. E.P. Pokatilov , V.A. Fonoberov , V.M. Fomin , and J.T. Devreese , Phys. Rev. B 64 , 245329 ( 2001 ) .

87. T. Basché (private communications).

88. R.G. Neuhauser , K.T. Shimizu , W.K. Woo , S.A. Empedocles , and M.G. Bawendi , Phys. Rev. Lett. 8 5 ,

3301 ( 2000 ) .

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Slow Oscillation and Random Fluctuation in Quantum Dots:

Can we Overcome?

Raphael Tsu

University of North Carolina at Charlotte, Charlotte NC 28223, USA

12.1 Introduction

I would like to place the development of man-made solids, such as the introduction of superlat-

tice [1] followed by quantum wells [2 , 3] more than 30 years ago, evolving into the present with

quantum dots, into the category of quantum composites which is in contrast to classical composites

where mean-ﬁ eld approximation is adequate. First of all, let us deﬁ ne a size a when quantum

interference is important. Surely, it is dictated by the condition that a  Λ , the mean-free path in

bulk or more precisely the coherence length of an electron. On the other hand, one often encoun-

ters the notion that quantum conﬁ nement is important whenever a  R

, the Bohr radius. The lat-

ter condition is doubtful because only in bulk can the dielectric constant representing screening

be uniquely deﬁ ned. At low temperatures, Λ may be 100 nm.So that for a  100 nm, R

is practi-

cally the bulk value, therefore quantum interference is very much operative in a QD of 100 nm.

In any case, we shall use the former criterion to deﬁ ne the existence of a QD. Nicollian challenged

me to place many quantum dots under one large contact. Our ﬁ rst attempt was to crystallize a

thin layer of a Si followed by oxidation resulting in many islands of Si QD embedded in an oxide

layer. Our plan was to construct a functional device by utilizing several wells with separated dis-

crete quantum states produced by the high oxide barrier, being nearly 3.2 eV above the conduc-

tion band of silicon. We were pleased by our initial success in 1991 [4] of observing discrete

structures having a metal contact, indicating that there were several distinct quantum states

of the Si QD. However, strange conductance oscillations and switching were then observed [5] .

Typically in a contact of 40  4 0 μ m

, there were  1600 Si QDs of lateral dimension

 5–10 nm, so that these QDs are well separated. The conductance jump is in multiples of 39 μ S,

or Δ G  ge

/ h , with g , a degeneracy factor, usually 2, 4, etc. At this stage, we were sure that we

were dealing with complicated trappings, particularly the oscillation period, and switching time

is typically varying from a fraction of a second to as long as 20 s. This complexity discouraged us

from pursuing the study in a more aggressive manner, partly because we lost the NSF support

because our programme manager did not appreciate the complexity. I was also pursuing other

topics, e.g. porous silicon, very fashionable at the time, although I continued to analyze the vast

amount of data and applied a deeper interpretation and understanding. For example, it is only

recently that I discovered an understanding on the conductance spectrum. Meanwhile, I learned

about optical blinking in QDs which is quite similar to conductance switching and have decided

to present both cases in this chapter to show how similar the data are on electrical switching and

optical blinking.

CHAPTER 12

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372 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

12.2 What is a quantum dot?

Let us ﬁ rst describe what a quantum dot is:

1. A quantum dot should have the same coordination number. For example, silicon as a solid

has a coordination number of four, but six in the molten state. Clearly, a molecule of two

silicon atoms would have a coordination number of two. Taking the coordination number of

four as in a crystalline solid of silicon implies an Si QD deﬁ ned with four coordination num-

bers, this limits our consideration of a cluster of silicon atoms resembling solid silicon.

2. To characterize solid silicon, regardless of how good or, more precisely, how high the

mobility is, the energy band gap and the effective masses for the valence and conduction

bands are basically identical. Whereas for a quantum dot of silicon, apart from the separa-

tion of discrete states dictated by size, there is really no standard deﬁ ning two Si QDs. The

question is how can we choose a standard? For example, because of the tensor nature of

the effective masses, the states are far from the usual 1S, 2S, 2P, etc. quantum numbers

we use for the central ﬁ eld atomic states. Simple computation with two masses, one along

the longitudinal ellipsoid and the other transverse to the axis of the ellipsoid, leads to a dif-

ferent set of energy degeneracy factors for an Si QD. As size is reduced to  1 nm, clearly

the indirect nature of the band gap would have changed, leading to another complication

for the classiﬁ cation of Si QDs. Does it make sense to rely on ab initio computation? The

answer is no, because the computed energy structure, even if correct, gives us little physi-

cal meaning to serve as a guide for understanding the physics of an Si QD. My view of the

importance of ab initio computation lies in the conﬁ rmation of conventional approaches

with parameters. Once these parameters are in place, we can rely on the usual physical

models: energy bands, symmetry, effective masses, etc.

3. Let us see what we can do with the spectral linewidth of photoluminescence of a QD.

Before we delve into it, let us digress by considering what the linewidth is of optical transi-

tions involved in a QD. In solid, thermal broadening, k

T, represents the most important

factor in addition to other energy non-conserving processes. It is not quite recognized by

many workers in QDs that the primary broadening factor is still k

T , because while energy

barriers may serve to isolate the energy states of a QD, because of the similarity of the

dynamic force matrix, or simply the small difference in elastic constants of the barrier

conﬁ ning the QD from that of the QD, phonons are much less conﬁ ned, leading to efﬁ cient

energy losses in terms of phonons. At this stage of the discussion, I want to point out that

in an atom, unlike the vibronic states in a molecule, there is no broadening factor other

than the perturbation presented by the application of a vector potential to the atomic

Hamiltonian. Even at very low frequency such as near a dc electric ﬁ eld, the Stark effect is

sufﬁ cient to induce coupling between states allowing energy to be dispersed to states hav-

ing a probability of escaping from the atom, even though the probability is very small. In

short, the standard or the Figure of merit of a QD may be tied to the linewidth of the opti-

cal transition. Measuring the spectrum of a single InP QD, Bertram et al . [6] observed a

linewidth of 0.6 meV at 1.645 eV being equal to k

T at T  6.9 K, indicating the absence

of other energy non-conserving processes. Figure 12.1 shows their measurement.

4. Other parameters generally used to characterize a solid, such as size with AFM, XRD, high-

resolution TEM, and Raman scattering, are useful, but generally do not tell you the quality

of the QD. Most researchers in QDs still cling onto the terminology – quantum transport.

In fact, there is no such thing with QDs because only tunnelling and hopping between dots

can transport an electron through the dots.

5. Let us discuss the possibility of forming a 3D superlattice structure with QDs. Unlike

photonic crystals [7] where the periodic structure is characterized by the wavelength of

photons with the ﬁ rst Brillouin zone 50 times larger than the de Broglie wavelength of

electrons in typical QDs, allowing successful fabrication using available lithographic tools

as well as various schemes such as stacking of spheres [8] . Most QDs utilize the strain-

induced Stranski–Krastanov growth techniques, which is not possible at present to pro-

duce a periodic 3D structure of QDs. A single QD has the problem of contact for electronic

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Slow Oscillation and Random Fluctuation in Quantum Dots: Can we Overcome? 373

applications [9] . The fundamental issue is the lack of an equal potential surface for deﬁ n-

ing a voltage for input/output. Is there any advantage in having a distribution of QDs

either in electronic applications or in optical devices? Thus far it is doubtful that such

arrangement can outperform the present-day superlattice and quantum well structures.

After the presentation of random ﬂ uctuation, I shall return to make some comments and

judgements.

6. When the size of the QD is reduced to  1 nm, several things happen. First, the capaci-

tance is reduced [10] and the dielectric constant is lowered [11] to the point that Bohr

radius is much reduced with much increased ionization energy for dopants [12] . Even if

we can overcome the statistical problem of accurately introducing dopants into a QD, the

QD remains intrinsic at room temperature and below. On the other hand, since quantum

transport has no meaning, there is no need for doping. QD devices depend on carrier injec-

tion via tunnelling.

12.3 Slow oscillation and random switching instability in a distribution of QDs

There is a signiﬁ cant increase in interaction when the individual QD states are occupied consist-

ent with Pauli’s exclusion principle. Coupling of these QDs form a 2D-like system giving rise to

steps in conductance. This important many-body effect which explains why conductance steps

are usually led by conductance peaks was recognized only recently after LaFave and I [13a,b]

solved the effect of the discrete nature of electrons on the capacitance of a dielectric sphere, a

problem similar to the QD. In 1990 Nicollian and I decided to crystallize a -Si into Si QDs, sepa-

rated by thin oxide barriers and metallic contacts for I–V and G–V measurements [4] .

Figure 12.2 shows a schematic drawing of the modiﬁ ed MOSFET structure with nanoc-

rystallites shown as shaded circles embedded in an a -SiO

matrix. The substrate used has n 

3.5  1 0

c m

 3

silicon wafers. Starting with a thermally grown ﬁ eld of oxide 100 nm thick at

1050°C in dry oxygen, the active device, varying in size from 40  4 0 μ m

down to 10  1 0 μ m

is formed by etching a window photolithographically. A thin undoped a -Si layer,  1 5  18 nm,

is deposited by e-beam, followed by crystallization and oxidation at 800  900°C in a 3:1 dry

 O

at atmospheric pressure. The reason we did not anneal at 1000°C is to avoid oxidation

leading to deleterious effects. Typically, our samples consist of  5–10 nm crystallites surrounded

by 2.5  3 nm of oxide serving as barriers.

1.4 1.6 1.8 2.0 2.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

QD QW

(AIGa)AsGaAs

Multi-QD

T  6.9 K

Single

Multi-

Norm. PL intensity (arb. units)

1.62

1.64

1.66 1.68

Energy (eV)Energy (eV)

Figure 12.1 Photoluminescence at 5 K of InP SK QDs on 3 nm GaAs QW with Al

0.3

0.7

As barrier forming QDs

of 140 nm lateral dimension with an average density of 5 QDs/ μ m

. (a) Many QDs with line width  435 meV, and

(b) single QD with linewidth  0.6 meV at 1.66 eV, which is K

T at 6.9 K. From [6] .

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374 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

The device is electrically characterized by measuring dc I  V, capacitance and conductance

at 1 MHz as functions of gate bias V

using a voltage ramp generator, Model 410 C-V plot-

ter, at a rate of 1  10 mV/s, with a lock-in ampliﬁ er. An impedance transforming circuit with

 1 0 K Ω , and Z

out

 5 0 Ω , having an op-amp AD540K is designed to insure the output of the

voltage ramp capable of keeping pace with the demand at resonance. In usual resonant tunnel-

ling via quantum wells with n-doped contact, negative differential conductance, NDC, appears

whenever the applied voltage is such that the quantum state of the well moves below the source

of electrons from the contact into the forbidden gap. Current drops because source is cut off.

However, with a metal contact, the Fermi sphere is very large compared to all the quantum states

involved.

As shown in Fig. 12.3 , as the bias voltage sweeps the state E

below the conduction band mini-

mum, the supply of the electrons is always present with a metal contact and almost constant.

Therefore I  V is a step rather than a peak, tracing out the DOS.

Depletion layer

V

a – SiO

matrix with

Si nanoparticles

SiO

n-type Si

Figure 12.2 Modiﬁ ed MOSFET with nanocrystallites shown shaded circles in an a-SiO

matrix.

(a) (b)

Figure 12.3 Current peak with n-contact (a), conductance peak or current step with metal (b).

Empty lattice energy states of an Si QD were computed for a cube oriented in various sym-

metry directions as well as for a sphere using variational technique as shown in Fig. 12.4 [14] .

Resonant tunnelling via nano-crystalline silicon, nc-Si, embedded in an amorphous silicon

dioxide matrix has been exploited [4, 14, 15] . The smallest grain size,  300 nm, is obtained by

annealing at T  600°C of a thin layer of deposited a-Si at 30°C. Further size reduction is possi-

ble at lower deposition temperature. High resolution TEM showed almost perfect spherical shape

for the nano-crystallites [16] . However, with interfacial inhibition, a -SiO

crystallizes at the

annealing temperature above 900°C.

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Slow Oscillation and Random Fluctuation in Quantum Dots: Can we Overcome? 375

For a constant transition matrix, conductance is proportional to the density of states, DOS,

resulting in peaks for 0D and steps for 2D as shown in Fig. 12.5 . Let us ﬁ rst discuss why con-

ductance plays a leading role in the characterization of quantum dots, QDs, and quantum wells,

QWs. If the matrix element is constant, then the conductance is proportional to the density of

states, N ( E ) shown in Fig. 12.5 , for QD with 0D, Qwire with 1D and QW with 2D. For the para-

bolic energy band in 2D, DOS is a constant, m */ π , with m * being the appropriate transverse effec-

tive masses, when the energy of the level is reached for multiple levels in a QW, a series of steps

appears.

Let E

, E

, …, be the ﬁ rst, second … states in a QW. Since the conﬁ nement is in one direction,

the other two directions form a 2D system. Therefore the DOS, or N ( E ) consists of a staircase, or

steps, at the voltages corresponding to the energies E

, E

, …. With QD, the steps are replaced by

a series of δ function-like peaks. We need to be extremely cautious in dealing with input/output

for tunnelling via a QD. For layered superlattices, the equal potential surfaces representing the

input and output contacts are planes, therefore it is simple to deﬁ ne the Fermi levels for the two

contacts, and the incident and transmitted electrons are represented by plan waves. In micro-

wave transmission, the cavity resonator is basically a QD coupled to a waveguide, and via a horn

coupled to free space. What happens to radar systems? Take, for example, the scattering of plane

waves from a sphere; depending on the frequency, it may be that only the lowest dipole term is

signiﬁ cant. There is no such thing as a “ universal conductance ” or universal impedance because

Cube

(d  75 Å)

Sphere

(a  0.58d)

100

111

110

(111)

(112)

(113)

(121)

(113)

(211)

(122)

(121)

(211)

(122)

(212)

(221)

(123)

(213)

110

(011)

(012)

(013)

(111)

(112)

(213)

(123)

(212)

(122)

(211)

(121)

(113)

(112)

(111)

 0.19

m  0.916

Exact Variational

Calculated energy levels of 75

Si-cube and equivalent sphere (eV)

0.20

0.10

Figure 12.4 Calculated energy state for a d  7.5 nm Si cube, compared with a sphere of radius a  0.58 d

for aligning the ground staes. Right side gives the cube for other directions. The notation  100 

indicates the

orientation in  100  with degeneracy factor 6.

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376 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

the scattering matrix determines which incoming mode is scattered into which outgoing mode.

It is entirely similar for electron scattering, only that the problem is simpler because photons

have to contend with polarization. However, the bulk of the situation is identical; the most

important point is to recognize that no single number may serve as an impedance function. Even

for dipole terms at low frequency, each scattering term has at least four components to specify,

the incident and outgoing polarizations, as the incident and outgoing angles consist of two

sets of ( θ , φ ). Therefore one must set up the problem as in waveguides uniquely specifying the

input and output, with the QD as a plug in the waveguide. However, there is a further compli-

cation. Even in a waveguide it is generally not possible to deﬁ ne the input voltage and cur-

rent. However, one can use the Poynting vector at some plane considered as the input end, and

another plane at the output end. As shown on p. 301 of [15] , the wave impedance of the electron

waveguide is:





(12.1)

with

kkk



and





where the cutoff k

is the momentum in the transverse degree of freedom given by

kma nb

222

≡( ππ/(/))

for dimension a and b . Equation 12.1 looks similar to but not quite

the same as the waveguide case for photons. Furthermore, along the direction of propaga-

tion, unlike the sine dependences in the x and y directions, the wavefunction dependence on z

is exp( ik

z ), a propagating function, therefore periodic boundary conditions must be applied, i.e.

L  2 ᐉ π , with ᐉ being any integer. Then Eq. 12.1 becomes:

ZZ kk





1ᐉ [/]

(12.2)

with

Zhe

2 /.

(12.3)

(E) N

(E)

Figure 12.5 N ( E ) for QD with 0D, 1D and QW with 2D.

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Slow Oscillation and Random Fluctuation in Quantum Dots: Can we Overcome? 377

At this point one might be tempted to call Z

the universal impedance. In Eq. 12.2 for

a  b   , and for ᐉ  1, obviously Z  Z

. Similarly, there is no such basis on which to call G

the universal conductance. It is simply the prefactor. Apart from what I have discussed thus

far, there is another very important consideration, the unique Fermi energy serving to char-

acterize conduction and propagation through a quantum dot. As discussed, without a deﬁ nite

equal potential, how do we assign the energy in the Fermi distribution function? We know that

the potential varies from point to point. This point is underlined when LaFave and I decided to

use an average potential [13b] . To help the reader understand, we need to realize the fact that the

geometrical boundary of a QD is not an equal potential surface except for a sphere with only one

electron! With this point made, let us move on to the data.

To arrive at very interesting data, I want to emphasize the need for electrical forming. The

idea came from the fact that dull I–V, after repeated measuring of the I–V, produced interesting

structures. Because of the Schottky barrier in the device structure, all our data were taken in the

reverse bias, because with forward bias, all that is present is a large current as in any Schottky

diode. However, the forward current is very repeatable, and Nicollian and I decided to past a cur-

rent in the forward direction until structures showed up in the reverse direction. This is precisely

what is referred to as electrical forming. Incidentally, almost all electroluminescent devices are

electrically formed! Forming may have resulted in the establishment of a ﬁ lamentary conduction

path. Figure 12.6 shows how Nicollian chose to electrically form in the forward direction where

I–V is represented by the typical forward direction of a Schottky diode.

I (mA)

f

14 12 0

V (Volts)

Figure 12.6 Electrical forming used to eliminate the unwanted tissue region of an annealed crystallized Si QD.

The path (a) and (b) indicates only minute conduction until path (c) is reached. Subsequent paths, (d)–(f), indicate

structures after forming with ( f  ) often showing hysteretics. After [5] .

In Figure 12.7 , two sharp conductance peaks at   11.1 V and  11.35 V are separated by

Δ V  0 . 2 5 V. Without electrical forming, usually no sharp peaks and steps were observed. The differ-

ence between the voltage V

 1

and V

per electron is

eV V e V E e C

nn nn

() E /





2≡

(12.4)

where C is the capacitance [17] . For QDs of  6 nm, E

 1

 E

 0.06 eV, and e

/ C  0.16 eV,

or  0.22 eV per electron, which is close to Δ V  0.25 V. This indicates that bulk of the voltage

 1 V represents the voltage across the deep depletion. (Unlike the oxide gate, holes are trapped at

the interface resulting in inversion.) Thus we know that only 20% of the applied voltage appears

across the quantum dot device, and the rest appears across the substrate due to deep depletion

as ﬁ rst pointed out in [4] . Note also in Fig. 12.7 that the ﬂ at part of the G–V gives Δ G  160 μ S,

close to 4 G

, showing that the degeneracy factor is 2, one for each spin for the ﬁ rst excited state.

Detailed discussions on peaks and steps will be given later. Figure 12.8 shows that the conductance

peak has thermal linewidth, indicating that the majority of the QDs are nearly what they

are supposed to be.

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378 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Figure 12.9 shows hyteresis, indicating the presence of trapping. We have identical G–V with-

out hysteresis indicating that hysteresis is from traps. Figure 12.10 shows that conductance

steps and peaks are not affected by size variation of the contact, indicating ﬁ lamentary rather

than area conduction. We know how to use electrical forming to select peaks and steps. Slower

forming results in steps, characterized by pseudo-2D coupled QDs, while rapid forming results

in peaks, typical for 0D QDs. Usually at lower bias, mostly below  15 V, conductance steps are

preceded by conductance peaks, while at high reverse bias, only conductance steps without the

leading conductance peak. As pointed out in [4] our interpretation is that at high bias involving

higher energy states, the wave-functions of adjacent QDs are not well localized which result in

coupling resembling a 2D system. In essence, the ﬁ lamentary conduction path gives us an ideal

11.8

11.7

11.6

V (volt)

11.5

11.4

200

400

600

G (μs)

Figure 12.8 Linewidth of a typical G–V is  k

T , at T  300 K.

T  300 K

Voltage (V)

14

200

400

600

800

1000

13 12 11 10 9 8 7 6

Current (μA)

800

600

400

200

Conductance at 1 MHz (μs)

Figure 12.7 Typical I  V and G  V for devices after electrical forming. Forming, a sort of controlled spot heating,

is performed with a current limiter usually for 15 minutes. Note the typical conductance steps with Δ G  160 μ S,

led by conductance peaks.

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