Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics

Подождите немного. Документ загружается.

Quantum Dot-based Mode-locked Lasers and Applications 619

48. G. Moreau , S. Azouigui , D.-Y. Cong , K. Merghem , A. Martinez , G. Patriarche , A. Ramdane ,

F. Lelarge , B. Rousseau , B. Dagens , F. Poingt , A. Accard , and F. Pommereau , Effect of layer stack-

ing and p -type doping on the performance of InAs/InP quantum-dash-in-a-well lasers emitting at

1.55 μ m , Appl. Phys. Lett. 89 , 241123 ( 2006 ) .

49. H. Dery , E. Benisty , A. Epstein , R. Alizon , V. Mikhelashvili , G. Eisenstein , R. Schwertberger , D. Gold ,

J.P. Reithmaier , and A. Forchel , On the nature of quantum dashes , J. Appl. Phys. 95 ( 11 ) , 6103 –

6111 ( 2004 ) .

50. J. Renaudier , R. Brenot , B. Dagens , F. Lelarge , B. Rousseau , F. Poingt , O. Legouezigou ,

F. Pommereau , A. Accard , P. Gallion , and G.-H. Duan , 45 GHz self-pulsation with narrow linew-

idth in quantum dot Fabry–Perot semiconductor lasers at 1.5 μ m , IEE Electron. Lett. 41 ( 1 8 ) ,

1007 – 1008 ( 2005 ) .

51. C. Gosset , K. Merghem , A. Martinez , G. Moreau , G. Patriarche , G. Aubin , J. Landreau , F. Lelarge ,

and A. Ramdane , Subpicosecond pulse generation at 134 GHz and low radiofrequency spectral

linewidth in quantum dash-based Fabry–Perot lasers emitting at 1.5 μ m , IEE Electron. Lett. 42 ( 2 ) ,

91 – 92 ( 2006 ) .

52. C. Gosset , K. Merghem , A. Martinez , G. Moreau , G. Patriarche , G. Aubin , and A. Ramdane ,

Subpicosecond pulse generation at 134 GHz using quantum dash-based Fabry–Pérot laser emit-

ting at 1.56 μ m , Appl. Phys. Lett. 88 , 241105 ( 2006 ) .

53. C. Gosset , K. Merghem , G. Moreau , A. Martinez , G. Aubin , J.-L. Oudar, A. Ramdane and F. Lelarge,

Phase-amplitude Characterization of a high repetition rate Quantum Dash Passively Mode-Locked

Laser , Optics Letters . 31 ( 12 ) , 1848 ( 2006 ) .

54. K. Merghem, C. Gosset, A. Martinez, G. Moreau, F. Lelarge, G. Aubin, and A. Ramdane, Effect of

spectrum ﬁ ltering on the performances of quantum-dash mode-locked lasers emitting at 1.55

micrometer, Techni. Digest Conference on Lasers and Electro-Optics/Europe-IQEC Conference , paper

CB13-5-THU, (2007).

55. G.A. Keeler , B.E. Nelson , D. Agarwal , C. Debaes , N.C. Helman , A. Bhatnagar , and D.A.B. Miller ,

The beneﬁ ts of ultrashort optical pulses in optically interconnected systems , IEEE J. Sel. Topics in

Quantum Electron. 9 ( 2 ) , 477 – 485 ( 2003 ) .

56. E.U. Cataluna , A.D. Rafailov , W. McRobbie , D.A. Sibbett , Livshits , and A.R. Kovsh , Stable mode-

locked operation up to 80°C from an InGaAs quantum-dot laser , IEEE Photon. Technol. Lett.

18 ( 14 ) , 1500 – 1502 ( 2006 ) .

57. D.G. Deppe , H. Huang , and O.B. Shchekin , Modulation characteristics of quantum-dot lasers:

the inﬂ uence of p-type doping and the electronic density of states on obtaining high speed , IEEE

J. Quantum Electron. 38 ( 12 ) , 1587 – 1593 ( 2002 ) .

58. M. Ishida , N. Hatori , K. Otsubo , T. Yamamoto , Y. Nakata , H. Ebe , M. Sugawara , and Y. Arakawa ,

Low-driving-current temperature-stable 10 Gbit/s operation of p-doped 1.3 μ m quantum dot

lasers between 20 and 90°C, IEE Electron. Lett.

43 ( 4 ) , 219 – 221 ( 2007 ) .

59. B. Dagens, O. Bertran-Pardo, M. Fischer, F. Gerschütz, J. Koeth, I. Krestnikov, A. Kovsh, O. Le

Gouezigou, and D. Make, Uncooled directly modulated quantum dot laser 10Gb/s transmission

at 1.3 μ m, with constant operation parameters, Techni. Digest of European Conference on Optical

Communications 2006, PD Paper Th4.5.7, (2006).

60. T.J. Badcock , R.J. Royce , D.J. Mowbray , M.S. Skolnick , H.Y. Liu , M. Hopkinson , K.M. Groom , and

Q. Jiang , Low threshold current density and negative characteristic temperature 1.3 μ m InAs self-

assembled quantum dot lasers, Appl. Phys. Lett. 90 , 111102 ( 2007 ) .

61. E.A. Cataluna , P. Viktorov , W. Mandel , D.A. Sibbett , J. Livshits , A.R. Weimert , J. Kovsh , and

E.U. Rafailov , Temperature dependence of pulse duration in mode-locked quantum-dot laser ,

Appl. Phys. Lett. 90 , 101102 ( 2007 ) .

62. J. Renaudier , B. Lavigne , F. Lelarge , M. Jourdan , B. Dagens , O. Legouezigou , P. Gallion , and

G.-H. Duan , Standard-compliant jitter transfer function of all-optical clock recovery at 40 GHz

based on a quantum-dot self-pulsating semiconductor laser , IEEE Photon. Tech. Lett. 18 ( 1 1 ) ,

1249 – 1250 ( 2006 ) .

63. G.-H. Duan, B. Lavigne, J. Renaudier, F. Lelarge, O. Legouezigou, B. Dagens, and A. Accard,

Importance of the beating spectral linewidth on the perfomance of the recovered clock at 40

Gb/s using self-pulsating lasers, Proc. Conference on Lasers and Electro-Optics 2006 , paper CMN3

(2006).

CH020-I046325.indd 619CH020-I046325.indd 619 6/25/2008 10:03:39 AM6/25/2008 10:03:39 AM

Quantum Dot Infrared Photodetectors by Metal-Organic

Chemical Vapour Deposition

Manijeh Razeghi , Wei Zhang , Hochul Lim, and Stanley Tsao

Center for Quantum Devices, Department of Electrical Engineering and Computer Science,

Northwestern University, Evanston, IL 60208

21.1 Introduction

The strong interest in low-dimensional semiconductor structures originates from their exciting

electronic properties, which have an important impact on the performance of high-speed elec-

tronic and photonic devices. The quantum dots (QDs), also known as quantum boxes, are nano-

metre scale islands in which electrons and holes are conﬁ ned in three-dimensional potential

boxes. They are expected to show a zero-dimensional, δ -function density of states and are able to

quantize an electron’s free motion by trapping it in a quasi-zero-dimensional potential conﬁ ne-

ment. As a result of the strong conﬁ nement imposed in all three spatial dimensions, quantum

dots are similar to atoms. They are frequently referred to as “ artiﬁ cial atoms ” . Due to this con-

ﬁ nement, novel physical properties will emerge, which can lead to new semiconductor devices as

well as drastically improved device performance.

As the particles are conﬁ ned in all three dimensions, there is no dispersion curve and the den-

sity of states is just dependent on the number of conﬁ ned levels. For one single dot, only two (spin-

degenerate) states exist at each energy level and the plot of the density of states versus energy will

be a series of δ -functions. Figure 21.1 shows the change of the density states from a bulk system

to the low-dimensional systems of quantum well (QWL), quantum wire (QWR) and QD.

In QDs, the width of the electron energy distribution is zero in an ideal case. This means that

electrons in those structures are distributed in certain discrete energy levels and the energy dis-

tribution width is fundamentally independent of temperature. In real semiconductor structures,

due to many interaction processes such as electron–electron and electron–phonon scattering

(which can also be reduced by QDs due to the lack of phonons to satisfy the energy conservation,

which is the so-called phonon bottleneck [1] ), a certain width in the electron energy distribution

exists; However, it is expected to be much smaller compared to bulk and QW.

QDs have generated great interest as a new material and structure that are expected to lead

to novel semiconductor devices or to improve current existing devices ’ performance. One such

example is QD lasers. The main advantages of QD lasers over the conventional QW lasers are

lower threshold current density, high gain, weak temperature dependence (high characteristic

temperature T

) and low chirp [2, 3] .

Another application of QDs is quantum dot infrared detectors (QDIPs), which is the main focus

of this chapter. Many applications require thermal imaging that is collected by infrared focal plane

arrays (FPAs). So far, most mid-wavelength infrared (MWIR) and long-wavelength infrared (LWIR)

photodetector FPAs are based either on HgCdTe (MCT) [4] or quantum well infrared photodetectors

CHAPTER 21

CH021-I046325.indd 620CH021-I046325.indd 620 6/27/2008 5:47:54 PM6/27/2008 5:47:54 PM

Quantum Dot Infrared Photodetectors by Metal-Organic Chemical Vapour Deposition 621

(QWIPs) [5] . MCT FPAs suffer from difﬁ cult material growth, device instability, high array non-

uniformity, and very high cost. QWIPs have been used successfully for FPAs. However, a limitation

of QWIPs is that, due to the transition selection rules, the most widely used n -type QWIPs are not

sensitive to normally incident light and typically have a narrow response range in the infrared. P -

type QWIPs are able to detect normal incident light due to the band mixture; however, they have

not found many practical applications due to their low detectivity.

An extension of QWIP is QDIP, which utilizes intersubband absorption between bound states

in the conduction/valence band in QDs. Given high uniformity and high density quantum dot

layer, QDIPs are predicted to outperform QWIPs due to their inherent sensitivity to normal inci-

dence radiation and reduced phonon scattering. Higher temperature operation and lower dark

current are also expected for this type of device [6] . So far, most QDIPs reported have showed

inferior performance compared to that of QWIPs with similar parameters. The major challenges

facing QDIPs are QD fabrication. To justify its potential advantages, QDIPs need high-uniform

and high-density QD layers. New device designs for QDIPs are also required to further improve

their performance as an infrared photodetector.

The condition for new electronic properties to occur in a QD device is that the lateral size of

their active region must be smaller than the coherence length and the elastic scattering length of

the carriers. Additional quantum-size effects require the structural features to be reduced to the

range of the de Broglie wavelength. The advantages in operation depend not only on the absolute

size of the nanostructures in the active region, but also on the uniformity of size. A large distri-

bution of sizes would “ smear ” the density of states of QDs thus making it more like that of bulk

material. Therefore, the repeatable fabrication of these nanometre three-dimensional quantum

structures requires methods with atomic scale accuracy, which is a major challenge for current

micro- or nanostructure material technologies.

The fabrication technique of quantum dots can be categorized into a “ top-down ” method

using lithography and etching, and a “ bottom-up ” method using self-assembly. The “ top-down ”

method usually includes electron beam lithography, dry etching, and sometimes regrowth

on patterned substrates. QDs can be etched from QW structures via low energy electron-beam

lithography. Another method of creating quantum dots is realized by applying voltage to nano-

electrodes. The spatially modulated electric ﬁ eld created by the voltage localizes the electrons

in a small area. Quantum dots can also be created through the selective growth of a narrow

gap semiconductor material on a patterned wide gap substrate. The problem of such a “ top-

down ” method is the low optical efﬁ ciency of such dots: high surface-to-volume ratios of these

DOS 

211

112

111

113

DOS 

QD QWR

QWL

Bulk

Figure 21.1 Density of state of 200-dimensional (upper-left), one-dimensional (upper-right), two-dimensional

(lower-left), and bulk (lower-right) systems.

CH021-I046325.indd 621CH021-I046325.indd 621 6/27/2008 5:47:55 PM6/27/2008 5:47:55 PM

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

nanostructures and associated high surface recombination rates, plus the damage introduced

during the fabrication, prevent the successful formation of high-quality QD devices.

A recent breakthrough in QD fabrication techniques is self-assembly based on the Stranski–

Krastanov growth mode [7] . In this method, the lattice constants of the substrate and the

crystallized material differ greatly, so only the ﬁ rst few deposited monolayers crystallize in the

form of epitaxial strained layers where the lattice constant is equal to that of the substrate.

When the critical thickness of the epitaxial layer is exceeded, the signiﬁ cant strain that occurs

in the epitaxial layer leads to the break down of this ordered structure and to the spontaneous

creation of randomly distributed islets (e.g. quantum dots) of regular shapes and similar sizes.

The small sizes of the self-assembled quantum dots, the homogeneity of their shapes and sizes

in a macroscopic scale, the perfect crystal structure, and the fairly convenient growth process,

without the necessity to precisely deposit electrodes or etching, are among this method’s greatest

advantages.

This rest of the chapter is organized as follows: in section 21.2 we discuss the theoretical calcu-

lations and modelling of QDIPs; in section 21.3, the MOCVD growth and characterization of the

materials for QDIP device are presented. We show that the size, shape, and density of the InAs

QDs are strongly affected by growth conditions such as temperature, V/III ratio, growth rate, rip-

ening time and the matrix materials. Section 21.4 focuses on the fabrication and measurement

of QDIP devices. Next, FPAs based on our QDIPs are presented in section 21.5. Finally, we will

summarize and discuss brieﬂ y how to further improve the QDIP device performance.

21.2 Theoretical modelling of quantum dot infrared detectors

It is important to understand the correlations between the design parameters and the device char-

acteristics for QDIPs. However, in order to make correlations between them, one needs to know

what is going on inside the device and should have certain pictures about the physics of the device.

In this section, the modelling of QDIPs will be presented based on the semi-phenomenological

theory. The modelling includes the calculation of energy levels and oscillator strengths, respon-

sivity, dark current, gain and detectivity.

21.2.1 Single-band effective mass envelope function method

The simplest model for the quantum dot energy level calculation is when a quantum dot is sur-

rounded by an inﬁ nite potential barrier. Even though it is not realistic, a general idea about the

discrete energy levels and other properties can be obtained. The wavefunctions of a quantum dot

surrounded by an inﬁ nite potential barrier are used for the basic functions of the single-band

ﬁ nite potential problem. In the case where the quantum dot has cylindrical symmetry, the basis

wavefunctions can be obtained by solving Schrödinger’s equation in the cylindrical inﬁ nite

potential wall.





Figure 21.2 One particle inside a cylindrical inﬁ nite potential box.

CH021-I046325.indd 622CH021-I046325.indd 622 6/27/2008 5:47:55 PM6/27/2008 5:47:55 PM

Quantum Dot Infrared Photodetectors by Metal-Organic Chemical Vapour Deposition 623

The cylinder has radius a and height b . The Hamiltonian has the cylindrical symmetry which

is azimuthal around the z axis and is given as:

ˆˆ



⎛

⎝

⎜

⎞

⎠

⎟

(21.1)

where

.pL

zρ

ρρ ρ φ

22 22

 

∂

⎛

⎝

⎜

⎞

⎠

⎟

∂

and

The boundary condition for this problem is:

sin kb J K a

qmmn

()0

(21.2)

where J

is the m -th order of Bessel function and K

is the n -th zero of J

After solving the Hamiltonian with the boundary condition above, the eigenfunctions and

eigenenergies can be calculated as follows:

qmn

mmn

mmn q

xyz

baJ K a

JK kze(, ,)

[( )]

( )sin

′

(21.3)

qmn mn q





()

(21.4)

For the design of a real QDIP device, it is desirable to have such a tool that the energy levels

and their transitions can be easily calculated and can be applied to the design of the device struc-

ture. The process between the design and the growth should be quick. As a further approxima-

tion to multi-band effective mass theory, the single-band effective mass envelope function method

can be used for the energy levels of quantum dots.

Gershoni et al . [8] developed a numerical method in which they expand the envelope function

of a rectangular quantum wire which is a 2D conﬁ ned system using a complete orthonormal set

(COS) of periodic functions, which are solutions for a rectangular wire with an inﬁ nite barrier

height and suitably chosen dimensions. The advantage of this method is that it can be applied

to structures of arbitrary shape. Moreover, all the matrix elements can be calculated analyti-

cally. Gangopadhyay and Nag [9] extended this method to study 3D conﬁ ned structures such as

pararellepipeds and cylinders.

Wetting layer

Quantum dot

Barrier

X 20,000 nm/dlv

210,000 nm/dlv

Figure 21.3 Left : AFM image of uncapped InAs quantum dot on InP. Right : The calculation model of a capped

quantum dot.

CH021-I046325.indd 623CH021-I046325.indd 623 6/27/2008 5:47:55 PM6/27/2008 5:47:55 PM

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

The aim of this section is to extend Gershoni et al. ’s method to determine the energy levels of

current quantum dots under discussion.

Our InAs QDs on InP grown by MOCVD have lens-like shape as shown in Fig. 21.3 . In order

to model the quantum dot energy levels, it is necessary to simplify the geometry of quantum

dots into perfect lens shape as shown on the right of Fig. 21.3 . The quantum dot is varied inside

the barrier cylinder which has inﬁ nite potential wall. The size of the barrier cylinder should

be independent of the calculation results when it is bigger than a certain size. Usually the size of

the barrier cylinder is chosen so that four times QD height is the height of the cylinder and four

times QD radius is the radius of the cylinder. The dimensions of the lens-shaped quantum dot are

usually taken from AFM (atomic force microscope) measurement. For example, one of the InAs

QDs on InP is shown on the left of Fig. 21.3 .

Schrödinger’s equation for the envelope function in the effective mass approximation can be

written as:

∇∇

⎛

⎝

⎜

⎞

⎠

⎟

rVrr Er

()

() () () ()



 

ΨΨΨ

(21.5)

The unit length is the Bohr radius

()  /

0.529 Å and the unit energy is the Rydberg con-

stant Ry (  M

/2

) 13.6 eV. This Schrödinger’s equation is Hermitian and its wavefunctions

are orthogonal and the probability current is conserved at the interface of the heterojunction.

The envelope function of the quantum dot with a lens shape Ψ ( x , y , z ) is then expanded in terms

of a complete orthonormal set of solutions ψ

lmn

( x , y , z ) of the cylindrical problem with inﬁ nite

barrier height (see Eq. 21.3):

Ψ(, ,) (, ,)xyz a xyz

lmn



∑

(21.6)

πρ ρ

lmn

mmn

xyz

HJk

Jkr l

(, ,)

[()]

( )sin



01 0

⎛

⎝

⎜

⎞

⎠

⎟

⎟⎟

⎟

⎡

⎣

⎢

⎤

⎦

⎥

imφ

(21.7)

The boundary condition is J

( k

)  0. We have chosen the domains [  H/2 , H/2 ] and [0, ρ

]

for the variation z and r . This approach does not need explicit matching wavefunctions across the

boundary between the barrier and dot materials. This method is easily applicable to an arbitrary

conﬁ ning potential. Substituting Eq. 21.6 into Eq. 21.7, multiplying on the left by

lmn

, and

ﬁ nally integrating over the cylinder, yields the matrix equation:

()AE a

lmnl m n mm nn ll

lmn

′′′ ′ ′ ′

δδδ 0

(21.8)

The matrix element A

lmnl



is given by:

lmnl m n l m n

lmn

′′′ ′′′

′′′

∇∇

∫





ψψυ

()

cylinder



lmn

rVr rd

() () ()

cylinder

∫

 

ψυ

(21.9)

The problem in above equation is the discontinuity of effective mass in passing from the well

region into barrier region. In order to overcome this problem, the integral can be split into three

parts, within each of which the effective mass is constant [10] . First, we take an integral with

m *  m

over the whole cylinder which also includes the quantum dot and wetting layer (the

well region). Second, we subtract the integral with m *  m

over the well region and third, we

CH021-I046325.indd 624CH021-I046325.indd 624 6/27/2008 5:47:56 PM6/27/2008 5:47:56 PM

Quantum Dot Infrared Photodetectors by Metal-Organic Chemical Vapour Deposition 625

add the integral with m *  m

over the well region. The same procedure has been adopted for

the integral containing the potential. The ﬁ nal expression for the matrix element is:

lmnl m n

′′′

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤



⎦⎦

⎥

⎛

⎝

⎜

⎞

⎠

⎟

∇∇

′′′

δδ δ

ψψ

ll mm nn

lmn

rr

() (



))

() ()

Vrrd

lmn

ψψυ

QD WI





∫

′′′



(21.10)

where the subscript QD  WL in the integrals means that the integration is over the quantum dot

and wetting layer inside a cylinder. The ﬁ rst term of Eq. 21.10 is simply the free particle energy

inside a cylinder.

In order to calculate the matrix element A

lmnl



, the integrals need to be calculated over the

quantum dot and wetting layer. But these volume integrations can be done analytically for the lens

shape geometry.

In this work, we have used 70 sine functions and 10 Bessel functions as basis functions for

expanding the envelope functions. Equation 21.10 is a 700  700 matrix and can be solved

numerically. This method which we have discussed is not the most accurate of the available

methods [11, 12] , but it is simple to use, versatile, and good enough for our modelling for QDIPs.

21.2.2 Oscillator strength

In order to calculate the optical absorption spectrum or analyse the photocurrent spectrum in

the QDIP, one requires the energy levels and the oscillator strengths for transitions between the

various states. The oscillator strength is the measure of the interaction between the light and

electrons (or holes). When the incoming light enters the quantum dots, the electrons in the dis-

crete energy levels gain the energy and experience dipole transitions. The rate of the dipole tran-

sition can be obtained from Fermi’s golden rule. The oscillator strength for a transition from a

level i to a level j is given by:

fipjm

ij ij

2

|| | |〈〉



ηω(* )

(21.11)

where

|i〉

and

|j〉

are wavefunctions of the quantum dot,



is the photon polarization,



is the

electron momentum operator, and ω

is the transition frequency. It may be noted that a spheri-

cal QD of cubic material is optically isotropic, and the oscillator strength is independent of the

polarization of the incoming light. In reality, the quantum dot has asymmetric shape and there-

fore the oscillator strength of the quantum dot has strong dependence on polarization. For the

normal incidence of light, which is perpendicular to the growth plane, the incoming light has

in-plane polarization (or s -polarization) which is parallel to the growth plane. If the quantum

dot has rotational symmetry, the wavefunctions of the quantum dot also has rotation symme-

try. In such a case, the in-plane coordinates such as x and y do not make any difference. For the

x -polarization (or s -polarization),



η  p

is    /  x and for the z -polarization, it is  i   /  z . In order

to calculate the oscillator strength numerically, it is necessary to know the wavefunctions and

their derivatives. In a later section, the results of the calculation of the oscillator strength for the

quantum dots in our QDIPs will be discussed.

21.2.3 Absorption

The absorption of the light in the QDIPs mostly happens in the quantum dots. The absorption

coefﬁ cient α ( ω ) can be written as

εε ω









mc E

nnf

dop

gege

()

⎡

⎣

⎢

⎤

⎦

⎥

(21.12)

CH021-I046325.indd 625CH021-I046325.indd 625 6/27/2008 5:47:56 PM6/27/2008 5:47:56 PM

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

where n

is the refractive index, c is the velocity of light, and Γ is the total level width. The

absorption coefﬁ cient also involves the following quantities: (i) the dot density N

, (ii) the oscil-

lator strength f

, and (iii) the probability n

that the carriers remain in the initial state and the

probability n

that the carriers stay in the excited state. As you can see, the absorption can be

increased by increasing the dot density and the oscillator strength and maximizing n

(1  n

). In

order to increase the dot density, the growth condition of the quantum dot should be optimized.

Increasing the oscillator strength is not a simple problem and will be discussed later. The occupa-

tion probabilities n

and n

can be calculated if the energy levels of the quantum dot are known.

Assuming Boltzmann statistics for convenience, n

can be written:

de e d f N

EkT









∑∑

∫

ερ ε ε

()()/

(21.13)

where the E

are the quantum dot energy levels, d

the degeneracy, the “ t ” sum is over traps

including the new eigenstates formed by electron–phonon resonance; ρ ( ε ) is the band density of

states and f ( ε ) the Fermi function. As we can see, the absorption of the quantum dot depends on

the occupation probabilities of the levels. Experimentally these occupation probabilities can be

controlled through the doping of the quantum dot.

21.2.4 Modelling of responsivity and photocurrent

When the incoming infrared light is absorbed in the QDIP, the electrons inside quantum dots are

photoexcited into the continuum state under the bias, and those electrons can be collected as a

photocurrent. The photoexcited electrons should escape from quantum dots to be detected. This

escape process involves tunnelling and thermal activation. If R is the responsivity as a function

of temperature T and applied bias V , then the photocurrent I

ﬂ owing is given by:

IARTVP

 (),

(21.14)

where P

is the optical power per unit area, and A is the illuminated area of the device. Collecting

together the terms, we can write the peak responsivity in terms of the quantum efﬁ ciency η and

the gain g :



ω

(21.15)

We can deﬁ ne the quantum efﬁ ciency η

ηα

νν ν







EkT

eff

⎡

⎣

⎢

⎤

⎦

⎥

(21.16)

where the ﬁ rst factor α L is the absorbance, L is the device length, and α ( ω ) the absorption coef-

ﬁ cient. The factor “ g ” in Eq. 21.15 is the gain deﬁ ned as the ratio of the recombination time over

the transit time:



(21.17)

where μ F / L is the transit time, and C

the capture rate of electrons from the continuum band

into the bound excited state of the quantum dot. The third factor in Eq. 21.16 is the ratio of the

escape rate W

(  ν

exp[  E

eff

/ kT ]) out of the excited state to the continuum and the inverse life-

time of the excited state ( ν

). In this section, we will concentrate on this factor which is related to

the escape rate, and the quantum efﬁ ciency and gain will be discussed in a later section.

CH021-I046325.indd 626CH021-I046325.indd 626 6/27/2008 5:47:57 PM6/27/2008 5:47:57 PM

Quantum Dot Infrared Photodetectors by Metal-Organic Chemical Vapour Deposition 627

21.2.5 Escape rate

In the dot, the electrons can tunnel out back to the extended state outside the quantum dot if the

electric ﬁ eld is applied. Transmission out of the dot through a triangular barrier which is created

by an electric ﬁ eld, known as Fowler–Nordheim tunnelling, is given as follows:

TmEeFxdx

 exp



().

′′

⎡

⎣

⎢

⎤

⎦

⎥

∫

(21.18)

Another process involved for the escape process is thermal activation. In Fig. 21.4 , the physical

picture is that (1) the electron in the excited state can thermally activate to the continuum state

or (2) directly tunnel out or (3) thermally activate and tunnel to continuum state. In order to

estimate the escape rate, the sum of all the possible paths should be required.

max

(1)

(2)

(3)

Figure 21.4 Escape paths out of the excited state to the continuum state.

If the longest tunnelling path is divided into a lattice constant a , the possible escape occurs at

the point whose distance from the well is multiple lattice constants ( na ) ( Fig. 21.4 ).

The escape rate through one path is given by:

WEeFxdx

EeFna

nec ec

2  



νγexp exp()

′′

⎡

⎣

⎢

⎤

⎦

⎥

⎛

⎝

⎜

∫

⎞⎞

⎠

⎟

(21.19)

where ν

is the attempt frequency, F is the applied electric ﬁ eld,

γ  2m*/

and E

is the height

of the escape barrier. The total escape rate is the sum of the all the escape rates through each

path. With appropriate approximation, the escape rate is ﬁ nally given by:

WEFkT

EkT aE

ec ec

ec c

ec ec







exp /

exp / exp ]

eff

[()]

[][2

eexp / exp

exp exp /

[][/]

[][]





EeF eFakT

aE eFa kT

(21.20)

where g

is the density of ﬁ nal states of escaping charge at the band edge which is reach-

able within kT. ν

varies between (a) the value of a phonon frequency multiplied by the prob-

ability of ﬁ nding the charge at a given site in the quantum dot localized state, giving ν

 10

to 10

Hz, and (b) the excited state pure tunnelling attempt frequency  E

/ h. g

is given by

 10

( kT / e )

3/2

 10, assuming three-dimensional plane wave-like states. Even though clearly

is not a constant but depends on the path chosen, we shall assume that the product ν

is a

ﬁ t parameter varying between 10

and 10

H z .

21.2.6 Gain

There are two different deﬁ nitions of gain in the QWIP which can also be applied to the QDIP. One

is the noise gain and the other is photoconductive gain. The gain factor g in Eq. 21.15, which is the

CH021-I046325.indd 627CH021-I046325.indd 627 6/27/2008 5:47:57 PM6/27/2008 5:47:57 PM

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

photoconductive gain ( g

) cannot be deduced easily from the experiment. Later, in order to ﬁ t the

experimental responsivity to a theoretical one, we should use the noise gain instead of the photo-

conductive gain. The assumption that the photoconductive gain is equal to the noise gain should

be checked before using the approximation. Borrowing from the QWIP case, one uses the genera-

tion–recombination (G–R) noise formula which says that the noise power spectrum or the square

of the noise current

is proportional to the dark current I

dark

at given bandwidth Δ f

. But

this equation can be also expressed with the correctional factor F the photoconductive gain [14] :

IegIfegFIf

nn e

44

dark dark

ΔΔ.

(21.21)

Here

Fr r pp p

cc c

   () ( )11 1/ln

where p

is the capture probability and r  t

/ t

where t

and t

are transit times across one well and one period, respectively. For the case when r is close to

1, the correction is small for all p

. For the small capture probability ( p

 0.2), the correction fac-

tor F is greater than 0.9 and relatively independent of r (see Fig. 21.5 ).For the small p

, the pho-

toconductive gain can be equal to the noise gain to a good approximation. The photoconductive

gain in the QDIP can be expressed in terms of the capture probability p

, the number of quantum

dot layers N , and the ﬁ ll factor of the quantum dots Q .

QNp



12/

(21.22)

with Eq. 21.22, the capture probability of the electrons in the QDIP can be extracted.

Identifying the proportionality factor “ g

” with gain as deﬁ ned by Eq. 21.21 implies that the

current really is one of generating and recombining carriers in a band. Note also that some

authors, in analogy to QWIPs, assume that the recombination in QDIPs is drift limited [15] . From

this it would follow that the drift velocity dependence in the gain drops out and the gain can be a

constant. This is, however, not justiﬁ ed in our devices which we are going to discuss later, where

the wetting layer scatters and reduces the band mobility, but is too thin to capture charge.

0.5

Correction factor F

0 0.2 0.4 0.6 0.8 1

Capture probability p

0.2

0.5

 0.9

Figure 21.5 The correction factor F as a function of capture probability p

f o r r  0, 0.2, 0.5, and 0.9. The

straight solid line is predicted by the theory presented by Beck ( [16] , Ref. 14).

21.2.7 Dark current

Let’s consider now the transport of the electrons in the dark situation [17–19] . This can be under-

stood as follows: electrons are emitted from a dot by ﬁ eld assisted tunnelling and thermal activa-

tion on a timescale which depends on the local temperature and bias, and on the eigenstates.

A charge already in the top most excited state will, for example, be emitted with an escape

rate with Eq. 21.20. In the dark current, this time has to be lengthened by dividing it with the

probability that the level is occupied.

CH021-I046325.indd 628CH021-I046325.indd 628 6/27/2008 5:47:58 PM6/27/2008 5:47:58 PM