CHAPTER 10
Optoelectronic Properties of Self-Assembled InAs/InGaAs Quantum Dots
SANJAY KRISHNA
Center for High Technology Materials, Electrical and Computer Engineering Department, University of New Mexico, 1313 Goddard SE, Albuquerque, NM 87106
1 Introduction
1.1 Background on Quantum Dots
1.2 Quantum Dots: A Historical Perspective
1.3 Advantages of Quantum Dots
1.4 Conditions for Quantum Confinement
2 Self-Assembled Quantum Dots: Growth, Strain Distribution and Band Structure
2.1 Self-Assembly of Nanostructures
2.2 Energetics of Quasi-Zero-Dimensional Coherently Strained Islands
2.3 Self-Assembled Quantum Dots
3 Intersubband Carrier Dynamics in Self-Assembled Quantum Dots
3.1 Background
3.2 Dynamics of Hot-Carriers: Carrier Relaxation and Phonon Bottleneck
3.3 Application of Favorable Dynamics to Intersubband Devices
4 The Intersubband Quantum Dot Detector
4.1 Overview
4.2 Vertical Quantum Dot Detectors
4.3 Lateral Quantum Dot Detectors
5 Conclusion
ABSTRACT
In the past decade, self-assembled quantum dots (SAQDs) grown by epitaxial techniques such as molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) have become a topic of extensive research not only for the fundamental understanding of fascinating physics that exists in zero-dimensional systems but also for their application in electronic and optoelectronic devices. In this article, the basic properties and device applications of SAQDs are reviewed. After a brief introduction, which includes a historical perspective of the work undertaken on SAQDs, the thermodynamics of the formation of these dots in the “Stranski-Krastonow” growth are discussed. The carrier dynamics and relaxation mechanisms present in these dots are then reviewed. Finally, the properties of an important device that is expected to exploit the favorable properties of SAQDs, namely, the intersubband QD detector, are discussed.
1. INTRODUCTION
1.1Background on Quantum Dots
The development and application of compound semiconductor based diodes for lasers and photodetectors has revolutionized the ever-changing face of technology, a fact that was acknowledged by the Royal Swedish Academy of Sciences by awarding the year 2000 Nobel prize in Physics "for development of semiconductor heterostructures used in high-speed- and opto-electronics". In particular, with the advent of optical fibers and the ever increasing need for high-speed, large-volume data transmission amply fed by the rising internet traffic [1], the communications market has more than doubled in the past year [2]. To cope up with this astounding increase in bandwidth, novel technologies are being researched on a large scale [3].
One such novel technology is the development of quantum dots for semiconductor lasers, detectors and modulators. It has been known for a while that incorporation of quantum dots in the active region of optoelectronic devices would drastically improve device performance [4,5], primarily due to the enhanced Dirac-delta-like density of states function. The density of states, which is a measure of the total number of quantum mechanically allowed energy states per unit volume, is a very important parameter in solid-state physics and a large density of states is highly desirable for optoelectronic devices such as lasers and detectors. As one goes from a bulk material in which there is no quantum confinement, to a system in which the carriers are allowed to move in two dimensions (quantum well), one dimension (quantum wire) or in quasi-zero dimensions (quantum dots), the density of states function increases in magnitude and becomes discretized. The density of states in a semiconductor for various levels of confinement is shown in Figure. 1. Some of the major advantages of lasers with quantum dots in the active region are:
? A significant decrease in the threshold current [6] and in the temperature dependence of the threshold current [7]
? A large increase in the differential gain and modulation bandwidth [8]
? A vastly reduced chirp and low linewidth-enhancement factor [9-10]
But the fabrication of these ultra- small (~10-100?) objects in the laboratory has been a major challenge. Even when these objects were formed, by using direct patterning or otherwise [11], they lacked not only the large areal density that was required to make them feasible for devices but also the high optical quality that was an essential prerequisite for optoelectronic devices such as lasers, modulators and detectors.
It was only a decade back, that a novel technique, called self-organization or self-assembly, enabled the realization of coherently strained three-dimensional islands or “quantum dots” [12-15]. It was observed that under certain conditions, during the heteroepitaxy of lattice-
mismatched systems, either by molecular beam epitaxy (MBE) or by metal-organic chemical vapor deposition (MOCVD), these dots were formed. Soon after, various electronic and optoelectronic devices such as lasers [16-18], mid infrared detectors [19-23], emitters [24-27], single electron transistors [28,29] and electro-optic devices [30] were reported using quantum dot active regions. In the past 5 years, various research efforts have focused on the optimization of the growth of the dots and quantum dot devices have demonstrated rapid progress. This is epitomized by the fact that the two present day technologies that are competing for the commercial viability of the technologically important GaAs-based 1.3 μm laser are the self-assembled quantum dots and dilute nitride-based alloys such as InGaAsN, which will be addressed by Pan (Chapter 8) and Leibiger et al. (Chapter 9) in this book .
Fig.1 Density of states, bandstructure and carrier distribution (as shown in the shaded area) for quantum confined heterostructures (a) bulk, (b) quantum well, (c) quantum wire and (d) quantum dots.
Bulk E-E c E c Quantum Well c
c Quantum Wire Quantum Dot c (a) (b) (c) (d) 1.2 Quantum Dots: A Historical Perspective
By the late 1980’s the important properties of quantum wells and superlattices were fairly
well understood and the interest shifted towards structures with further reduced dimensionality. It was known that the reduction of the remaining two dimensions of a quantum well would lead to carrier localization in all three dimensions and to a breakdown of the classical band structure model of a continuous dispersion curve. Perhaps the first realization of quantum dots were nano-size semiconductor inclusions, like CdSe, in glass [31]. Quantum confinement in this system was
verified experimentally by Ekimov and Onushenko [32]. The electrically isolating matrix, however, inhibited electric injection of carriers.
The first successful demonstration of three-dimensional quantum confinement was in field-effect-confined quantum dots fabricated by direct patterning using high-resolution lithography [33,34]. However, the reactive ion etching process induced severe damages to the active dot material, which rendered it optically dead. A variation of this approach was “implantation induced intermixing”, which used a quantum well heterostructure and lithographically defined masks to define the dot regions [11,35]. However the maximum achievable resolution of the intermixing process and the inability to obtain high degree of intermixing of the quantum wells proved to be the limitation of this technique.
One highly successful method of fabrication of quantum wires and dots was through epitaxial growth on substrates patterned with V-grooves [36,37]. In this technique, a net preferential migration of adatoms away from patterned sloped sidewalls to the bottom of the V-groove results in more material at the bottom and crescent shaped quantum wires along the V-grooves. Madhukar et al. have successfully fabricated quantum dots through epitaxial growth on size reducing mesas [38,39]. The main problem with this technique lies in the added complexity of careful growth control and pre-growth patterning. The dot spacing of these structures also needs to be kept an order of magnitude larger than the dot size, resulting in a lower dot density. Other techniques such as growth on patterned substrate through MOCVD selective area growth have been demonstrated by Arakawa et al. [40,41]. But this technique also requires careful pre-growth patterning and the dot density is limited.
One of the most successful techniques for forming in-situ quantum dots is through strained layer epitaxy in the Stranski-Krastonow growth mode [42,43]. This technique has demonstrated the formation of high-density arrays of 3-D islands that are coherently strained. This technique is discussed in detail in the next section. Before proceeding further, it must be mentioned that quantum dots have also been grown by in-situ droplet epitaxy in strained and unstrained systems [44,45]. In this technique, alternating fluxes of group III and group V molecular beams are supplied during growth. This leads to the initial formation of droplets of the group III element which subsequently reacts with the group V element flux to form well-defined islands of the III-V compound. Droplet epitaxy is a promising method to realize GaAs (and other III-V) based devices on Si substrates.
1.3 Advantages of Quantum Dots
The main advantages of quantum dots stem from the discrete nature of the density of states, which is expected to vastly improve the performance of electronic and optoelectronic devices. The large density of states is expected to improve the performance of semiconductor lasers by
reducing the threshold current and temperature dependence of threshold current, increasing the differential gain and reducing chirp. Quantum dots are also expected to have very large non-linear electro-optic coefficients, a fact that can be exploited in optical modulators and other devices. Moreover, single electron devices can be fabricated using QDs and the unique physics of zero-dimensional systems can be investigated.
Classes of devices that can also exploit the unique properties of quantum dots are far
infrared (8-14 μm) emitters [24-27,] and detectors [19-23,46-48] based on intersubband transitions in quantum dots. Unlike interband processes that involve an electron in the conduction band interacting with a hole in the valence band, intersubband processes occur only in the conduction or valence band. In such processes, electrons or holes are excited to higher quantum confined energy states, or excited electrons or holes radiatively relax to a lower energy state. Since the intersubband separation in self-assembled dots is about 80-100 meV (~12-15 μm), they can be used for making far infrared sources and detectors that are in great demand for chemical spectroscopy, thermal imaging and night vision applications. The unique carrier dynamics of the dots suggest the operation of these devices at elevated temperatures, a feature that has not been achievable using competing technologies.
1.4 Conditions for Quantum Confinement
For quantum dots to be useful in room temperature devices, they must exhibit the following
properties, a) sufficiently deep localizing potential, b) high uniformity and high volume fill factor, c) coherent i.e. without defects and dislocations. In the next section the restrictions that these conditions impose on the dimensions and properties of quantum dots will be discussed.
1.4.1 Size of quantum dots
The lower size limit of the quantum dot is given by the condition that there exists at least
one bound energy level of an electron or a hole. We assume that the dot is spherical and solve the Schr?dinger equation with the following Hamiltonian,
with V m H )()(*2)(2
r r r ψψ??????+??=h ???≥≤?=)0000(R r R r V V r (1)
The wavefunction can be separated into radial and angular components ),()()(φθψlm nlm Y r R =r ,
where Y lm ’s are the spherical harmonic functions. From the boundary conditions that R(r) and (1/m *)[?R(r)/ ?r] are continuous across the interface at r=R 0, the transcendental equation
)1(1)cot(02100R m m kR kR κ+?= (2)
is obtained, where, m 1, m 2; k ,κ; are the effective masses and corresponding wavevectors in the dot and the barrier respectively. Using this formula, the energy of the single particle bound states in a spherical quantum dot can be obtained. For a given radius, the potential needs to have a minimum value, V 0,min , such that there exists at least one bound state. This value can be determined from the equation
20
2
2min ,0*8R m V h π< (3)
for m 1=m 2=m*.
In the case of a quantum dot buried in a type-I heterojunction, the confinement potential is
ΔE c and this gives the minimum dimension of the dot as
D =min (4)
where is the effective electron mass (note Eq. (2) and (3) refer to a general particle in a box
whereas Eq. (4) is for an electron). Assuming a conduction band offset value of ~0.3eV for GaAs/Al 0.4Ga 0.6As heterostructures, the diameter of the quantum dot must be larger than 4 nm . This constitutes the lower limit of the quantum dot size. For the InAs/AlGaAs the conduction
band offset is much larger but the effective mass is smaller, so the product remains
comparable and the critical size is about 3-5 nm depending on the non-parabolicity effects in the InAs conduction band [*e m c e E m Δ*243].
There also exists a limit for the maximum size of the dot. A thermal population of higher-
lying levels is undesirable for devices like interband lasers and intersubband detectors. The condition to limit the thermal population of higher-lying levels to 5% (i.e. ~e -3) can be written as
kT E E ≥?)(3
112 (5)
where and are the energies of the first and second excited levels in the QD respectively. This equation establishes an upper limit at room temperature for the size of the QDs. This limit is ~12 nm for GaAs/AlGaAs QDs and ~20 nm for InAs/AlGaAs QDs . It is important to note that this limit depends on the operating temperature of the device.
2E 1E Consideration of the hole quantization leads to size limits, which are very different from the
ones derived for electrons. For both material systems, the maximum dot size assuming hole
quantization is 5-6 nm. This means that quantum dots with large sublevel separation for both electrons and holes are difficult to realize. The range of dimensions needed to observe quantum confinement in different material system is summarized in Fig. 2.
1.4.2 Uniformity
Uniformity among the dots is essential in order to attain similar characteristics for different devices. In most cases, equisized and equishaped quantum dots are desired. In principle, all structural parameters of the dots, such as size, shape, and chemical composition are subject to random fluctuations. This is especially true for dots formed by self-assembled epitaxy.
050100150200
Dot Diameter (A)
Fig. 2 Range of dimensions needed to observe quantum confinement at T=300K in spherical quantum dots with different material systems.
The main impact of the size fluctuation is a variation in the energy position of the electronic levels. This is reflected in an inhomogeneous broadening of the photoluminescence spectrum and the gain curve. In devices such as a quantum dot laser that rely on the integrated gain in a narrow energy range, the inhomogeneous broadening should be as small as possible. Self-assembled QDs usually have a size fluctuation of about 10-15%, which results in an inhomogeneously broadened linewidth of about 40-60 meV. A novel approach based on strain patterning of the growth front by buried higher band gap stressor dots has been undertaken. Using this technique, linewidths as low as 19 meV have been observed at T=17K[49].
Another requirement for most electronic and optoelectronic devices is a large areal density and volume fill factor. In QD lasers, typically a minimum number of dots need to attain “population inversion” to overcome the cavity losses and obtain gain in the medium.
1.4.3 Material Quality
High optical quality material with a large radiative efficiency is one of the most essential prerequisites for quantum dots. The density of defects in a QD material and its interface should be as low as possible. Self-assembled QDs are expected to have very few defects because of the presence of the strain field around them that deflects the dislocations away from the dots.
2. SELF-ASSEMBLED QUANTUM DOTS: GROWTH, STRAIN
DISTRIBUTION AND BAND STRUCTURE
2.1 Self-Assembly of Nanostructures
Spontaneous formation of periodically ordered domains in solids with a periodicity much larger than the lattice parameter is a general phenomenon that is at the origin of a large variety of different domain structures. This process, also called self-assembly or self-organization, can occur if the homogenous state of the system is thermodynamically unstable and the system undergoes a phase transition into an inhomogeneous state [13,50]. In an inhomogeneous state, a coordinate-dependant order parameter, such as strain, is the source of a long-range field. Therefore, a multi-domain state may be energetically more favorable than the single-domain state, since the former provides compensation of the long-range field at large distances outside the domain structure. The long-range field is responsible for the periodic ordering of the equilibrium domain structure that meets the conditions of the total Helmholtz free energy minimum. The free energy can be written as a sum of three distinct contributions:
F total = F domain +F boundaries + E long-range (6)
where F domain is the free energy of the domains, F boundaries is the free energy of the domain boundaries and E long-range is the energy of a long-range field, which includes interaction between domains.
2.2 Energetics of Quasi-Zero Dimensional Coherently Strained Islands
There are three well-known modes of heteroepitaxial growth that occur depending on the lattice mismatch between the epitaxial layer and the substrate. In unstrained or very lightly strained systems (e.g. GaAs/Al(Ga)As), the growth proceeds via a two-dimensional layer-by-
layer formation, which is referred to as Frank-van der Merwe growth mode [51]. In systems where the lattice mismatch exceeds 1.8% [13], there is an initial formation of a two dimensional “wetting” layer, after which coherently strained three dimensional islands are formed. This is known as the Stranski-Krastanow growth mode [42]. Most of the quantum dots for the present day devices are grown in the Stranski-Krastonow mode. For example, the InAs/GaAs system has a lattice mismatch of about 7%. If the strain exceeds about 12%, three-dimensional islands are directly formed on the growth without the formation of the wetting layer. This is known as the Vollmer-Weber growth mode [52]. The three growth modes are shown schematically in Fig. 3.
Frank-van der Merwe 2d layer by layer Stranski-Krastanow Initial 2d growth, later 3d island growth Vollmer-Weber 3d island growth Increasing Strain
Fig. 3 The three growth modes of heteroepitaxial growth in strained layer epitaxy.
2.2.1 Thermodynamic model
The growth mode in InGaAs/GaAs systems for different values of the lattice mismatch was
studied by Berger et al. [13] using solid source MBE. Simple thermodynamic calculations were used to estimate the critical thickness above which the SK growth dominates. The energy difference between the two extremal cases, a) in which the atoms on the growth front occupy an atomically flat surface and b) when they are arranged in three-dimensional islands was minimized to obtain
))((20213c
d R W W n ? (7)
where n 2 is the height of the island in number of monolayers, W 1 and W 2 are the nearest and second nearest neighbor bond energy respectively, R 0 is the monolayer distance (not to be confused with R 0 defined in Eq. (1) and (2)) and d c is the critical thickness. Since W 1/W 2 ~10-15, this equation suggests that if d c <20 monolayers (strain > 2%), the strained epilayer will minimize its energy by forming a three-dimensional island surface. For growth of InGaAs on GaAs, the island formation would be expected to occur for In compositions greater than 25-30%.
2.2.2 Volume Relaxation and Dislocations
An independent theory developed by Vanderbilt and Wickham [53] compares the two
mechanisms of elastic relaxation, namely the volume elastic relaxation of the coherently strained islands and the formation of dislocations. These two competing processes yield a phase diagram of a lattice-mismatched system where all possible morphologies are present, i.e. 2D films, dislocated islands and coherent islands. The phase diagram is shown in Figure 4. The formation of an island from a uniform 2D film is accompanied by a relaxation of the elastic energy, , and by a reduction of surface area, ?A < 0. The size of the corresponding change of the magnitude of surface energy change depends on the formation of side facets of the islands and on the disappearance of certain areas of the planar surface. It is usually believed that the change of the surface energy caused by the formation of islands is positive, ?E surf > 0. It was shown by Vanderbilt and Wickham [0<ΔV elastic E 53] that the morphology of the mismatched system is
determined by the relation between ?E surf and the energy of the dislocated interface . The
ratio of these two energies, denoted by disc erf E int surf disc erf E E =Γint , is the control parameter that governs
the morphological phase diagram of Fig. 4.
If ?E surf is positive and large, or the energy of the dislocated interface is relatively small, the
corresponding value Γ on the phase diagram is smaller than Γ0. Then formation of coherently strained islands is not favorable. With the increase of the amount Q of the deposited material, a transition occurs from a uniform 2D film to dislocated islands and coherently strained islands are not formed.
Г
Fig. 4 Phase diagram depicting the preferred morphology as a function of the amount of deposited material, Q, and Γ, the ratio of the energy of a dislocated interface to the change of surface energy due to island formation.
However, if ?E surf is positive and small, or the energy of the dislocated interface is relatively
large, the corresponding value Γ on the phase diagram is larger than Γ0. With the increase of the amount of the deposited material, a transition from a uniform 2d film to coherent islands occurs. Further deposition of material may cause the onset of dislocations. A similar theory was developed by Ratsch and Zangwill for pyramidal shaped islands [54].
2.3 Self-Assembled Quantum Dots
The self-assembled three-dimensional islands formed by epitaxial growth are typically
pyramidal to lens-shaped with a base dimension of 10-20 nm and a height of 4-8 nm with an areal density determined to be 5x1010cm -2 using Atomic Force Microscopy (AFM). A cross-sectional TEM image of a single InAs quantum dot is shown in Figure 5. The details of the growth conditions of the dots will be discussed in the next section. Since the first demonstration of the quantum sized effects in these 3-dimensional islands in the In(Ga)As/GaAs system, interest has soared leading to the observation of SK growth mode in InAs/In(Ga,Al)As/InP
[55,56], (In,Ga,Al)Sb/GaAs [57,58], InP/InGaP[59,60], SiGe/Si [61,62] and InAs/Si [63].
2.3.1 Strain Distribution: VFF model
As discussed earlier, the strain mismatch of ~7% in the InAs/GaAs system leads to the
formation of coherent islands for indium compositions greater than 0.35. Various models have been used for examining the strain distribution in solids, such as the continuum mechanical model [64], valence force field model [65,66] and the density functional techniques [67].
In this section, we will briefly review the strain distribution calculated by Jiang and Singh
[68,69]. In their calculation, the bond stretching (α) and bond bending (β) parameters are calculated using the valence force field (VFF) model of Keating [65] and Martin [66]. To find the strain tensor in the InAs/GaAs dots, an arbitrary choice of the atomic positions was made and the system energy was minimized using the Hamiltonian given by
))(4
321)(4341,0,02,0,02,02,02ik ij ik ij ik ij ijk i k j ij ij ij ij ij d d d d d d d V +?+?=∑∑∑≠d d α (8)
where i runs over all the atomic sites, j and k run over the nearest-neighbor sites of i, is the vector joining the sites i and j, is the length of the bond, is the corresponding length in the
ij d ij d ij d ,0
(a) Fig. 5 (a) Cross-sectional transmission electron microscopy image of a single InAs quantum dot, (b) schematic of the pyramidal shape dot depicting the various crystal facets. (TEM by P. Rotella) (b)
binary constituents, and ij αand ijk β are the bond stretching and bond bending constants, respectively.
The results of this approach are summarized here. The strain distribution shows that there is
a relaxation at the top of the dot, which has been borne out experimentally [70]. The biaxial strain changes sign from the bottom to the top of the dot, which means that there is more confinement for heavy holes at the bottom and more confinement for light holes at the top of the dot. The shear strain is also very large. Moreover, there is a strong intermixing between the light holes and the heavy holes for the valence-band bound states.
2.3.2 Electronic Band Structure: 8-Band k·p Formalism
The problem of band structure in semiconductor quantum structures has usually been solved
by decoupling the conduction and valence band problems. The conduction band problem is typically solved using a scalar effective mass approach while the valence band problem is solved using a k·p approach, which includes the heavy hole and light hole coupling. This decoupling of valence band states from conduction band states is possible because in problems typically encountered (e.g. quantum wells and wires or quantum dots with small strain), there is minimal remote band influence on the conduction band. However, in self-assembled quantum dots, this is not true. It is important to note the following observations: (i) the bandgap of bulk InAs is 0.4 eV while the effective gap of the dot is ~1.1 eV; (ii) the nature of strain tensor is such that there is strong spatial variation in strain; (iii) the strain components are very large and the resultant splittings in the bands are comparable to the interband separations in the bulk material. All these considerations suggest that the simple decoupled conduction-valence band picture and the effective mass description may not be adequate.
In the 8-band k·p model used by Jiang and Singh [68] to calculate the dot band structure, the
influence of remote bands on the conduction and valence band states is included. In the presence of strain, the Hamiltonian has the form,
str t H H H +=0 (9)
where is the kinetic energy term and the is the strain term. The spin-orbit dependent deformation potential is neglected in this approach. Using a finite-difference method the equation is solved numerically, with the distance between two grid points chosen to be equal to the lattice constant of GaAs (5.6533?). The details of the calculation are discussed elsewhere
[0H str H 69]. The summary of the results is presented here.
For pyramidal shaped quantum dots, with a base width of 124 ? and a height of 64 ?, the
electronic spectrum is solved using the 8-band model and is shown in Fig. 6. These dimensions
are very close to the experimentally observed dimensions obtained from cross-sectional TEM
[70]. There are a number of excited states in the conduction band. The presence of the excited
state is clearly seen from photoluminescence [71], far infrared absorption [20] and capacitance measurements [72]. The first and second transitions observed in the photoluminescence measurement are due to radiative recombination between the ground electronic states and ground hole states and between excited electronic states and excited hole states. An interesting feature of the calculated spectra is the near four-fold degeneracy of the second excited states.
These states have an exact double degeneracy due to time-reflection symmetry and are almost degenerate due to different excitation directions. The second excited electronic state is 62 meV higher than the ground state. There is considerable mixture of wetting layer states in this state because it is less confined. There are also many confined hole states. The splitting between the ground and excited hole states vary from 22 to 30 meV. It is important to note the following two points. Firstly, the energy separation between the electronic ground and excited states (62 meV) lies in the far-infrared (8-20 μm). This suggests that intersubband transitions in QDs could be used to fabricate far infrared sources and detectors. Secondly, since the intersubband spacing is larger than the optical phonon energy in GaAs (36 meV), optical phonon scattering, which constitutes the major scattering mechanism in quantum wells, is suppressed in the dots [73]. This prevents the carriers in the excited state from relaxing to the ground state. This effect, referred to as the “phonon bottleneck”, can be used to create population inversion [74], between the excited and the ground states and thus fabricate an intersubband QD laser [24-27]. The phonon bottleneck also enables photo-excited carriers that are generated in an intersubband detector to live longer in the excited state and thus increases the probability of photo-excited carriers getting swept out as photocurrent. The detailed carrier dynamics of the quantum dots, and the fabrication and characterization of intersubband QD detectors will be discussed in Sections 3 and
4 respectively.
Fig. 6 Conduction and valence band energy levels for an InAs–GaAs quantum dot with base width 124? and height 62? calculated using the 8-band k · p model (Figure reproduced from Jiang et al. [33] with permission).
3. INTERSUBBAND CARRIER DYNAMICS IN
SELF-ASSEMBLED QUANTUM DOTS
3.1 Background
Quantum dots are expected to display very unique carrier dynamics compared to other sub-
three-dimensional quantum confined structures such as quantum wells and quantum wires. The
main difference arises from the three-dimensional confinement in QDs that leads to the
formation of discrete atom-like energy levels. In contrast, an energy continuum exists along one
and two dimensions in a quantum wire and a quantum well, respectively. Extensive studies have
been undertaken to understand the dynamics of hot carriers in these systems and to elucidate the
scattering mechanisms that may explain the experimentally observed carrier relaxation phenomenon [72-75]. The carrier dynamics in quantum wells is reasonably well understood and
the intersubband relaxation time, attributed to phonon scattering, is relatively short [76].
However, there is confusion about the recombination dynamics within the QD energy levels
due to the large variation in reported data from many laboratories. In most applications, electrically injected or photo-excited carriers are created in a GaAs barrier and captured in the 2-
D wetting layer (WL) surrounding the QDs. The capture of carriers from the WL and their
relaxation through the discrete energy levels of the QDs are particularly sensitive to growth
method as the reported time scales for such processes vary from a picosecond to several hundred picoseconds [73,77]. The reports of slow relaxation times claim evidence of a phonon bottleneck
[77,78,79]. Because the QD energy level spacings are not typically equal to a single phonon
energy, ~33 meV, some researchers report that single phonon-carrier interactions are suppressed
in QDs. It is likely that single phonon-carrier interactions are suppressed in some QDs but not in
others. Boggess and coworkers [80] have reported enhanced relaxation rates in large QDs in
which electron energy level spacings may be close to 33 meV. Other explanation for faster
capture times include multiphonon interactions [81], electron-hole scattering [77], and Auger processes [73]. Competing factors such as thermal re-excitation [82] and tunneling [83] complicate the carrier capture time, especially near 300K. Due to the existence of discrete
energy levels in the conduction and valence band wells of quantum dots, carrier relaxation of
electrons (or holes) between such levels by phonon- or other carrier-mediated scattering processes would have to satisfy strict energy conservation rules. Therefore phonon-mediated scattering can be inhibited. In the context of device applications, it is of interest to examine the
possibility of intersubband population inversion and eventually stimulated emission in QDs.
The objective of this section is to discuss the unique carrier dynamics that exist in self-
organized In(Ga)As/GaAs quantum dots that has been measured in the recent past [80-86]. This
includes carrier capture times extracted from high frequency electrical impedance measurements
on quantum dot lasers [84] and intersubband relaxation times obtained by direct pump-probe spectroscopy [85,86].
3.2 Dynamics of Hot-Carriers: Carrier Relaxation and Phonon Bottleneck
All the results reported in this section have been obtained using 4-layer vertically coupled In0.4Ga0.6As/GaAs self-organized quantum dot sample grown by solid source MBE. Since the separation between the mini- bands formed around the ground state and the excited state are greater than the LO phonon energy, the LO phonon scattering mechanism, which is mainly responsible for the rapid carrier relaxation in bulk semiconductors and quantum wells, is suppressed. We therefore expect to observe long carrier relaxation times in quantum dots.
3.2.1 High Frequency Electrical Impedance Measurements on QD lasers
The first indication of a slow carrier relaxation was observed when high frequency electrical impedance measurements were performed on multi-layer In(Ga)As quantum dot laser diodes [76]. In this technique, the magnitude and the phase of the electrical impedance were measured as a function of frequency using a network analyzer. Using appropriate carrier and photon rate equations, the capture times were extracted. This technique has been very successfully used in the past to determine capture times in quantum well lasers [87,88]. Weisser et al.[87] have obtained capture times of 2-5 ps for In0.35Ga0.65As/GaAs quantum wells. Using this technique, we have obtained capture times ranging from 30-100 ps in quantum dot interband lasers depending on the injection current and the heterostructure used. This long capture time is believed to be responsible for the low small-signal modulation bandwidths (5-7 GHz) of single-mode QD lasers at room temperature. However, from this measurement alone, it cannot be ascertained whether the slow relaxation occurs from the barrier to dot excited state or from the excited state to the ground state. To measure these relaxation times directly, the technique of differential pump-probe spectroscopy is generally used.
3.2.2 Pump-Probe Differential Transmission Spectroscopy
Femtosecond pump-probe differential transmission spectroscopy (DTS) measurements on 4-layer In0.4Ga0.6As quantum dot heterostructures at 10K and higher temperatures for a range of excitation levels have been performed in Prof. Norris’s group at the University of Michigan. Approximately 2 electron-hole pairs per dot are generated in the barrier region of the dots using a 800-nm pump beam, and the differential transmission signals at the ground and excited state
transition energies were measured as a function of the delay between the pump and probe pulses. Since the DT signal is proportional to the occupation number of each level, the relaxation times are obtained directly using this technique. The short rise time (~2 ps) of the excited state signal is a measure of the capture time from the barrier to the excited states. The decay of the excited state response, which matches the rise of the ground state signal, is a measure of intersubband relaxation. From a rate equation fit to the data, intersubband relaxation times of 5.2 ps and 0.6 ps were obtained for the electrons and holes, respectively. The fast carrier relaxation is consistent with electron-hole scattering [70]. The electron capture time from the barrier to the excited state is determined to be about 2-3 ps. Note that the phonon-mediated hole relaxation results in a broadening of the hole levels, which helps to relax the energy conservation requirement in the electron-hole scattering process. The fast hole relaxation is a result of band mixing, strong anisotropy and high density of states of the valence band states.
Results from temperature-dependent carrier-capture measurements are shown in Figs. 7(a) and 7(b). The experiments were conducted in the same way, with less than one electron-hole pair per dot, except that the temperature was varied from 10K to room temperature. At 40K, the n=2 DT scan shows a carrier capture and decay behavior that is consistent with the model discussed above. The n=1 scan in Fig. 7 (b) shows a corresponding increase indicating that carriers in the excited states relax down to the ground state. As the temperature is raised to 80 K, the fast component (~8ps) of the n=2 scan begins to diminish while the slow component (~100 ps) remains constant in magnitude. It is believed that the fast component diminishes because at higher temperatures the carriers occupy the energy states in the barrier continuum due to the large density of states within the continuum and this reduces the effectiveness of the electron-
Fig. 7 Temperature dependent differential transmission spectroscopy scans of In0.4Ga0.6As/GaAs quantum dots for (a) first excited state and (b) ground state (Measurements done by J. Urayama and T. Norris).
Above 100K, the n=2 reveals a rapid direct capture of carriers into the excited states at low temperatures, but the fast decay component disappears, leaving only the slow component with a time constant of >100 ps. At temperatures of 100K and above, the slow decay of the n=2 scan is attributed to the thermal equilibrium that is reached among the carriers in the high lying electronic states. As the temperature is increased above 100K, the n=2 DT signal decreases in magnitude because a higher proportion of the carriers are in the barrier region due to the high density of states; nonetheless the rise time remains fast (~2 ps) due to the rapid thermalization between the barrier and quantum dot excited states. The n=1 scan at 150 K shows a diminished signal due to carrier thermalization as mentioned above and has a rise time (~10-15 ps) that does not reflect the decay of the n=2 population. It is believed that this time constant reflects the overall thermalization time of electrons from the higher lying states to the ground state in the dot, by a combination of the mechanisms discussed above. Since the n=2 states are essentially in thermal equilibrium with the barrier states, the n=2 population does not show a decay that matches the rise of the n=1 signal.
3.3 Application of Favorable Dynamics to Intersubband Devices
The large intersubband relaxation time poses a fundamental limit to the modulation bandwidth of quantum dot lasers. The largest modulation bandwidth in QD lasers has been measured to be about 7-8 GHz in contrast to the 30-40 GHz bandwidth obtained from GaAs based quantum well lasers. Alternative injection mechanisms like resonant tunneling into the ground state of the dots have been proposed. However, the long intersubband relaxation times in quantum dots can be favorably applied to devices that are based on intersubband transitions in the dots, such as far infrared sources and detectors.
The long intersubband relaxation time ensures the presence of a large number of carriers in the excited state of the dot and this allows for the possibility of population inversion between the ground state and the excited states, provided the ground state carriers are depleted rapidly. This feature is exploited to realize intersubband spontaneous and stimulated emission. The details of these devices is discussed in the next section.
The lifetime of the excited state in the dot can be made even longer (~ns) using a unipolar device, in which there is essentially no electron-hole scattering. This can be very useful in intersubband detectors, which require a long-lived excited state that does not allow the rapid relaxation of photogenerated carriers. This enables their efficient contribution to the photocurrent. In quantum well infrared photodetectors (QWIPs), the excited state carriers relax rapidly to the ground state (~2-5 ps) and many of the photogenerated carriers do not contribute to the photocurrent.
In conclusion, the intersubband carrier relaxation in self-organized InGaAs/GaAs quantum dots is determined from high frequency electrical impedance (HFEI) measurements on single mode QD lasers and by direct pump-probe differential transmission (DT) spectroscopy at various temperatures. The hole relaxation time is very small (0.6-0.7 ps). Although the electrons relax rapidly from the barrier to the excited state (1-2 ps), the electron relaxation time from the excited state to the ground state exhibits two time constants (6-8 ps and >100ps). The possible scattering mechanisms responsible for these long relaxation times are discussed. It is believed that the carrier dynamics in this regime are very favorable for intersubband devices such as for the mid infrared detectors discussed in the next section.
4. THE INTERSUBBAND QUANTUM DOT DETECTOR
4.1 Overview
Infrared (IR) detectors are important for a variety of applications, including night vision, targeting and tracking, medical diagnosis, law enforcement, environmental monitoring, and space science. High performance IR systems and focal plane arrays require detectors that can demonstrate low dark current, high detectivity, high temperature operation, and low-cost fabrication. At the present time, HgCdTe (MCT) interband IR detectors lead the technology [89,90]. However, there are still problems in epitaxial growth of HgCdTe based materials due to the presence of large interface instabilities, etch-pit and void defect densities and this is reflected in the high uncertainties and fluctuations in the value of D* [91]. QWIPs are an alternative technology that uses intersubband optical transitions in quantum wells as the detection mechanism [92-94]. They benefit from a mature III-V growth and processing capability and have been incorporated in IR camera systems and large focal plane arrays. However, QWIPs have to operate at temperatures lower than MCT devices because of a very large rate of thermionic emission of photo-excited electrons from the quantum wells. Another disadvantage is that QWIPs cannot detect normally incident light due to polarization selection rules. This drawback is overcome by incorporating random reflectors on the top surface of the devices. Because of these shortcomings, alternative technologies are being investigated in III-V materials. One such technology is the strained layer superlattice with a type II band alignment (e.g. InAs/GaSb) [95,96,97], which is expected to reduce Auger recombination rates in the detectors, thereby leading to increased operating temperatures. However, the development of these devices is still in its infancy.
A promising device that has emerged in the recent past is the quantum dot infrared photodetector (QDIP), which, like the QWIPs, are based on optical transitions between bound
states in the conduction (valence) band in quantum dots. Also, like the QWIPs, they benefit from a mature technology with large-bandgap semiconductors. There are many advantages offered by QD detectors. First, QDs inherently allow sensitivity to normal excitation. The electron relaxation times between the discrete bound states (separated by 50-70 meV) are larger than in quantum wells due to a phonon bottleneck. This promises high temperature operation of QDIP. The three-dimensional confinement of carriers results in decreased thermionic emission and a lower dark current. Uncooled IR detectors will significantly reduce the size and operating costs of arrays and imaging systems for a variety of applications. This section presents intersubband QD detectors as a promising technology for tunable and multi-wavelength IR detection.
4.2 Vertical Quantum Dot Detectors
4.2.1 MBE Growth, Heterostructure Engineering, and Fabrication
Self-assembled InAs/Ga(Al)As quantum dot heterostructures are grown by solid source molecular beam epitaxy (MBE) on semi-insulating GaAs (100) substrates. The vertical n-i-n structure typically has ten layers of doped InAs/Ga(Al)As QDs (n = 0.5-1 x 1018 cm-3) separated by 0.025-0.1 μm of undoped GaAs spacers [48]. A variation of this detector is the dots in a well detector (DWELL), in which the InAs dots are placed in a thin In0.15Ga0.85As matrix. This lowers the ground state of the dot thereby reducing thermionic emission. This entire absorption region is then sandwiched between heavily-doped GaAs contact layers to complete the n-i-n device. The general heterostructure of a typical InAs/GaAs vertical n-i-n detector is shown in Fig. 8.
(a) (b)
Fig. 8 Heterostructure schematic of a 10 layer quantum dot detectors with (a) InAs/GaAs QDs and (b) with InAs/In0.15Ga0.85As dots in well design.
The InAs/Ga(Al)As quantum dot layers are uniformly and directly doped with silicon to provide carriers for absorption. AFM measurements conducted on samples with surface quantum dots grown under similar conditions indicate a dot density of 1010-1011 dots/cm 2. Thus, the doping level corresponds to ~ 0.5-1 dopant atoms/dot. The doping levels in the dots and the potential barrier encountered by the carriers trapped in the dots are the two most critical parameters in the design of the vertical QDIP structure. The doping level needs to be optimized to provide 1-2 carriers per dot, so that when the device is not illuminated, the carriers are confined to the ground state of the dot and do not occupy the excited states or the continuum levels in the wetting layer. The absence of free carriers in the WL ensures a lower dark current.
A simple, three-step photolithography and wet-etching process is used to fabricate the vertical QDIPs. The first step consists of Ni/Ge/Au/Ti/Au metal evaporation for the top ring contact. Next, a mesa etch (~ 1 μm) is performed around the top contact to define the active region for a single pixel. Finally, the metal evaporation is repeated for the bottom ring contact, which is deposited around the device mesa, and the contacts are annealed. The area of the detector exposed to IR radiation is determined by the inner radius of the top ring contact. The devices are wire bonded to a leadless chip carrier (LCC) for characterization.
00.250.50.7512468Wavelength (um)N o r m a l i z e d R e s p o n s i v i t y (a .u .)
00.20.40.60.81345678910Wavelength (μm) S p e c t r a l R e s p o n s e
(a) (b) Fig. 9 Normal incidence spectral response of (a) a QD detector with a single 30% AlGaAs barrier in the mid-wavelength IR, and (b) InAs/InGaAs dots in a well detectors. 4.2.2 Spectral Response and Tuning
The spectral response of normal incidence mid-wave infrared (MWIR) and long-wave infrared (LWIR) QD detectors are shown in Fig. 9. The MWIR InAs/GaAs quantum dot