Overview of Semiconductor Quantum Dot Laser

Self-assembled quantum confined nanostructures, in the form of quantum dots, have been paid lots of attention due to its near delta-function of density of states (DOS) [28]. Ultralow threshold current density Jth [29], better temperature stability T0 [30], high modulation bandwidth [31], low chirp [32], and small linewidth enhancement factor have been demonstrated [33]. Limited by the number of quantum dot (QD), the gain in QD saturates rapidly at a certain level, gst, with increasing current density [34], which is the main difference of QD laser behavior from quantum well (QW) system. After the gain saturation at ground state (GS), carriers start accumulating at excited states (ES) and then achieve its threshold condition. For a specific laser device, GS

Fig.1.5 Optical bandwidth, interference signal and point spread function using a typical SLD and a femtosecond Ti:Al2O3 laser. For ultra-high resolution OCT, a broadband femtosecond laser with spectrum bandwidth ~260 nm can achieve a free space axial resolution of 1.5 μm.

and/or ES lasing will be determined by its cavity length, density of QD, driving current, and other structural factors.

Broad-spectrum, high-power and highly efficient semiconductor light sources are strongly desirable for sensing and low-coherence imaging applications such as OCT. With higher power than SLD and more compactness than femtosecond laser, broadband semiconductor laser is more competitive if real supercontinuum broad bandwidth can be performed. Nevertheless, because of active material growth technology and fundamental physics, a conventional QW laser generally produces a narrow spectrum with a spectral width of the subnanometer order for a single-frequency laser to a few nanometers for multi-longitudinal mode lasers. Recently, simultaneous two-state lasing from the GS and ES has been observed from QD lasers with well-separated wavelength emissions [35]. Except for incomplete gain clamping at threshold stated as before, this behavior is also attributed to the retarded carrier relaxation process in QDs, which is also known as the phonon-bottleneck [36]. On the other hand, to achieve real broad laser spectrum of tens of nanometer scale from the QD laser, high dispersion in QD size and small energy spacing between quantized energy states that lead to the broadened optical gain characteristics play important roles as well. In fact, optical line broadening of the emission spectrum of a transition depends on both homogeneous and inhomogeneous broadening mechanisms. Besides the contribution of large dot size fluctuation towards inhomogeneous broadening phenomenon, mechanisms such as acoustic and optical phonon-carrier interaction, lifetime broadening, and carrier-carrier interaction greatly affect the homogeneous linewidth in semiconductors [37]. Lately, NL Nanosemiconductor demonstrated the extended lasing spectral width with uniform intensity distribution (only a 3dB modulation) of about 75 nm [38~39]. A broad lasing spectrum of 40 nm was also

demonstrated by H. S. Djie et al [40].

In this study, specially designed chirped-multilayered QD (CMQD) lasers with three kinds of different wavelength InAs-InGaAs QD stacks were grown. Besides, we have set up a fiber-based, delay-tunable Mach-Zehnder interferometer to measure the interferograms or autocorrelation function which can be used to quantitatively determine the coherence length of a light source. Except for achieving low threshold current density and high saturated modal gain, we demonstrate broad spectrum of 29 nm in this structure and expect that it could be improved to satisfy all the demands for a suitable OCT light source someday.

Chapter 2

Theoretical Fundamentals

2.1 Coherence Theory

In optics, the original sense of the word coherence was attributed to the ability of radiation to produce interference phenomena. Nowadays, the term of coherence is defined more specifically by the correlation properties between quantities of an optical field. Common interference is the simplest phenomenon showing correlations between light waves. It is convenient to divide coherence effects into two classifications, temporal and spatial. The former relates directly to the finite bandwidth of the light source, the latter to its finite extent in space. Experimentally, Michelson introduced a technique for measuring the temporal coherence by Michelson interferometer. Spatial coherence is illustrated by the double-slit experiment of Young. When it comes to coherence length, what we mainly consider here is temporal coherence.

2.1.1 Temporal Coherence

First of all, we consider the path of rays in a Michelson interferometer which is shown in Fig. 2.1. The incident light is divided into two beams by a beam splitter. One beam is reflected back onto itself by a fixed mirror, the other one is also reflected back by a mirror which can be shifted along the beam. Both reflected beams are divided again into two by the beam splitter, whereby one beam from each mirror propagates to a photodetector (PD). The idea of this arrangement is to superimpose a light wave with a time-shifted copy of itself.

Let E be the light wave that reaches the screen via the fixed mirror ₁ and E be the one that reaches the PD via the movable mirror. Then we ₂ have at a point on the PD, when the incoming wave has been split evenly at the beam splitter, mirror displacement d according to

c 2d

τ = . (2.2) On the PD, the interference of both waves given by the superposition of the wave amplitudes: This superposition is not directly visible. A photodetector with a finite integration time detects the superposed light beams and provides the integrated/averaged intensity:

Fig.2.1 The Michelson interferometer.

} the first wave and I of the second wave and an additional term, the ₂ interference term. The meaningful information is contained in the expression 〈E₁^*E₂〉. With Eq. (2.1) this gives rise to the definition:

Γ is known as the complex self coherence function. It is the autocorrelation function of the complex light wave E₁(t). We then get

)}

The magnitude γ(τ) is called the complex degree of self coherence.

Because Γ(0)=I₁ is always real and is the largest value that occurs when we take the modulus of the autocorrelation function Γ(τ), we have

1 easy to determine the contrast K between interference fringes. This quantity has been used by Michelson who called it visibility and defined

it via the maximum and minimum intensity I_max and I_min as

Obviously, the maximum and the minimum intensity of the interference fringes do not occur at the same time shift of the light waves; that is, K is a function of τ . Let τ₁ and τ₂, τ₂ > , be the time shifts belonging τ₁

τ − corresponds to half a mean wavelength, which is small compared to the duration of the wave train to be investigated. Only in this case does the definition make sense. According to Eq. (2.6), the maximum intensity is attained at maximum Re{Γ(τ)}, occurring at τ₁, and the minimum intensity at minimum Re{Γ(τ)}, occurring at τ₂. It follows, for τ taken from this interval, that

) ( )}

(

Re{Γτ₁ = Γτ and Re{Γ(τ₂)}=−Γ(τ). (2.12) This leads to the intensities

) and to the contrast function

) The visibility then is identical to the modulus of the complex degree of

coherence. This is valid for two waves of equal intensity, otherwise some prefactors will arise. Fig.2.2 shows the self coherence function Γ(τ) in the complex plane (up) and contrast function K(τ) (down) for different light sources. Light in the large range in between the two limiting cases, completely coherent and completely incoherent, is called partially coherent. The following cases are distinguished (τ ≠0, γ(0) =1) :

All real light sources are partially coherent. Most part of natural and artificial light sources have a monotonously decreasing contrast function;

for instance, the light from a mercury lamp. To characterize the decay of the contrast function, the coherence time τ_c is introduced. It is defined as the time delay when the contrast function has dropped to 21/ or possibly 1 of its maximum value. In other words, it specifies the /e duration of time that a source maintains its phase. Equivalently to the coherence time, the coherence length

τc

Completely coherent Completely incoherent Partially incoherent

Fig.2.2 Graph of the self coherence function Γ(τ), and the contrast function )K(τ for light from completely coherent, completely incoherent, and partially coherent light, respectively.

is used for characterizing the interference properties of light. Typical values of the coherence length are some micrometers for incandescent light and some kilometers for single-mode laser light.

In a word, to determine the coherence length of some light source experimentally, all we have to do is to measure the interferogram or autocorrelation with enough path difference by a delay-tunable interferometer. After extracting the AC component and then detecting its envelope, we can define the FWHM of this profile as the coherence time.

Coherence length will be obtained after multiplying coherence time by the light speed, c .

2.1.2 Wiener-Khinchin Theorem

When it comes to coherence of a light source, its spectral bandwidth plays an important role. A connection between the autocorrelation of a function and its Fourier transform is established by the Wiener-Khinchin theorem. It states that the autocorrelation and the power spectrum of a function are Fourier transforms of one another. In the time domain, if

) (t

f is a radiated electric field, f(t)² is proportional to the radiant flux or power, and the total emitted energy is proportional to f t dt

2

0 ( )

∫

In the frequency domain, F(ω) is the Fourier transform of f(t), which is called the field spectrum. F(ω)² =F^*(ω)F(ω) must be a measure of the radiated energy per unit frequency interval, which is sometimes called the power spectrum. In other words, F(ω)² is just what we measure from the optical spectrum analyzer (OSA).

According to the definition, the autocorrelation of f(t) is denoted by

dt t

f t

∫

=

Γ(τ) _- ^*( ) ( τ) (2.17)

0

Let )F(ω be the Fourier transform of f(t), denoted by FT{ tf( )}. The autocorrelation can be rewrite to

∫ ∫

Changing the order of integration, we obtain

ω

In the last integral, notice that

}

IFT means inverse Fourier transform. Then, Eq. (2.19) becomes ω and both sides are functions of the parameter τ. If we take the transform of both sides, Eq. (2.20) then becomes

)2 This is a form of the Wiener-Khinchin theorem. It allows for determination of the spectrum by way of the autocorrelation of the generation function. By the way, Wiener-Khinchin theorem is the main idea in Fourier-Transform Spectroscopy (FTS). The variable part of the interference function, on Fourier transformation, yields the spectrum. It is now used widely in the infrared region because of the improved signal-to-noise (S/N) ratio, high throughput, and high resolution possible with it.

2.2 Semiconductor Quantum Dots

In this chapter, I will discuss about the density of state (DOS) in semiconductor, the formation of quantum dots and the carrier transition between excited states and ground state in quantum dots.

2.2.1 Ideal Quantum Systems

In semiconductor, an interaction between electrons of neighboring atoms is sufficiently strong. This interaction splits an atomic energy level to as many sublevels as the number of atoms contained in the system.

These sublevels are grouped in bands. The width of a band is controlled by the strength of interaction of electronic shells, that is, interatomic spacing. In semiconducting crystals, where the interatomic spacing is set by the lattice constant in the range of few angstroms, the width of the order of several electron volts is typical for the valence (the last occupied) and conduction (the first empty) bands. The energy gap, which exists between these two allowed bands, contains no electronic states and is known as the forbidden band. Its width ranges from several electron volts (insulators) to zero (metals).

The number of allowed states within a given band is roughly equal to the number of atoms inside the crystal. It is as many as few 10 ²² electrons per cubic centimeter. However, these states are not distributed uniformly within the band. To characterize this distribution, a DOS is introduced. DOS is a function of energy ρ(E) and it is a measure of allowed electronic states within unit energy interval per unit volume of crystal:

) / )(

/ 1 ( )

(E = V dN dE

ρ . (2.22)

DOS has dimensionality of energy^-1 volume^-1 and shows how many states are available within unit energy interval around some energy E in a crystal of unit volume. Eq. (2.23)~(2.30) are the corresponding dispersion relation and DOS for four kinds of ideal quantum systems:

Bulk: where E is the energy of the band edge of the corresponding band; ₀

) (E −E_n

Θ is step-function which returns zero if E <E_n or unity otherwise; δ(E − E_{n, lm,} ) is the delta-function; and n_Qwire (cm^-1) and n_QD

(cm^-2) are the surface density of quantum wires and QDs, respectively.

Fig. 2.3 illustrates the schematic view and DOS of various ideal quantum nanostructures.

2.2.2 Structural Characteristics of Quantum Dots

To realize quantum dot structure, various techniques based on self-organization have been intensively investigated recently instead of conventional lithography-based or regrowth technology. The self-organized quantum dots have high crystal quality, and they are small enough to exhibit quantum confined effects. Among several techniques, the Stranski-Krastanow (S-K) growth in highly strained heteroepitaxial systems is most studied by using molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD). Cluster formation occurred during epitaxial growth of a semiconductor material on top of another one that has a lattice constant several percent smaller, for instance, InAs on GaAs substrate. For the first few atomic layers, the atoms arrange themselves in a planar layer called the wetting layer. As the

3D : Bulk 2D : QW 1D : QWire 0D : QD

Fig. 2.3 Schematic view and DOS for various quantum nanostructures.

epitaxial overgrowth proceeds, the atoms tend to bunch up and form clusters (quantum dots). It is energetic favorable, because the lattice can elastically relax the compressive strain and thus reduce strain energy within the islands.

The size of the islands can be adjusted within a certain range by changing the amount of deposited dot material. The change in size corresponds to an energy shift of the emitted light, which is due to the altered carrier confinement and strain in the dots. On the other hand, to obtain the information about structural and morphological characteristics of QDs, atomic force microscopy (AFM), scanning tunneling microscopy (STM), transmission electron microscopy (TEM), and photoluminescence (PL) spectroscopy are the most well-known and widely adopted methods.

In fact, except for its shape and size, the energetic property of QDs also depends on the thickness and the highness of potential barrier of the capping layer.

2.2.3 Carrier transition in Quantum Dots

To achieve population inversion, fast intraband relaxation time is required. In quantum well systems, the typical intraband relaxation lifetime is about 0.1~1 picoseconds (ps). However, due to the lack of phonons needed to satisfy the energy conservation rule, the carrier relaxation into the discrete ground state is significantly slowed down in QDs (the so-called phonon bottleneck). Retarded carrier relaxation, with a lifetime ranging from several tens to hundreds of picoseconds, has been observed in many experiments using time-resolved PL. Even though Auger-like process and carrier-carrier scattering have been suggested for possible fast relaxation channels with a lifetime of 10 ps or less, the bottleneck effect still affects QD laser performance to some extent, such

as the degradation of threshold current and the external quantum efficiency.

2.3 Quantum Dot Lasers

Laser is an acronym meaning “light amplification by stimulated emission of radiation”. A semiconductor, owing to a possibility of direct injection of electrons to the active region, gives an opportunity to make compact and efficient laser device. Basically, laser consists of the light-amplifying region where inverse population takes place and the positive optical feedback.

2.3.1 Basic Principles of Diode Lasers

In semiconductor, photon emission due to electron return from the conduction band to the valence band can be considered as radiative recombination of electron-hole pair. However, electron can also lose its energy non-radiatively. For any light-emitting device, too non-radiative recombination results in additional consumption of excited electrons and has to be minimized. Actually, there are two possible mechanisms when photon emission occurs. One is spontaneous emission with arbitrary phase, arbitrary direction and at an unpredictable instant of time.

Otherwise, photon emission can be triggered by another photon, the so-called stimulated emission, and holds all the properties of the initial one.

Under condition of thermal equilibrium, light emission is balanced by the light absorption. The following equation connects the electron, n , and hole, p , concentrations:

2

ni

np= (2.32) Here n is intrinsic concentration governed by temperature. To _i overcome this loss of photons, larger population of the upper level than that of the ground level is required. This situation, the so-called population inversion, is obviously non-equilibrium and can be reached by some external influence, such as illumination or current injection.

2

ni

np> (2.33) To describe concentrations of electron and hole, their non-equilibrium Fermi levels, F and _C F are introduced. A condition of inverse _V population means that electron population probability of the conduction band state f_C(E_C) having energy E is higher than that of the valence _C where population probability of carriers in semiconductor follows Fermi-Dirac distribution: Taking into account these expressions, the condition of inverse population comes to the following inequality:

g It means that energy separation of electron and hole Fermi levels has to be larger than the band gap of a semiconductor, E . At this moment, so-called transparency takes place. Material becomes transparent for light of the given energy ηω =E_C −E_V . The corresponding density of injection current is known as the transparency current density.

g

2.3.2 Optical Gain and Laser Threshold

When light propagates through a medium, its intensity, Φ , will be attenuated due to light absorption, which is characterized by absorption coefficient α. Mechanisms which contribute to optical loss have various origins. They usually are divided into the output (mirror) loss, α_m, caused by light output from the cavity and the internal loss, α_i, which combines the effect of all the other mechanisms. However, if population inversion takes place in the medium, the absorption coefficient changes sign to negative due to stimulated emission dominates the light absorption.

To emphasize these two different cases of light absorption or light amplification, optical gain coefficient, G , is introduced. Now we consider the output loss inside a Fabry-Perrot cavity as shown in Fig. 2.4.

Let us assume that light propagating from the left to the right facet has intensity Φ at its initial position at the left facet. Over the length L ₀ it becomes to Φ₀exp[(G−α_i)L]. After partial reflection at right facet having reflectivity R , a beam of intensity ₁ R₁Φ₀exp[(G −α_i)L] starts from right to left. In the similar manner, the intensity of the light becomes equal to R₁R₂Φ₀exp[2(G−α_i)L] at its initial position after a round trip, where R is the left-facet reflectivity. In steady state (at threshold), this ₂

Fig. 2.4 Light propagation and its intensity in a round-trip cycle inside the Fabry-Perrot cavity.

round trip results in no change of the light intensity.

G is the threshold gain. At this moment, the injected current density is th

called the threshold current density J_th. The optical confinement factor Γ should be introduced when we consider the interaction between charge carriers and the light. It is defined as the ratio of the integrated light intensity with the region of inversion population to the total optical intensity throughout the laser structure. Different transverse optical modes will have different overlap with the active region. Therefore, the optical gain in Eq. (2.38) corresponds to a certain mode and is known as the modal gain as opposed to the material gain. Specifically, the modal gain is a product of the Γ - factor of the given q mode and the material _th

2.3.3 Characteristics of Quantum Dot Lasers

For an individual self-organized QD, its electronic structure is very similar to the ideal case. There is a set of atomic-like quantum levels composed of several excited states and ground state. However, in contrast to the ideal array of identical QDs, self-organized islands differ greatly from each other in their size , shape, and other parameters affecting the energy of the quantum level. Namely, the quantum energy of real QD arrays is statistically determined rather than the zero-dimensional DOS of

For an individual self-organized QD, its electronic structure is very similar to the ideal case. There is a set of atomic-like quantum levels composed of several excited states and ground state. However, in contrast to the ideal array of identical QDs, self-organized islands differ greatly from each other in their size , shape, and other parameters affecting the energy of the quantum level. Namely, the quantum energy of real QD arrays is statistically determined rather than the zero-dimensional DOS of