CHAPTER 1 INTRODUCTION
1.2 Motivation and background
The reliability of the wide-gap nitride LDs is still an important issue because of their short lifetime caused by high threshold current. In comparison with conventional zinc-blende GaAs-based lasers, the high threshold current of the wurtzite nitride lasers may be attributed to several factors, including immature material preparation, the intrinsic large density of states in the valence bands, and large leakage current in the device structures not yet optimized.
1.2.1 Spillover effects
The leakage current can be regarded as composed mainly of three components according to their different origins: (1) the component caused by nonradiative recombination of electrons and holes in the active region, (2) the one due to electron leakage from the active region to the p-type cladding layers,[23] and (3) the one caused by the interband transition of high-energy carriers in the neighborhood of the active region. [23] The leakage current due to nonradiative recombination has been considerably alleviated in the GaN lasers by the reduction of defects with the progress of material growth and device processing technologies.[24,25] As to the electron leakage into the p-type cladding layer, it has been commonly found from recent works that such leakage can be reduced significantly by the insertion of an AlGaN electron blocking layer (EBL) between the active region and the p-type layer.[26-30] Furthermore, it has been demonstrated that this leakage can be made very low compared to other leakages by optimization on the structure with EBL.[27-30] The hole leakage out of the active region can be neglected because of the large effective mass of the inertial holes.
The EBL, however, cannot suppress the spillover of energetic carriers into the
continuous subband states above the barriers in energy surrounding the quantum wells (QWs) of the active region. Interband transition involving the high-energy spillover carriers usually gives a negligibly small contribution to the optical peak gain of the QW active region, but may cause significant consumption of electric current. Therefore, the recombination of spillover carriers can be regarded as one of the paths for the leakage current. Such a problem of carrier spillover depends on temperature and is particularly serious for electrons in the conduction bands because of the small electron effective mass, the large asymmetry between the densities of states of the conduction and valence bands, and the narrow QWs usually used in the nitride LDs. There are quite few literatures discussing leakage due to the recombination of spillover carriers,[23] and so far, the influences of the optical transitions from the spillover carriers on the shapes of gain and spontaneous emission spectra have not been discussed and analyzed in detail. Furthermore, there has not been any research work on this problem for the short-wavelength nitride LDs.
In this dissertation, we present the calculation results of the carrier spillover effects on the optical gain, the spontaneous emission, and the threshold current for InGaN/GaN QW LDs. The influences of the temperature are also discussed. We found that the electron spillover can broaden the gain and the spontaneous emission spectra, deteriorating the threshold of QW LDs. An optimized multi-QW structure is then proposed to solve the problem.
1.2.2 Modulation p-type doping in the active region
In addition to the optimization on the structure of the active region, more efforts can be made for further reducing the threshold current. For example, one can introduce dopants into the active region. Actually, n-type doping with Si in the InGaN/GaN quantum wells (QWs) has been performed in some experimental works.[31-34] It is found that the Si
doping can improve the crystalline quality of GaN layers and the InGaN/GaN interface.[35,36] It is well known that the optimized temperature for the high crystalline quality of InGaN wells is lower than that suitable for the growth of GaN barriers. In order to get high quality of InGaN wells, the GaN layers are generally compelled to be grown at low temperature. Hence, this forms the island-like spiral defects initiated by threading dislocations existing in the underlying GaN template. Fortunately, workers found that introducing Si impurities can dramatically suppress these island-like spiral structures of GaN layers.
However, the n-type doping causes a large amount of electrons occupying high energy states at threshold and leads to serious spillover of electrons. A high barrier of EBL is then needed to prevent the energetic electrons from leaking into the p-type cladding layer, but this gives rise to a reduction in optical confinement within the active region. From this point of view, the p-type doping should be a better choice than the n-type doping because it can reduce the number of spillover electrons at threshold. However, there are few experimental or theoretical works concerning the optical property of the p-type doped active region in InGaN/GaN QW LDs. In order to further improve the performance of the InGaN/GaN QW LDs, it is worthwhile to investigate the influence of p-type doping on the optical property of the active region.
So, we also theoretically investigate the influence of different species (the n-type and p-type) and various levels of doping to the active region on the spontaneous emission rate and the threshold current density of InGaN/GaN LDs. Our results demonstrate that the amount of spillover electrons at threshold can indeed strongly depend on the species and the level of doping. Accordingly, we can obtain a low threshold current by optimizing the doping level with a preferred doping species.
1.2.3 Vertical resistivity of p-type AlGaN/GaN superlattices
Besides the high threshold current, some difficulties obstructing further advancement still exist. For example, the p-type AlGaN cladding layers for LDs or LEDs have in general a high electric resistivity due to the high activation energy of acceptors doped in GaN and related nitride alloys.[37-41] To reduce the resistivity, one can insert an AlxGa1-xN/GaN superlattice (SL) into the cladding layer.[42-46] In the inserted SLs, the holes are easily ionized from the acceptors in the barriers to the wells since the acceptor levels in the barrier region are far below the Fermi level. Therefore, the spatially averaged density of holes can be much increased by inserting a SL in the cladding layer.[47]
However, the average vertical resistivity across the p-type AlxGa1-xN/GaN SL is still large because of the high barriers of the SL. Linearly graded barriers (AlxGa1-xN) with appropriate doping have been used to reduce the barrier height and hence obtain a low vertical electric resistance.[48,49] However, decreasing the barrier height may also cause a reduction of optical confinement by the cladding layers for light-emitting devices.
Furthermore, it is inconvenient to fabricate the graded layers with continuously varying composition.
In this dissertation, we also investigate theoretically the electrical properties of p-type SL structures without using the graded layers and find out the key factor that influences the vertical resistance dominantly. Accordingly, we propose a more preferred SL structure than the previous ones.
Chapter 2
Calculation Method
2.1 Piezoelectric and spontaneous polarization
The nature structure of the III-V nitrides is wurtzite, a hexagonal Bravais lattice with a basis of 2 diatomic molecules (for instance, GaN). The wurtzite structure is a noncentrosymmetric compound crystal exhibiting two different sequences of atomic layers laying in two opposite directions parallel to the crystallographic c-axis, and consequently crystallographic polarity along the c-axis can be observed. In wurtzite structure, the [0001]
(also defined as the z axis here) is conventionally defined as the direction pointing from the cation to the nearest anion along the c-axis. The polarization mentioned above is so called spontaneous polarization Psp. The orientation of the spontaneous polarization is defined assuming that the positive direction goes along [0001], which depends on the sequence of the atomic layers. We take GaN as an example. For binary GaN compounds with wurtzite structure, the sequence of the atomic layers of the constituents Ga is along the [0001]
direction, which is reversed to that of N along [000 1 ], as shown in Fig. 2.1. The corresponding (0001) and (000 1 ) faces are respectively the Ga-face and N-face, as indicated in the figure. So, we can define Psp =P zspˆ for Ga-face GaN and for N-face GaN. The GaN films grown by MOCVD are generally Ga-face while they are commonly N-face by MBE.
sp = −P zspˆ P
Since the lattice mismatch is an inherent problem in preparing nitrides, the nitride films are generally stressed. The components of the piezoelectric tensor are not all vanished in
wurtzite structure under stress. So, we have piezoelectric polarization as well, and taking GaN as the example, it is expressed as[50]
2 and InGaN over the whole range of compositions, the orientation of the piezoelectric polarization is parallel to the spontaneous polarization in the case of tensile strained layers, but anti-parallel in compressive strained ones.
If we consider a epitaxial films grown on the top of a thick relaxed GaN buffer layer, the discontinuity of the polarization (Psp+Ppz) induces sheet charge density at the interface, as indicated by σ in Fig. 2.2, and free carriers will be attracted and accumulate[51].
Fig. 2.1 Schematic drawing of the crystal structure of wurtzite Ga-face and N-face GaN
Fig. 2.2 Orientation of the spontaneous and piezoelectric polarization in pseudomorphic grown wurtzite AlGaN/GaN and InGaN/GaN heterostructures with Ga- or N-face polarity.[51]
2.2 Band structure of quantized heterojunction
2.2.1 Luttinger-Kohn Hamiltonian for holes
In this section, we derive the Luttinger-Kohn Hamiltonian for holes in the wurtzite structure.
The method introduced by Bardeen and Seitz is used here.[52,53] Consider the Hamiltonian near the zone center with the spin-orbit interaction,
⋅ theorem, the original Schroodinger equation for an electron wave function
m0 c
Here, is the cell periodic function. Further, we can obtain the Schroodinger equation for :
Note that the last term on the left-hand side is the k-dependent spin-orbit interaction. Since the electron velocity in the atomic orbit is much larger than the velocity of the wave packet with the wave vectors near the zone center, the last term is much smaller than the other terms and can be neglected.
For the wide band-gap semiconductors, the coupling of valence bands to the two degenerate conduction bands can be ignored. So we can treat the six valence bands in class A, which we are interested in, and put the other bands in class B. The upper valence bands of bulk wurtzite semiconductors at the zone center (k = 0) are described mostly by p-like functions:
We choose these six valence-band wave functions as the bases for the states of interest (class A) and write the band-edge wave function as
(2.6)
Use Loowdin’s method. The six-by-six Hamiltonian matrix for the valence bands can be written as
(2.7)
6 6jj ( )ˆ 6 6jj ( 0) 6 6jj ( ), H ×′ k =H ×′ k= +D ×′ k
where the operation version is used. The first term on the right-hand side is corresponds to the Hamiltonian obtained from Kane’s model at k = 0.[54] Use the six wave functions in Eq. (2.5) as the bases, the Kane Hamiltonian takes the form
1 2
The parameters account for either the crystal-field split energy ( ) or the spin-orbit interactions ( and
( 1, 2, 3)
i i
Δ = Δ1
Δ2 Δ ). E3 v is the reference energy. The second term comes from the Loowdin’s perturbation theory. But here, the Burt’s exact envelope-function theory (EEFT) is considered to make the envelop function behave smoothly and continuously even
at abrupt interfaces.[55] Using the notation of Stavrinou and van Dalen,[56] the k-dependent second term is given by
2 2
Note that the ˆkα operators must have the specific ordering indicated here. Use the six wave functions in Eq. (2.5) as the bases, the D6 6× matrix takes the form,[57] nonvanishing inner products of the perturbation.[57] The full Hamiltonian H6 6× ( )kˆ is
obtained by adding Eq. (2.8) and (2.10).After replacing the operators in H6 6× ( )kˆ with their corresponding wave numbers, we can relate the parameters , , σ σ σ δ πz xz, , , and πz to the Rashba-Sheka-Pikus (RSP) parameters Aj’s and get
1
2.2.2 Hamiltonian applicable to strained QW along [0001]
We consider a typical III-V nitride QW structure grown along the [0001] direction (also defined as z axis here). In such a case, the potential of hole is only confined along the z axis, and since the structure is perfectly bulklike along the xy plane of the QW structure, ˆk and x
can be replaced by their wave numbers. Using the essential results from Pikus-Bir Hamiltonian for a strained semiconductor, the strain effects can be easily included by the same symmetry consideration and a straightforward addition of corresponding terms:
ˆy
k
k kα β →εαβ.[58,59] By means of the usual unitary transformation of basis functions with the unitary matrix Hamiltonian can be block diagonalized into one consisting of two 3×3 blocks,[57]
ˆ ( ; )ˆ 0
with are deformation potential constants. The normal strain components ε⊥ and ε in the QW region are given by
where and are the lattice constants of undeformed materials making up the substrate and the QWs, respectively. and are stiffness constants of the QW material.
a0 a
C13 C33
2.2.3 Wave function of valence states
Based on the k p⋅ model, the wave functions of valence band states can be expressed as
3
which, together with their energy E, can be solved by the effective-mass equation,
(2.19)
and are the in-plane position vector and the wave vector of the particle, respectively;
rt kt
0( )
E zv is the z-dependent valence band edge of the undeformed materials composing the heterostructure. Again, it is noticed that the order of the operators in Eq. (2.16) is of importance to the correct boundary conditions for matching the envelope functions.[57]
2.2.4 Conduction states and wave functions
For the conduction band states confined in the QW structure along the [0001] direction, we use the single-band effective-mass equation,
2 effective mass in the direction along (transverse to) the growth direction;
mz mt
0( )
E zc is the z-dependent conduction band edge of the undeformed materials composing the heterostructure; Pcε is the hydrostatic energy shift in the conduction band. Neglecting the small spin-splitting effect, we can write down the wave functions of conduction band states for spin up and spin down as simply a product of the envelope part and the Bloch function part,
( ; )t 1 ei t t ( ; ) ,z k u A ⋅ ϕ
Φ± r k = k r t c± (2.21)
where uc± are the conduction band Bloch functions at the Γ point for spin up ( ) and spin +
down ( ). −
2.2.5 Carrier concentration
Once the wave functions are explicit, one can obtain the carrier concentrations along the z axis
where the factor 2 in Eq. (2.22) accounts for the spin degeneracy in the conduction band.
is the quantum well width. The functions
Lw f F kic( , )c t and fjσ( , )F kv t are the Fermi-Dirac distribution functions for the probabilities of electrons occupying the states
( ; ) 1 i t t ( ; )
The sheet carrier concentrations are useful in this dissertation and are shown here as well. For given quasi-Fermi levels of the conduction bands and the valence bands , one can obtain the sheet carrier concentrations as functions of quasi-Fermi levels by the integrals
2
When we consider the charge concentration in the active region, the charge neutrality allows us to write donors, and ionized acceptors, respectively. Here we use the sheet concentrations instead of the volume concentrations to avoid the complexity arising from the penetration of the bound wave functions into the barriers. The dopants are assumed to be introduced into the barriers around the QW. For convenience, we also assume the carriers released from the dopants are all relaxed to the well region.
2.3 Calculation method for optical properties
2.3.1 Optical gain and spontaneous emission rate
Consider the electron-photon interaction. The Hamiltonian can be expressed as
( )
where A is the vector potential of the electromagnetic field. Expanding the square term,neglecting the q2A2/ 2 0 because of eA p , and using the Coulomb gauge
H ′ can be consider as the perturbation term due to the photon. Assume the vector potential in the direction of the electromagnetic field. Hence, we get
ˆe
According to the Fermi’s golden rule, the transition rate per unit volume in the crystal Ra b→ of electrons from state a with energy Ea to state b with energy Eb (assuming Eb > Ea) due to the absorption of a photon with energy Eop can be expressed as
2
where the prefactor 2 accounts for the spins, V is the total volume of the crystal, and f is the Fermi-Dirac distribution. Similarly, when the electrons transit from state b to state a because of the emission of a photon with energy ω, the transition rate per unit volume in the crystal Rb a→ has the form is defined as the net number of photons absorbed per second per unit volume divided by the number of total photons injected per second per unit area, can be written as
2
where S is the magnitude of the optical intensity and can be substituted by . Here, n
2 2
0 0 / 2
n cr ε ω A
r is the refractive index, and c is the speed of light in free space. Using the Eq. (2.30), we get
Since the speed of light in material and the density of states for the photons per unit volume per energy interval around Eop are and , respectively, the stimulated and spontaneous emission rates per unit volume per energy interval around E
/ r
2.3.2 Formulas for quantum structure along [0001]
In a wurtzite heterojunction structure quantized along the z axis, the quantum numbers corresponding to the x-y plane for the quantized states can be expressed by kt, where
2
t x
k = k +ky2 . The summations over the quantum numbers k becomes summations over (kt, i, j), where i and j index respectively the ith conduction subband and the jth valence
subband. So we get
where σ corresponds to the upper (+) and lower (-) block Hamiltonian. The functions fic and fjσ are the Fermi-Dirac distribution functions for the probabilities of electrons occupying the states 1 the -component (e = x, y, z.) of the momentum-matrix element for interband transition
between the states and
for the TE-polarization component, and
2 0 2
for the TM-polarization component, where the parameters Epx and Epz are defined as
2
Change the summation to the integral form by 2 1
t 2
where Lij is the width of the region in which the interband process occurs between subbands i and j. When the scattering relaxation is included, the delta function is replaced by a
Lorentzian function with a intraband relaxation time γ , assumed to be 0.1 ps here, and we
where is the interband transition energy between the conduction
subband state with energy
( ) ( )
Similarly, we can obtain the optical gain g( ω) and the spontaneous emission rate rsp( ω) for the wurtzite quantum well structure as below
( )
2.3.3 Optical radiative current density
The component of the spontaneous emission rate is due to the recombination of electrons in conduction subband and holes in valence subband
sp,ij
r
i j . It has the meaning of
the number of emitting photons due to the recombinations per unit time per unit volume per unit photon energy interval at energy ω. The corresponding component of the resulting radiative recombination current density can be written as
sp, ( )
ij ij ij
J =qL r
∫
ω d ω. (2.45)The total current density J =
∑
J , which is the sum of all current density components.2.3.4 Threshold condition
We are mainly interested in the effects of carrier spillover on threshold. For the threshold condition, we use the formula
max (g ) ,
ω ω α
Γ = (2.46)
where α is the total cavity loss, which includes internal loss and mirror loss, is the optical confinement factor, and
Γ
max (ω g ω) is the peak gain. From the threshold condition, we first obtain the quasi-Fermi levels, and , at threshold. With and , we obtain the carrier distribution in energy space, based on which we further calculate the gain spectra, the spontaneous emission rates, and the recombination current densities at threshold.
Fc Fv Fc
Fv
2.4 Current transport in p-type layers
When we consider the case of current injection in p-type layers, three mechanisms are used here, including the drift-diffusion, tunneling and thermionic emission. As illustrated in Fig.
2.3 with valence band profile of GaN/AlGaN SL as the example, the drift-diffusion model is used for current across the wells (W1 and W2) and bulk GaN regions whereas for current through the barriers (BB12) we use only the tunneling model for traveling holes with energy
2.3 with valence band profile of GaN/AlGaN SL as the example, the drift-diffusion model is used for current across the wells (W1 and W2) and bulk GaN regions whereas for current through the barriers (BB12) we use only the tunneling model for traveling holes with energy