Material Characteristics of Zinc-Oxide

In recent years, the desire for blue and UV diode lasers and light emitting diodes has prompted enormous research efforts into II–VI and III–V wideband gap semiconductors. Among the well-known semiconductor materials employed in various technical applications, two unique positions are held by gallium nitride (GaN) and zinc oxide (ZnO) in the wide direct band gap semiconductor. In the material property, both GaN and ZnO have many similar aspects, such as material structure, lattice constant, energy band gap,… etc.. In the difference of which, the remarkable property of ZnO better than GaN is

exciton binding energy of 60 meV, which is only 30 meV for GaN. Owing to the larger exciton binding energy, more excitons exist in the room temperature, resulting in higher luminescence than GaN. Furthermore, ZnO can be grown at lower temperature on the cheaper substrate and lead to low cost of growth. However, because of more intrinsic defects, the hard growth of p-type ZnO to achieve the p-i-n junctions, and the degradation of material quality, the current commercial blue and UV LEDs are primitively composed of GaN. However, GaN-based LEDs still face some problems of the luminescence, such as more defects in the material and low electron-hole recombination of c-direction growth. Therefore, it is worth making the further researches on the material of ZnO and GaN on purpose of possessing well-performed LEDs and LDs.

In materials science, ZnO is often called a II-VI semiconductor because zinc and oxygen belong to the 2nd and 6th groups of the periodic table, respectively. This semiconductor has several favorable properties: good transparency, high electron mobility, wide band-gap, strong room-temperature luminescence, etc.

Zinc oxide crystallizes is hexagonal wurtzite, as shown in Fig1.12(a). The hexagonal structure has a point group 6°mm or C_6v, and the space group is P6₃mc or C_6v⁴. The lattice constants are a = 3.25 Å and c = 5.2 Å; their ratio c/a ~ 1.60 is close to the ideal value for hexagonal cell c/a = 1.633. As in most II-VI materials, the bonding in ZnO is largely ionic, which explains its strong piezoelectricity.

Due to this ionicity, zinc and oxygen planes bear electric charge (positive and negative, respectively).

ZnO has a relatively large direct band gap of ~3.3 eV at room temperature; therefore, pure ZnO is

colorless and transparent. Advantages associated with a large band gap include higher breakdown voltages, ability to sustain large electric fields, lower electronic noise, and high-temperature and high-power operation. The bandgap of ZnO can further be tuned from ~3–4 eV by its alloying with magnesium oxide or cadmium oxide.

Most ZnO has n-type character, even in the absence of intentional doping. Native defects such as oxygen vacancies or zinc interstitials are often assumed to be the origin of this, but the subject remains controversial. An alternative explanation has been proposed, based on theoretical calculations, that unintentional substitutional hydrogen impurities are responsible. Controllable n-type doping is easily achieved by substituting Zn with group-III elements Al, Ga, In or by substituting oxygen with group-VII elements chlorine or iodine. Reliable p-type doping of ZnO remains difficult. This problem originates from low solubility of p-type dopants and their compensation by abundant n-type impurities, and it is pertinent not only to ZnO, but also to similar compounds GaN and ZnSe. Measurement of p-type in "intrinsically" n-type material is also not easy because in-homogeneity results in spurious signals.

Current absence of p-type ZnO does limit its electronic and optoelectronic applications which usually require junctions of n-type and p-type material. Known p-type dopants include group-I elements Li, Na, K; group-V elements N, P and As well as copper and silver. However, many of these form deep acceptors and do not produce significant p-type conduction at room temperature.

ZnO has wide direct band gap (3.37 eV or 375 nm at room temperature). Therefore, it’s most

common potential applications are in laser diodes and light emitting diodes (LEDs). Some optoelectronic applications of ZnO overlap with that of GaN, which has a similar bandgap (~3.4 eV at room temperature). Compared to GaN, ZnO has a larger exciton binding energy (~60 meV, 2.4 times of the room-temperature thermal energy), which results in bright room-temperature emission from ZnO. Recent studies of ZnO epilayers have observed spontaneous emission from free-exciton (FE) radiative recombination as well as stimulated emission from exciton-exciton scattering (EES) and electron-hole-plasma (EHP) radiative recombination at temperature up to ~550K.[3] Other properties of ZnO favorable for electronic applications include its stability to high-energy radiation and to wet chemical etching. The pointed tips of ZnO nanorods result in a strong enhancement of an electric field.

Therefore, they can be used as field emitters. Transparent thin-film transistors (TTFT) can be produced with ZnO. As field-effect transistors, they even may not need a p–n junction, thus avoiding the p-type doping problem of ZnO. Some of the field-effect transistors even use ZnO nanorods as conducting channels.

Fig 1.1 (a) Before condensation, the atoms look like fuzzy balls. (b) After condensation, the atoms lie exactly on top of each other . (c) Schematic of the apparatus. Six laser beams intersect in a glass cell, creating a magneto-optical trap (MOT). (d) Bose-Einstein Condensation at 400, 200, and 50 nK.

kz k E

Eexc

0 E=

LP

can’t couple to light

optical

cone UP

couples to light

Eexc

k_ E

LP

can’t couple to light

(a) (b)

Fig 1.2 (a) the dispersion of bulk polariton (b) the dispersion of microcavity polariton

Fig 1.3 The different types of microcavity (a) planar microcavities (b) pillar microcavities [13]

((c) Whispering-gallery microdisk resonator [14] d) photonic-crystal cavity [15]

250 300 350 400 450 500

Fig 1.4 DBR reflectivity spectrum with different wavelength

300 350 400 450 500

Fig 1.5 Microcavity reflectivity spectrum with different wavelength

2 1

E −E = ω E2

E1

ω

Fig 1.6 Schematic of two level system

Probability

Fig 1.7. Probability for finding the atom in either the upper or lower level in the strong-field limit

in the absence of damping. The electron oscillates back and forth between the two levels at the Rabi angular frequency, ΩR. This phenomenon is either called Rabi flopping or Rabi oscillation, [16].

Fig1.8
Polariton lasing and photon lasing. (a) The emission energy vs pump power for a N=12 multi-QW planar microcavity. (b) The dispersion characteristics of polariton BEC (green diamond)

and photon lasing
(pink triangle) as well as the linear dispersion of UP (red square) and LP (blue circle) at low pump power, [36].

Fig 1.9 A negative conductance polariton amplifier. (a) The LP emission intensity taken as a function of energy and in-plane wave vector. The system is above the quantum degeneracy threshold at the bottleneck. The solid line and the dashed line indicate the theoretical dispersion (b) Observed probe emission for pump only, probe only, and simultaneous pump and probe excitation, [37].

Fig 1.10(a) Semi-logarithmic plot displaying RT emission spectra at average pump power densities ranging from 0.16 to 28.8 W/cm², shifted for clarity. C and X are also reported (arrows). (b)

Three-dimensional representation of the far-field emission with emission intensity displayed on the vertical axis
linear vertical scale above threshold. C and X are also reported
(white lines).

Fig. 1.11 (a) Angle-resolved PL spectra at RT in the range of 0 º to 40 º for a ZnO hybrid MC. The dotted line is the exciton mode. The solid lines are guides to the eye. (b) Experimental cavity olariton dispersion curve. The dashed lines represent the cavity and exciton modes. (Courtesy of R. Shimada).

Fig 1.12 (a) Structure of wurtzite, which is a member of the hexagonal crystal system and consists of tetrahedrally coordinated zinc and sulfur atoms that are stacked in an ABABAB pattern. (b) the bangap structure of wurtzite structure

systems atomic gases excitons polaritons

lifetime 1ms/1s~10^-3 1ps/1ns~10^-2 (1~10ps)/(1~10ps)

=0.1~10 Table 1.1: Parameter Comparison of BEC Systems

Material Bandgap Exciton

binding energy

Rabi splitting

Advantages Drawbacks

GaAs 1.519eV ~10meV 4meV lattice-match DBR crystal quality

small exciton binding energy

GaN 3.507eV ~26meV(Bulk)

~40meV(QW)

Table 1.2: Comparison of material use in exciton-polariton BEC Systems

Chapter2.

THE COUPLING BETWEEN EXITON WITH PHOTON 2.1 Quasi-particle model

2.1.1 Properties of Wannier-Mott Exciton

A solid consists of 10²³ atoms. Instead of describing the 10²³ atoms and their constituents in full detail, the common approach is to treat the stable ground state of an isolated system as a quasi-vacuum ( the state with filled valence band and empty conduction band for a semiconductor ) and to introduce quasi-particles as a unit of elementary excitation, which only weakly interact with each other. An exciton is a typical example of such a quasi-particle, consisting of an electron and a hole bound by the Coulomb interaction. The quasi-vacuum of a semiconductor is the state with filled valence band and empty conduction band. When an electron with charge − is excited from the valence band into the e conduction band, the vacancy it leaves in the valence band can be described as a quasi-particle call a

‘hold ’. A hole in the valence band has charge + , and an effective mass defined by e

hole and an electron at p~ 0 interacts with each other via Coulomb interaction and form a bound pair (an exciton) analogous to a hydrogen atom where an electron is bound to a proton. The envelope wavefunction of an exciton is also analogous to that of a hydrogen atom. However, due to the strong dielectric screening in solids and a small effective mass ratio of the hole to the electron, the binding energy of an exciton in GaAs, GaN, and ZnO is on the order of 10 meV, 26meV, and 60meV, respectively, three orders of magnitude smaller than that of hydrogen atoms, and the radius of an exciton is about extending over tens of atomic sites in the crystal (Wannier-Mott exciton).

An exciton can be classified into Wannier-Mott exciton and Frenkel exciton, depending on the properties of the material in question, as shown in Fig 2.1. In Wannier excitons, typically observed in covalent semiconductors and insulators, the electron and hole are separated by a distance much larger than the atomic spacing, so that the effect of the crystal lattice on the exciton can be taken into account primarily via an average permittivity.

An exciton is a typical example of such a quasi-particle, consisting of an electron and a hole bound by the Coulomb interaction. Therefore, an effective is that of a hydrogen-like atom formed by

an electron and a hole interacting, though simplified picture of the exciton state. The energy of exciton

* 2

1

n y

E R⁼ ⋅n Ry is the Rydberg energy.Following Hanamura and Haug [64], the Hamiltonian of the

electronic system of a direct two-band semiconductor is:

( ) ( ) ( )

potential, and ψ is the field operator for electrons expanded in terms of the electron eigenfunctions

( )

is the number of unit cells of the lattice. aˆ_kj is the fermionic annihilation operator for an electron. It

obeys the commutation relations

{

^{a aˆˆ ,k l†}

}

^{=δk l, and}

{

^{aˆˆ ,kj alj}

}

⁼⁰. For the valence band, we introduce

the hole creation operator ˆb_−k to replace the electron annihilation operator:

† ˆ

we can now simplify the denotation as:

ˆˆ_{kc k}

a = a Eq(2.5)

Substitute eqn to eqn into eqn, neglecting number non-conserving term, we obtain:

( ) ( )

V are the direct and exchange interactions among

electrons and holes due to the Coulomb potential ˆV :

( )

1 2 3 4 1, 2 ˆ 3 , 4 , , , ,

i j m n k k k k

V = k i k j V k m k n i j m n=c v Eq(2.8)

For Wannier-Mott excitons, the plane wave factor in eqn and the Coulomb potential ˆV are

slowly varying functions which charge vary little in one unit cell of the lattice, hence eqn can be calculated by first integrating the Block functions in a unit cell Ω , then summing over all unit cells _i

weighted by the planar wave factors. We also nitice that:

( ) ( )

And we find that eqn can be simplified to a form, for example:

( ) ( )

²

Now we consider the general wavefunction of an electron-hole state:

††

'ˆ ˆ; kk k k

p =

∑

C a b vac Eq(2.11)

Where vac is the quasi-vacuum with a full valence band and an empty conduction band. From

the eigenvalue equation H p =E p , we obtain the equation for the amplitude C : _kk'

( ) ( )

(

^{E ke} +^{Eh k'} −^{E C}

)

^kk'−

∑ (

^{Vkcv v}− −^{l' k l'c}−^{Vkcv c v}− − −^{l' l k'}

)

^Cu =⁰ Eq(2.12)

Takinf a Fourier transsfrom of eqn,use eqn, we obtain the Wannier equations [65] for an exciton:

( ) ( )

We separate the center of mass motion and the relative motion by introducting the new

e h, e e h h

r=x −x R=β x −β x Eq(2.15)

Where β =_e m M_e ,β =_h m M_h ,M =m_e+m_h. Then eqn become:

(

x xe^, h

)

n

( )

r ^exp

(

iK R

)

V

φ =φ ⋅ Eq(2.16)

And the equation of relative motion is:

2 2

( )

It has the same form as the equation of relative motion for a Hydrogen atom, but the reduced mass _r m m^{e h}

m = M is normally four orders of magnitude less than the hydrogen atom mass, and the Coulomb interaction is screened and reduced by a factor ε . These lead to a much larger Bohr radius ₀ and much weaker binding energy of an exciton compared to an hydrogen atom. The total energy of the

pair is:

And the Bohr radius of the 1s exciton is:

2

operators are:

Hence excitons can be considered as boson in the low density regime when n_exc<<a_B⁻³, or, when the exciton inter-particle spacing is much large than its Bohr radius.

The electron and hole in an exciton form a dipole which interacts with electromagnetic fields of light. The interband optical transition matrix element is given by the Fermi's golden rule:

( )

consider an uncorrelated electron-hole pair, the matrix element is given by:

( )

Notice that for the lowest energy interband transition

A convenient material parameter that characterizes the exciton-photon coupling is the exciton

oscillator strength f defined analogous to the atomic oscillator strength as:

2

The optical transition matrix element M can be expressed in terms of f as:

e f2

M m V

π ε ^∗

= Γ Eq(2.27)

Γ are the first three terms in eqn which depends on the selection rules and geometric properties

of the semiconductor.

2.1.2 Introduction to coupling between photon and exciton

Typical microcavities have a thickness of a few integer multiples of half the photon wavelength at the exciton resonance frequency. Consequently, an exciton is coupled to a single cavity mode according to the in-plane wave-vector conservation.

Using the rotating wave approximation, the linear Hamiltonian of the system is written in the second quantization form as:

wavenumberk_c = ⋅k z determined by the cavity resonance. ˆ eˆ_k^†

 is the exciton creation operators with in-plane wavenumber k_. Ω is the exciton-photon dipole interaction given by the exciton optical

transition matrix element M, and we used the condition that M is non-zero only between modes with the same k_. The above Hamiltonian can be diagonalized by the transformation:

ˆˆˆ system. They are called the lower (LP) and upper polaritons (UP), corresponding to the lower and upper branches of the eigen energies. A polariton is a linear superposition of an exciton and a photon with the same inplane wavenumber k_. Since both excitons and photons are bosons, so are the

polaritons. The exciton and photon fractions in each lower polariton (and vice versa for upper polaritons) are given by the amplitude squared of X_k

 and C_k

( )

The energies of the polaritons, which are the eigenenergies of the Hamiltonian, are deduced from the diagonalization procedure as:

When the un-coupled exciton and photon are at resonance, E_exc =E_cav, lower and upper polariton energies have the minimum separation E_UP−E_LP =2 , which is often called the `Rabi splitting' in Ω analogy to the atomic cavity Rabi splitting. Due to the coupling between the exciton and photon modes, the new polariton energies anti-cross when the cavity energy is tuned across the exciton energy. This is one of the signatures of 'strong coupling'. When E_exc−E_cav   , the polariton energies reduce to the Ω same as photon and exciton energies due to the very large detuning between the two modes, and polariton is no longer a useful concept. So the detuning is assumed to be comparable to or less than the coupling strength in our discussions unless specified.

We use ∆ as the exciton and photon energy detuning at k_ =0:

2 dispersions. At k_  , the dispersions are approximately parabolic: k_c

( ) ( )

^{2 2}

The polariton effective mass is the weighted harmonic mean of the mass of its exciton

and photon components: mass of its center of mass motion, and m_cav is the effective cavity photon masses given by eqn. Since

mcav is much smaller than m_exc,

The very small effective mass of LPs at k_ 0 determines the very high critical temperature of

phase transitions for the system. At large k_k_c, ^Ecav

( )

^kk ^−Eexc

( )

^k  ^Ω, dispersions of the LP and UP converge to the exciton and photon dispersions respectively, and LP has an effective mass

( )

LP c exc

m k_ k_ m . Hence the LP's effective mass changes by four order of magnitude from k_ 0 to large k_. This peculiar shape has important implications in the energy relaxation dynamics of

different ∆ are given if fig 2.2. When taking into account the finite lifetime of the cavity photon and exciton, the eigen-energy equation is modified as:

( ) ( )

^{2 2}

( )

²

,

1 4

LP UP 2 exc cav cav exc exc cav cav exc

E k^ = ^E +E +i γ +γ ±  Ω +E −E +i γ −γ  ^ Eq(2.39) Here γ_cav is the out-coupling rate of a cavity photon due to imperfect mirrors, and γ_exc is the non-radiative decay rate of an exciton. Thus the coupling strength must be larger than half of the difference in decay rates to exhibit anti-crossing, i.e., to have polaritons as the new eigen modes. In another word, an excitation must be able to coherently transfer between a photon and an exciton at

least once. When  Ω

(

γcav −γexc

)

², we call the system in the strong coupling regime. In the opposite limit when excitons and photons instead are the eigenmodes, the system is called to be in the weak coupling regime, and the radiative decay rate of an exciton is given by the optical transition matrix element. We are mostly interested in microcavities with γ_excγ_cav   , then equation gives Ω an accurate approximation of the polariton energies.

As a linear superposition of an exciton and a photon, the lifetime of the polaritons is directly

determined by γ_exc and γ as: _cav

In general, the polariton lifetime is mainly determined by the cavity photon lifetime:

2

LP C cav

γ _ γ . Polariton decays in the form of emitting a photon with the same k_ and total

energyω =E_{LP UP,} , The one-to-one correspondence between the internal polariton mode and the

external out-coupled photon mode lends great convenience to experimental access to the system. The

external emitted photon field carry direct information of the internal polaritons, such as the energy dispersion, population per mode, and statistics of the polaritons. It is mainly through the emitted photons that we study the internal polaritons.

2.1.3 Quasi-Particle Model simulation in MatLAB The Hamiltonian for free exciton and cavity photon is:

( ) ( )

The in-plane dispersion relation of exciton and cavity photon is

( )

^{ =}

^{( )}

^{0 +}₂^{2 2 − Γ}₂^

( )

^ phenomenological decay rate the accounts for finite linewidth of the exciton resonance, and Γ_C is the cavity photon escape rate taking into account the possibility of photon tunneling across the cavity mirror.

0

Where n is the cavity refractive index, and _C λ is the resonant wavelength of the cavity. ₀ The linear exciton-photon coupling formally reads:

0 =

∑

_ ^Ω k^{+ }k + . .

k

H a b H c , Eq(2.45)

where Ω is the coupling strength between an exciton and a photon, which is proportional to the exciton oscillator strength and to the number of QWs embedded in the cavity.

If we only consider only one exciton and one photon interaction, the problem can be reduced to a two-level system. As result, the eigenenergies of resulting states can be found by diagonalisation of the

matrix:

The eigencalues of this matrix are given by

2 2

(EX −λ)(EC−λ)= Ω Eq(2.47)

Solutions of the equation are

( ) ( ) ( )

¹

( ( ) ( ) )

^{2 4 2 2}

E and U E are the energies of the upper and lower polariton branches, respectively. _L

From the solution of eigenvector, we can define the Hopfield coefficients X_{U L( )} and C_{U L( )} for

the upper (lower) polariton branch

( )

^{( )}

( )

^{( )}

X 2,C² describe weights of excitonic and photonic parts in each branch. As result, the sum of X 2 and C² equal one.

However, we have to deal with more complicated system in practice. Generally, for III-V nitride and II-VI zincoxide semiconductors with a wurtize structure belonging to the space group C , the _6v⁴ conduction band minimum is located at the center of the Brillouin zone and its vicinity is almost

However, we have to deal with more complicated system in practice. Generally, for III-V nitride and II-VI zincoxide semiconductors with a wurtize structure belonging to the space group C , the _6v⁴ conduction band minimum is located at the center of the Brillouin zone and its vicinity is almost

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