de Dead space for electron dh Dead space for hole q Electron charge Cj Junction capacitance εo Vacuum permitivity εr Dielectric constant dw Depletion width A Device area n Ideality factor Id Dark current Io Saturation current kB Boltzmann’s constant Rs Series Resistance Itunn Tunnelling current m* Effective electron mass h Plank’s constant V Applied voltage SMU Source measure unit AC Alternating current DC Direct current LIA Lock-in Amplifier Jgr Generation recombination current density Jd Diffusion current density L0 Diffusion length of minority carrier θsig Phase of measured signal θref Phase of reference signal Iph Photocurrent TIA Transimpedence amplifier TLM Transmission line model Rsys System resistance Rc Contact resistance Rsemiconductor Semiconductor resistance GSG Ground signal ground DUT Device under Test
been derived [,]. AlSb has been studied as being a potential material for high energy photon detection. It is an indirect bandgap material, and the calculated band structure is shown in (a), the minimum conduction band is at X valley with the bandgap of 1.6 eV []. (b) shows the band structure of AlAs, calculated by a k×p method []. It is an indirect bandgap material with the ordering of X-L-Γ conduction band valley minima. The minimum bandgap is calculated as 2.19 eV. For all the ternary III-V semiconductor compounds, the dependence of energy bandgap on alloy composition is assumed to fit to the equation of [] (2.6) where Cbow is the so called bowing parameter which accounts for the deviation from the linear interpolation between the bandgaps of the two binaries A and B. The bowing parameter is typically positive so that the alloy bandgap is smaller than the interpolation result and can be temperature dependent. Here we only focus on the parameter at room temperature. The recommended bowing parameter in AlAsSb for EgΓ, EgX, EgL are 0.8, 0.28 and 0.28 respectively []. The bandgap value of AlAs0.56Sb0.44 can be estimated from AlAs and AsSb [], as listed in . The minimum indirect X valley bandgap is calculated as 1.84 eV, while there is uncertainty in the bowing parameters that also introduces uncertainty in bandgap estimation of AlAs0.56Sb0.44. Responsivity measurements indicated that the cutoff wavelength of AlAs0.56Sb0.44 is around 750 nm which corresponding to the minimum bandgap of 1.65 eV. AlAs (eV) AlSb (eV) Cbow AlAs0.56Sb0.44 (eV) EgΓ 3 2.3 0.8 2.49 EgX 2.16 1.6 0.28 1.84 EgL 2.35 2.21 0.28 2.22 2.1 Bandgaps of estimated AlAs, AlSb and AlAs0.56Sb0.44
non-local models [] which accounted for dead space have been introduced to allow extraction of α and β. The dead space de (dh) is defined as the minimum distance that an electron (hole) has to travel before impact ionising, The random ionisation path length of an electron xe is described by its probability density function (PDF) Pe(xe) is given by (2.22) where α* is the ionisation coefficient after the dead space and de is the dead space. The integration is the probability of a carrier undergoing impact ionisation at the position xe. A similar expression is made for holes by replacing α* by β*.A simple dead space estimation using the threshold energies, Ethe for electron and Ethh for hole, can be given by the following equations. (2.23) (2.24) where q is the electron charge and E is the electric field. Conventionally α* can be deduced from the local ionisation coefficient α and the dead space as following (2.25) Using equation 2.22, the survival probability Se(xe) of an electron after travelling a distance xe is given by (2.26) By substituting Se(xe) to be a random number r between 0 and 1, xe could be derived as (2.27) The same expression also applies to the holes. Each injected carrier produces multiplied carriers with a multiplication factor M. The avalanche process is a stochastic process and the
region is far from the photocarriers generated region, the diffusion time can limit the bandwidth. The diffusion time is given by [] (2.33) where x is the diffusion distance and Dc is the minority carrier diffusion coefficient. The diffusion time limited bandwidth is described by (2.34) Avalanche limited bandwidth Avalanche limited bandwidth is the main limitation of an APD’s bandwidth at high gain due to feedback of carriers in building up the avalanche gain and the time required for the avalanche process to This will limit the response time in an APD. As studied by Emmons [], a model was presented to simulate the bandwidth versus multiplication with different values of ionisation coefficient ratio β/α. Assuming the photocarriers’ diffusion is negligible, the -3 dB bandwidth is mainly limited by the avalanche duration time. The -3 dB bandwidth is decreasing with avalanche gain due to time required for the avalanche duration. shows the calculated -3 dB bandwidth versus avalanche gain results presented by Emmons. It is indicated that as β/α decreases, the avalanche gain is less effective in reducing the frequency response. A curve of M = β/α is superimposed on those curves. When M>β/α, the multiplication has little effect on the bandwidth as the bandwidth is largely determined by the transit time and is less dependent on gain. While when M<β/α, the bandwidth is decreasing with avalanche multiplication. The avalanche duration time τa is dependent on the value of ionisation coefficient ratio k= β/α, and is given by [] (2.35) wa is the absorption layer thickness, M is the avalanche gain value, k is the ionisation coefficient ratio, w is the avalanche region thickness and vs is the saturated carrier velocity. It is important to note that at very large M, M/τa is a constant so that the gain bandwidth product has a constant value at very large gain.
The sensitive technique, adopted in the photo-multiplication measurement, requires a reference signal at a fixed frequency. This is used in the lock-in amplifier to generate a reference signal at the same frequency. The diagram illustrating the signals in a phase sensitive detection is shown in . θsig and θref are the phases of the measured signal and the lock-in amplifier reference signal. The measured signal and lock-in amplifier reference signal have the expression as follows. (3.4) (3.5) Asig and Aref are the amplitudes of the measured signal and reference signals. The output of this sensitive detection is the product of the two sine waves, given by (3.6) When the output Vo passed through a low pass filter, the AC component in equation 3.6 with the frequency at ωsig + ωref will be filtered out, leaving the low frequency component with frequency at ωsig - ωref. When ωsig = ωref, the output would be a DC signal Asig Aref cos(θsig - θref). Therefore, the measured signal with same frequency component to the reference signal will be isolated with output amplitude proportional to the measured signal amplitude. Figure 3.4 Diagram of phase sensitive technique When the measured signal is a square wave with amplitude of Ap at frequency f, it can be considered to be sum of many harmonic sinusoid at multiples off as (3.7) The first sinusoid wave component will be measured and displayed in its RMS value of 2Ap/π × /2 = Ap/π. This technique could avoid any low frequency noise from high leakage or tunnelling current. The avalanche gain, dependent on voltage can be derived by
Appendices Appendix A Wafer details Internal wafer identification Thesis identification Location Description Observation Rev1 PIN1 Chapter 5 p-i-n structure with 80 nm undoped region, semi-insulating substrate Surface current dominant Rev2 PIN2 Chapter 5 p-i-n structure with 230 nm undoped region, semi-insulating substrate Surface current dominant UCL1 PIN3 Chapter 6 p-i-n structure with 30 nm undoped region, conducting substrate Negligible tunnelling current IQE SAM APD1 Chapter 7 InGaAs/AlAsSb SAM APD, with InAlAs in the charge sheet layers Bulk current dominant, with high tunnelling current from InGaAs absorption layer, and edge breakdown SAM APD2 Chapter 7 InGaAs/AlAsSb SAM APD, with AlAsSb in the charge sheet layers Bulk current dominant, with high tunnelling current from InGaAs absorption layer M3698 SAM APD3 Appendix F InGaAs/InAlAs SAM APD, semi-insulating substrate Dark current of 10-100 nA
In a conventional p-i-n structure, assume the light is fully absorbed in p+ cladding layer, dark carriers are uniformly generated in i region for simplicity. In the local model, the position dependent multiplication for primary carriers generated within w is given by So that the gain of pure injection as carriers injected at position x = 0 is given by The mean gain from the multiplied dark carriers and photo carriers is denoted as Mt, the parameters used to calculate Mt is explained in the following Np: total photo generated carriers injected at x = 0 Gi: dark carriers generated in the i region at x position, since dark carriers are uniformly generated in i region so that Gi is independent of position, the total dark carriers generated in the i region would be z: the ratio of the total primary dark carriers over the photo generated carriers, k: ionisation coefficients ratio, k=β/α The mean gain concerning multiplied dark carriers and photo carriers, which is namely the gain measured under DC condition, is given by Here M(0) is the gain of pure electron injection, which is namely the gain measured under AC condition. If z is very small, which means, , Mt will not be deviated a lot from M(0), if z cannot be ignored, here we consider two cases, k=0 and k=1 i) k=1 if k=1, α= β, M(x)=M(0), it suggests that when k=1, the dark carriers and photo carriers will have similar gain values and the gain measurement will not be affected either under DC or AC condition. ii) k=0 so that β=0,M(0)=exp(αw), the mean gain can be expressed as