Combined optical and electric simulation of metal- semiconductor-metal photodetectors

T.O.Korner

Abstract: The combined optical and electric simulation of a metal-semiconductor-metal (MSM) photodetector is presented. The generalised multipole technique (GMT) for rigorous modelling of the light propagation is combined with electric device simulation methods based on a drift- diffusion charge carrier transport model. Different optical designs are compared with respect to the detector performance that can be achieved. It is demonstrated that the use of incoupling gratings can significantly increase the detector efficiency, compared to designs using antireflective layers. Numerical experiments carried out for illumination under varying angles of incidence also show that not only the amount of light that reaches the semiconducting substrate but also the exact spatial intensity distribution have non-negligible effects on the detector efficiency and response times. This in tum proves the necessity of rigorous optical simulation.

1 Introduction

Metal-semiconductor-metal (MSM) detectors play an increasingly important role in the detectiodgeneration of ultrashort optical/electric pulses. Examples of their uses include optical communication systems and chip-to-chip connections, where the low dark current, fast response and high sensitivity achievable in MSMs are important advan- tages. While traditionally mainly 111-V semiconductor materials have been used for optoelectronic applications, a number of silicon-based MSM designs for ultrafast pulse detection at visible wavelengths have recently been presented [l, 21. The advantages of such designs are the lower technology cost, potentially higher integration density and the possible integration with conventional, silicon-based circuits and devices.

The response times and, as a result, the bandwidth of an MSM detector depend on the electrode spacing. The smaller the spacing, the shorter are the response times that can be achieved due to the shorter distances the charge carriers need to travel to the electrodes from where they are generated. On the other hand, a decrease in the electrode spacing leads to a decrease in the sensitivity, since the electrodes shade the incoming radiation. As a result, a trade-off has to be found between efficiency and response time.

Various strategies exist for influencing the efficiency for a given electrode spacing. Antireflection layers are commonly used to eliminate the high reflection losses at the surface of the semiconductor due its high refractive index. Also, microlenses or incoupling gratings can be applied. The latter have been suggested as a potential way of increasing the detector efficiency [3].

0 IEE, 2000 IEE Proceedings online no. 20000293 DOI: 10.1049/ip-opt:20000293 Paper received 23rd September 1999 The author is with ISE Integrated Systems Engineering AG, Belgristr. 102, 8008, Zurich, Switzerland

IEE Proc-Optoelectron., Vol. 147, No. 2, April 2000

Numerical simulations can be of great help in finding the optimum design for a given application, as they eliminate the need to actually manufacture all the designs in ques- tion, which is both time-consuming and expensive. Hence, a huge number of computational models have been presented in the recent literature that apply state-of-the- art techniques to the electric aspects of MSM behaviour. In many cases, however, the aspects of light propagation are treated in a strongly simplified manner by assuming a ‘shadowed exponential decay’, i.e. a laterally uniform intensity in the substrate region underneath the grooves that decays exponentially with depth, and zero intensity underneath the metal electrodes.

While such approximate models may give sufficiently accurate results for structures that are large compared to the wavelength, the MSM electrode spacing is normally of the order of magnitude of the illumination wavelength, in many cases smaller. As is well known from microscopic and interferometric metrology, rigorous electromagnetic theory must be applied to accurately model the light propagation in this so-called resonance regime 114-71, and thus to find the optimum material and geometric para- meters of the light-guiding structures to be used.

Recently, a number of approaches have been published that use more sophisticated theories to model the light propagation within the detector. In [3], a rigorous theory of grating diffraction [8] has been applied to show that dielectric gratings placed on top of an MSM can increase the detector efficiency. Green function formalisms for surface optics were applied to model the experimentally observed wavelength and polarisation dependence [9]. A model using a combination of a TEM waveguide mode in the electrode region and the surface equivalence theorem to express the intensity distribution in the substrate was used in [lo] to account for the effects of interference and diffraction in the simulation.

In this paper, the generalised multipole technique (GMT) will be used to rigorously model the light propaga- tion. This method is capable of handling complex geome- tries, e.g. the MSM design with the incoupling grating, with relative ease for arbitrary angles of incidence of the

89

incoming radiation. The GMT will be discussed in detail in the next section.

2 Generalised multipole technique (GMT)

As was already pointed out in the introduction, for typical MSM electrode spacings which are in the order of magni- tude of the illumination wavelength, the Maxwell equa- tions, which in the frequency domain can be reduced to the Helmholtz equation, need to be solved rigorously in order to obtain accurate results.

2.1 Field expansion in homogeneous domains These partial differential equations can be solved by numerical methods. In this article, the so-called general- ised multipole technique [I 1, 121 will be used. In this frequency-domain method, the computational domain D is divided into a number of homogeneous subdomains Vi, i = 1,. . . , n with constant material properties (i.e. permit- tivity E ; , permeability ,U; and conductivity oi). Assuming that the sources are outside the subdomain Vi, the Helm- holtz equations in Vi are obtained

where kj and Hi are the electric and magnetic field phasors in that domain and ki = 06 is the wave number in the corresponding medium.

Withjn each subdomain, the electromagnetic vector fields Vi are expressed by a, linear combination ofAexpa?- sion functions Vi,/ (where V may represent either E or H)

ea+ of which fulfills the Helmholtz equation, (eqn. l), i.e. V2Vj:.,/ + k;? Vi,/ = 0. Note that the summation in eqn. 2 will in general extend to infinity and will therefore need to be truncated at finite Ni for the purpose of computational implementation. However, in some situations it may be finite. For perfect electric conductors, for example, no electromagnetic field $an penetrate so that in the expan- sions for the E- and H- field one has N=O.

For the case of a two-dimensional structure with peri- odicity in one direction, as discussed here, the field inside the structure is modelled with cylindrical Bessel functions, and the fields above and in the substrate are expressed in terms of Rayleigh expansions [ 131.

2.2 Boundary conditions To determine the expansion coefficients c;,), the Maxwell equations have to be evaluated on the boundaries aV, = aVi n aDj of all the neighbouring domains (i, j = 1, . . . , n). The boundary conditions

90

are obtained, where it is assumed that no surface charges or currents are present. Inserting eqn. 2 into eqn. 3 yields a system of equations

r 1 r 1

cl, / ';,/I ,,= b ~ , v ' j ,k ' j , k J (4) k I'

containing expansion functions and coefficients. The index v = 1,2,3 refers to the spatial components x, y and z. The constant b,,, is derived from eqn. 3 and depends on the material parameters in the adjacent dom$ns i and j and on the component v of the field quantity V under considera- tion. Note that a homogeneous material region may be divided into more than one subdomain in an actual simula- tion, resulting in at least two adjacent subdomains with equal material properties. If this is the case for two subdomains io and j,, this results in brO,JO,v = 1 for all the coefficients in eqn. 4.

The system of equations in eqn. 4 is homogeneous. For a unique, nontrivial solution to exist, an inhomogeneous system is required. Eqn. 4 can be turned into an inhoqo- geneous system by introducing a known incident field VIn. Owing to the superposition principle, the total field Vot is then given by

The incident field may be defined in one or more subdo- main(s) and may itself be ,expressed by a series expansion. In the following, let V? denote the incident field in subdomains Di where such a field is defined. In all other subdomains, Dj, V y = 0 is assumed.

2.3 Generalised point matching To actually compute the c j , / , a system of linear equations must be determined. This can be achieved in different ways. GMT uses the generalised point matching method. In this method, a number of matching points rm (m = 1, . . . ,Mi,.) are selected on the domain boundaries. Evaluating eqn. 4 for all rm, a system of M, linear equations

(6) with Nu = Ni + Nj unknowns c is obtained on each bound- ary between adjacent subdomains i and j . In the point matching or collocation method, M y = N , is chosen. In generalised point matching, MiJ > N,, corresponding to an over-determined system of equations in eqn. 6. Allowing a residual error e(rm) and introducing a positive real weight function w(r) on the matching points gives

N,

tY(rm> =z ~ ( r m ) PP + x c i , / k i , / ( r m ) i /=I

\-I

This system can be solved in a least squares sense, i.e. by minimising the sum of the squares of all the residuals.

2.4 Expansion functions As was pointed out, the expansion functions Vi,! used in eqn. 2 need to be solutions to the Helmholtz equation. One possible approach to finding such solutions is by separa- tion, which is possible in several co-ordinate systems. A

IEE Proc.-Optoelectron., Vol. 147, No. 2, April 2000

very simple example is the plane wave, which is obtained by separation in Cartesian co-ordinates. Important exam- ples are the cylindrical and spherical expansions, which are found by separating the Helmholtz equation in the respec- tive co-ordinate systems. The radial dependence of these expansions is given by cylindrical/spherical Bessel func- tions. For cylindrical co-ordinates, there is a harmonic dependence on the polar angle 4 and the z-dependence is governed by a complex exponential fimction of yz, with y the propagation constant. For the spherical case, the exponential z-dependence has to be replaced by the asso- ciated Legendre functions of the cosine of the azimuthal angle, cos 8. The cylindrical and spherical expansions are frequently used for modelling in 2D and 3D, respectively. A practical advantage of these expansions is that both the Bessel and Legendre functions can be evaluated recur- sively in an efficient way. For a more comprehensive overview and discussion of the various expansions and functions admittable, see [ 14-16].

One key issue for analytical computation is complete- ness, ensuring that the solution can be approximated to any desired accuracy as N + 00 . For numerical modelling, it is more important that, with as few terms in the expansion as possible, a solution is found with an error below a given tolerable, finite limit. Practical experience with GMT simulations has shown that in many cases the use of multiple finite multipoles with low maximum order at different origins can more efficiently approximate the electromagnetic fields than a single expansion with high maximum order, especially for domains of very complex shape. However, when using this approach, care has to be taken that the overlap between each of the two expansions is sufficiently small, i.e. appropriate expansion functions and maximum orders have to be chosen [17].

3 Optical simulation results

In this section, simulation results for the light propagation within three different types of MSM Schottky contact diodes based on silicon will be presented. As was pointed out in the introduction, Si has recently received enhanced interest as a material for ultrafast photodetection. Fig. 1 shows a schematic diagram of the different designs: a basic MSM detector structure consisting simply of an Si substrate with the electrodes on top, a detector covered with an antireflection layer, and an MSM with an incou- pling grating [3]. The structures are approximated as infinitely periodic in 2D. The metal fingers are assumed to be perfectly conducting. For the GMT simulation of the first two detector structures, four subdomains are used. The first subdomain represents the substrate region. The fields

air, n=l 0 SiO,, , SiNor SiO, ; air n=1.8 n=1.0

&A=l.64pm- , I I i t A=1.64pm+ &A=1.64pn-[

a ‘ b ’ C

Fig. 1 Three dzfferent MSM detector designs. The metal electrodes are assumed to be perfectly conducting and have a width and spacing of 0.41pm and a thickness of 0.126pm a Basic MSM detector structure b MSM detector with antireflective layer c MSM detector with incoupling grating

IEE Proc.-Optoelectron., Vol. 147, No. 2, April 2000

X X

X X X ......................................................................................................................................................................................

i x ................. ................. x i i X x : !

x X : + . . . . . . . . . . . . . . . . . . . . x . . . . . . . . . . . >;( . . . . . . . . . . . . . . . 1 . . . . . 3 : X

X

Fig. 2 Example of the GMT model of the basic MSMstructure and the MSM with antirejective layer shown in Fig. 1. The crosses (x) show the locations of the expansions used to model thejield in the region close to the electrodes. The small black dots represent the matching points on which the boundary conditions between the second subdomain and the metal electro- des, second subdomain and the substrate and vacuum regions above and below, respectiveb, and the periodicity conditions (left and right sides} are enforced

in this region are expressed by means of Rayleigh expan- sions. The second subdomain is defined by the antireflec- tive layer region of the respective detector structure. For the basic detector structure, vacuum material properties are used for this subdomain. The fields in this subdomain are modelled using cylindrical multipole expansions. Their locations are shown in Fig. 2, together with the matching points on which the continuity equations are enforced. Note that the multipoles are singular at their origins, which therefore have to be located outside the subdomain in which they are to describe the fields. The region above the second subdomain forms the third subdomain. Here, the fields are again expressed by a Rayleigh series. The plane wave excitation is also defined in this subdomain. The metal electrodes are assumed to be perfectly conduct- ing, so no radiation can penetrate. Therefore, no field expansions are required for this fourth subdomain.

For GMT modelling of the MSM detector with the incoupling grating, an additional subdomain has to be introduced so that Rayleigh expansions can be used above and below the detector structure. Fig. 3 shows the two subdomains used to model the fields close to the electrodes and the incoupling grating. One subdomain corresponds to the material region formed by the incou- pling grating. The locations of the cylindrical multiple expansions used to express the electromagnetic field in this subdomain are depicted by crosses. Adjacent on top is a vacuum subdomain in which the field is modelled by cylindrical multipoles located at the positions of the black circles.

It must be pointed out that a formula for finding the unique locations of the expansion functions does not exist. Algorithms for automatically choosing these locations and the number of orders to be taken into account have been presented [17, 181. For more complex structures, however,

0

X

0 0 0 0 0 0 0 0 .................................................................................................................................................................

X ................................................... X I 0 ’f- 0 0 0 0 %x

i x / i o o i x ! I .................................. 3 ....................................

X i 0 0 I X

0

i x x- j

! x i ................................ : ....................................

$ + ............ x’ ....................................... ................................. ......................................................... i X x ( I X

Fig. 3 As in Fig. 2, butfor an MSM detector with incoupling grating. The circles indicate the positions ofthe expansions that express thejields in the additional intermediate subdomain above the grating that has been introducedfor this case

91

an iterative process involving user interaction is often necessary to obtain high accuracy results. This can be done by starting with an initial guess according to the strategies described in [ 17, 181. The matching errors &(U,) can then be computed and multipoles added or their maximum order increased in regions where unacceptably large errors occur. This will finally result in a configuration that gives sufficiently exact results [19]. Therefore, the locations shown in Figs. 2 and 3 are to be understood as one example of a number of potential solutions only.

For the basic structure, only a small fraction of the incident radiation reaches the substrate due to shading by the electrodes and the high reflectivity of Si at optical wavelengths. A practical measure for this fraction is the sum of the diffraction efficiencies [20] of all the trans- mitted orders qT. For an s-polarised, perpendicularly incident, monochromatic plane wave with wavelength Lo = 620 nm,

This problem can be partially overcome by covering the detector with an antireflection la er Ideally, such a layer has a refractive index of nar = J- 9?(ns,) (for a vacuum as the incident medium and %(ns,) >> 3 (nsl), here as, = 3.906 + 0.0221 [21]) and a thickness d = Lo (2k-t- 1)/(4nar) ( k c N), i.e. in our case, nar = 1.97 (e.g. S i 0 or SIN), and d=0.236pm is chosen [22]. In this way, the relative intensity in the substrate can be increased to

Another approach is to place an incoupling grating on top of the MSM. The optimum parameters for such a design can be estimated by means of the diffraction theory of gratings [3]; for rigorous optimisation, however, exact modelling of the complete structure is necessary. A grating consisting of 0.82 pm widelspaced and 0.391 pm thick material slabs with = 1.8 (SiO, with x 2 1) is assumed to be placed on top of a layer with the same index and a thickness of 0.442 pm. This way, qT = 69% can be achieved.

Fig. 4 shows the intensity distributions (time averaged Poynting vector) for the three designs. As expected, Fig. 4 reveals that not only is the fraction of light reaching the substrate influenced by the application of antireflective layers or incoupling gratings, but also the spatial distribu- tion of the intensity is altered. For example, the region of maximum intensity within the substrate moves from within the substrate (for the basic MSM) to the top of the substrate (for the two other structures). To determine the degree to which this influences the electric behaviour of the detector,

qT = 25% is obtained.

U], = 38% for the same parameters. C

2 . I_.. j -125 pW/pm

2 100 pW/pm

75 pWlpm

50 pWlpm

25 pWlpm

2

2

2

0

Fig. 4 GMT computation of intensity distributions for different MSM detector designs: basic structure (a), with antirejection layer (b) and with incoupling grating. (c) A n S-polarised. monochromatic plane wave (1, = 620 nm) perpendicularly incident.from the lefr

an additional electric simulation of the charge carrier transport is required. This aspect will be covered below, along with simulations for several additional illumination parameters.

The dependence of the fraction of intensity in the substrate on the angle of incidence 8 and vacuum wave- length I,, of the incoming radiation is shown in Fig. 5. For the structure with the incoupling grating, even higher efficiencies (1 80%) could be achieved at slightly different wavelengths/angles. This reflects the fact that the design parameters were derived by means of approximate theories only. This also holds for the antireflection layer, the parameters of which were computed assuming infinite homogeneous layers of material with plane boundaries.

. . . . c

*.: 0 I I I

0 10 20 30 40

angle of incidence, deg

600 650 700 750

wavelength in nm Fig. 5 0 = 0" (right) for the three different structures _ _ _ basic MSM structure . . . . . with antireflective layer

~ with incoupling grating

Dependence of the intensity reaching the substrate on the angle of incidence % at an illumination wavelength of ,l.o = 620nm (lefr) and on R, for

92 IEE Proc.-Optoelectron., Vol. 147, No. 2, April 2000

4 Electric device simulation

This section investigates the electric behaviour under illumination of the MSM detector types that were exam- ined above. To simulate the electric behaviour of a light- sensitive semiconductor element, the amount of charge carriers generated under illumination needs to be known. The optical generation rate Gopt is proportional to the intensity and the absorption coefficient. For a material with complex refractive index n’ + in’’

(8) 47tn” I

GOP‘ = __ y - ?LO iiw

where ;io is the vacuum wavelength of the radiation generating the carriers and ti the quantum yield, which gives the number of carrier pairs generated by a single photon. For generation in silicon at visible wavelengths, y = 1 can be assumed [23].

In addition to the GMT results presented in Section 3, optical simulations were also carried out for a wavelength of A. = 690 nm and for an angle of incidence of e = 30” at /lo = 620 nm. For the device simulation, it was supposed that each detector consists of five periods of the respective structures shown in Fig. 1 . Within each period, the inten- sity distribution obtained from the optical simulation of the infinitely periodic structure was assumed for simplicity. For the charge carrier generation, illumination was switched on for 1 ps at t = 0. Anode/cathode voltages are set to f: 1 V. For the intensity of the incident plane wave, 30pW/pm2 was assumed,

4.1 Electric device model The electric simulation was then carried out using DESSIS!,,, a multidimensional, mixed-mode device and circuit simulator available from ISE Integrated Systems Engineering AG, Zurich [24].

4.1. I Transport Model: For the investigations, we used DESSIS,,, to solve the semiconductor device equa- tions in 2D for the drift-diffusion charge carrier transport model, together with the external circuit equations required to properly model the connections of the positivehegative electrodes. The three governing equations of the drift- diffusion model are the Poisson equation and the two electron and hole continuity equations. The Poisson equa- tion can be stated as

V . cV$ = -q(p - n + N g - N i ) (9) where t is the electric permittivity, q is the elementary electronic charge, n and p are the electron and hole densities, and N& and NL are the number of ionised donors and acceptors, respectively. The electron and hole continuity equations are given by

v . ~ , , = q R + q - an and - V . J p = q R + q - aP (10) at at

where R is the net electron-hole recombination rate and Jn and Jp are the electron and hole current densities given as

Jn = -nqPu,Vqn and Jp = p q ~ p V ~ p (11)

Here, pn and pp are the electron and hole mobilities, and pn and qP the electron and hole quasi-Fermi potentials.

4.1.2 Mobility models: The model used for the charge carrier mobility takes the effects of lattice tempera- ture, doping concentration, carrier-carrier scattering and

IEE Proc-Optoelectron., Vol. 147, No. 2, April 2000

the saturation due to high electric fields into account. The constant mobility model according to Lombardi et al. [25] is used for the temperature dependence and gives the low- doping reference for the doping-dependent mobility, which in turn is modelled according to Masetti et al. [26]. Since a uniformly doped substrate with an arsenic concentration of 1 .O x I O l 5 cmP3 at a constant temperature of T= 300 K is assumed, this results in constant background mobilities ,U$’) = 1359 cm2/(Vs) for the electrons and ,U:) =461 cm2/ (Vs) for the holes. For the mobility contribution peh due to carrier-carrier scattering, we used a model based on Choo [27] and Fletcher [28], which uses the Conwell-Weisskopf screening theory, resulting in

where D = 1.04 x 102‘/(cmVs) and P=7.452 x 1013 cm-‘. The resulting low-field mobility plow is then given by the Matthiessen rule as

For the high-field saturation of the mobility, a model according to Canali et al. [29] was used. The overall mobility can thus be given by

with the gradient of the quasi-Fermi level V ~ I , , ~ taken for the driving force, i.e. F= lVqn,pl. In the Canali model, both p and the saturation velocity vSat in eqn. 14 are temperature dependent. At T= 300 K, one has Fe = 1.109, v , ,~ ,~ = 1.07 x lo7 cm/s for the electrons and Ph = 1.213, v , ,~ ,~ = 8.37 x lo6 cmis for the holes [29].

4.1.3 Recombination/generation models: For the net recombination rate R required for the drift-diffusion model, only the contributions due to Shockley-Read-Hall (SRH) mechanisms were taken into account. For recombi- nation via deep levels, the well-known Shockley-Read-Hall form

with

(16)

was used, where Etrap = 0 was assumed. Here, qeff is the effective intrinsic electron density. For the doping concen- tration and temperature used in our simulations, these equalled the intrinsic densities in Si at T=300K. The minority SRH lifetimes were determined from the Scharf- stetter relation

with z,,,,, = 1.0 x lop5 s for the electrons, z,,,,~ = 3.0 x s for the holes, zmin =0, Nref= 1 x 10’6cm-3 and

y = 1 [30].

93

Table 1 : Sum of diffraction efficiencies in transmitted orders CqT, maximum current Imax, time required for current to decrease to Imaxle for the three different designs and various parameters of the incoming radia- tion

I nm - 8 c VT

without anti- reflection with anti- reflection with incoupling grating

620 620 690

620 620 690

620 620 620 690

0" 30"

0"

0" 30" 0" 0"

+30" - 30"

0"

25% 18% 25%

38% 36% 23%

69% 23% 23% 58%

I,,, PA

0.640 0.407 0.393

0.958 0.820 0.365

1.777 0.521 0.576 0.937

8.77 8.96 9.72

8.56 8.45 9.69

8.41 8.75 8.04 8.94

2.0 1

--------____ ......................... ------___ ....................... 0.0 I I I I I

1 6 11 16 21

time, ps

Fig. 6 and angle of incidence 0 = 0" . . . . . without antireflection _ _ ~ with antireflection - with incoupling grating

MSM response as a function of time for wavelength I = 620nm

1 .o

0.8

- al

0 U

5 . 0.6

._ I - 0.4

c

0.2

n 1 .o 4.0 7.0 10.0 13.0

time, ps

a

Fig. 8 U = +30"

_.__ U = -30" - a=O"

94

Time dependence of electron/hole currents (a) and total current (b) _-_

4.2 Simulation results Table 1 gives an overview of the electric and optical simulation results for the various geometrics and illumina- tion parameters.

For the A = 620 nm perpendicular incidence case, the fractions of incident intensity reaching the substrate have already been given in Section 3. For this case, the incou- pling efficiency can be increased to a factor of 2.7 by means of the grating. Also shown are the electric pulse amplitudes, for which a similar increase results. The l le decay time was not influenced by the design modifications and therefore no decrease in bandwidth were to be expected. The currents as a function of time for the three designs are shown in Fig. 6.

For the A= 690 nm examples, weaker current pulses per incoupled energy and longer decay times can be observed. This was caused by the weaker absorption at this wave- length (nsi = 3.796 + O.O13i), which results in the carrier generation being more stretched out in depth.

As was already pointed out in the introduction, the electrical behaviour can be understood not only by the amount of light in the substrate but also by the exact spatial distribution of the generated charge carriers. To illustrate this, additional simulations more carried out for the detec- tor with the incoupling grating for illumination at two opposite angles of incidence, 8 = f 30"). Fig. 7 shows the

a b

Fig. 7 GMT computation of intensity distributions for opposite angles of incidence 8: +30" (a) and -30" (b). A n S-polarised, monochromatic plane wave (2 = 620nm) is incident from above from upper right for 0 = + 30')

1.0 4.0 7.0 10.0 13.0 16.0 19.0

time, ps

b

IEE Proc.-Optoelectron., Vol. 147, No. 2, April 2000

resulting intensity distributions. The electron and hole currents (normalised to their maxima) as functions of time that were obtained in the DESSISISE simulations are shown in Fig. 8. The different behaviour of the electrons and holes is caused by their different mobilities. Also shown is the total current, which consists of the electron/ hole current and an additional displacement current due to the change of the potentials in time. The absolute values for the maximum current I,,, and lie decay times are given in Table 1. Note that the sign of the incident angle significantly influences the decay time. This can be under- stood as follows: owing to the significantly lower mobility of the holes, the decay time will be dominated by the time it takes for the holes to travel to the negative electrode from where they have been generated. The response time will therefore be shorter the closer the carrier pairs are gener- ated to the cathode (within certain limits). For the 8 = - 30” case, the majority of the electron/hole pairs are generated closer to the cathode than for d = +30”. According to the above considerations, this configuration therefore has the shorter l/e decay time.

4

5 Conclusions

A combined optical and electric simulation of various MSM detector designs by means of rigorous electromag- netic theory and drift-diffusion transport models has been presented. The generalised multipole technique, which was used for modelling the light propagation in the device, has been discussed in some detail. The computations carried out have shown that the use of incoupling gratings can significantly increase the fraction of the incident radiation that reaches the device substrate. Electrical simulations revealed that this results in a similar increase in efficiency without any negative effect on the response times.

The effects of varying the illumination wavelength and angle of incidence have also been examined. Owing to the different mobilities of the electrons and holes, different response times result for o p p o s i t e angles of incidence. On the one hand, this emphasises the need to use rigorous optical simulations that allow exact computation of the spatial intensity distribution. On the other hand, it may give useful hints on how to shorten the detector response times by using designs that lead to non-symmetrical intensity distributions. This may be of special interest for III-IV- semiconductor based MSMs, where the difference in carrier mobilities is much more pronounced than in silicon. This will be the topic of future research.

6 Acknowledgments

A major part of this work was done when the author was with the Integrated Systems Laboratory at the Swiss Federal Institute of Technology (ETH) Zurich, Switzerland and funded by the Swiss Commission for Technology and Innovation (KTI), project 3193.1. The author wishes to thank W. Fichtner For allowing him to carry out this project at his institute and also U. Krumbein, who is now with

~ Infineon Technologies AG, for assisting him with the electric device simulations.

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