An Unique SPICE Model of Photodiode with Slowly Changeable Carriers Velocities

Transcription

1 J Infrare Milli Terahz Waves DOI.7/s An Unique SPICE Moel of Photoioe with Slowly Changeable Carriers Velocities Petar S. Matavulj & Miomira V. Lazović & Jovan B. Raunović Receive: 6 February 27 / Accepte: 2 November 2 # Springer Science+Business Meia, LLC 2 Abstract This paper eals with a relatively new SPICE moel of a P-i-N photoioe. The moel inclues a change of velocities of electrons an holes, ue to the voltage rop on the eges of the photoioe, which epens on the time form of input excitation. We have erive the moel of the P-i-N photoioe for igital input excitation i.e. for Heavisie s square wave time excitation. The moel is incorporate in SPICE program an simulate with it. The moel an the limitations of the moel itself are observe. The output results are compare with the similar ones. It is suggeste when it is practical to use the moel, an etermine the photoioe working omain regime when the moel gives accurate results. Keywors Equivalent circuits. P-i-N photoioe. Program SPICE. Quasi-linear working regime. Transient response Introuction In the last ecae there were a few significant trials to create the equivalent electrical moel of photoioe [ 5], as well as it is alreay one for a number of lasers [6 8]. The purpose of these observations is to establish universal electrical moel for the escription of the whole optical integrate circuit, an to introuce one program for solving the input-output epenences of all parts of the circuit. The aim is to give the moel with, as much as possible, relevant physical processes inclue in it, an to solve it using a simple an cheap program. All mentione resulte with the iea that it woul be very convenient to simulate a photoioe using an electrical moel, an solving it response with SPICE program. The previous works on this subject [ 5], eal with photoioes in linear working regime, where not a single effect of nonlinearity was observe, because of ifficulties in P. S. Matavulj (*) : J. B. Raunović Faculty of Electrical Engineering, University of Belgrae, Bulevar kralja Aleksanra 73, 2 Belgrae, Serbia matavulj@etf.rs M. V. Lazović Serbian Energy Efficiency Agency, Omlainskih brigaa, 7 Belgrae, Serbia

2 J Infrare Milli Terahz Waves escription of multiimensional transport equations with just one time imension use in electric circuits. But, nonlinearity in response becomes important fact when we eal with relatively high igital pulses in high-spee optical communications. In the paper we consiere an simulate a bounary situation, when the voltage rop on photoioe is enough high to generate nonlinearity in response, but is still not enough to generate space charge effect. The paper suggests possibly one new approach, in calculating photoioe response, which takes into account the effect of the change of carriers velocities. Velocities are changeable in the moel an that is the main ifference comparing to papers [ 5] which use constant saturation velocities. The moel gives more precise results in the working regimes wherein the carriers velocities epen a lot on the photoioe voltage i.e. wherein the electric fiel ecreases below kv/cm ue to the high input photon flux. Also, the moel is completely applicable for the linear regimes with the constant carriers velocities. The nonlinearity effect observe here happens only ue to the influence of photoioe voltage on carriers velocities [9 2]. The effect becomes relevant when the photoioe is not highly saturate, so the carriers velocities are not simple saturation velocities, an when the energy of light pulses varies (so the average carriers velocities alter from pulse to pulse). From one sie we propose physical moel how to calculate mean values for carriers concentrations an currents in the case of weak nonlinearity, an from other sie we propose how to implement it in SPICE. We think that it is almost impossible to solve the continuity equation irectly using SPICE because the program works with one inepenent variable time. So we partially solve continuity equation, an in an equivalent electrical circuit of a P-i-N photoioe we calculate the necessary concentrations of electrons an holes in the subcircuits. The moel is mainly evelope for Heavisie s square wave time input excitation, where first we solve analytically concentrations epenences on the applie voltage time form. We solve the continuity equations treating the velocities as almost as constant, but in calculating the currents we treat them as changeable. This is the correct step as long as the nonlinearity of the regime is not too high i.e. the velocities are almost constant comparing to the change of concentrations in continuity equations. The equivalent electrical moel, of the P-i-N photoioe, is incorporate into SPICE an results are compare with the similar results which o not take into account the changes of carriers velocities. Also, in FORTRAN is evelope a moel wherein the overpasse path is calculate by integral of voltage, with the aim of comparing the results with less accurate SPICE results. The last proceure enable us to efine the range of relevant working regime of the P-i-N photoioe, where our SPICE moel is fully applicable. 2 Equivalent moel We consiere P-i-N photoioe stanar electric circuit presente in Fig.. Several assumptions are mae: ) the with of the P region is much smaller than the with of the i region so there is no relevant generation of carriers in P region an the main current is rift current in i region; 2) the iffusion current in the N region is taken into account an in P is neglecte accoring to assumption ; 3) there is no space charge effect we consier the situation when the input signal is high but not that orer of magnitue to create space charge effect; 4) the ark current is neglecte because it is a few egrees of magnitue lower than the photocurrent; an 5) nonlinearity exists only as the consequence of the voltage rop on the loa resistance, []. These assumptions are use to simplify moel an

3 J Infrare Milli Terahz Waves -Vcc U(t) I(t) P I N light R I(t) Fig. The electric circuit with the P-i-N photoioe. they are vali if we eal with low, meium an high, but not overmuch high, incient light signal as in the case analyze in [, 2]. In establishing of the photoioe s equivalent electric circuit, which is going to be inserte into program SPICE, we start from the equation of the main electric circuit from Fig. : UðtÞ ¼V CC RIðtÞ: The above expression escribes the change of the photoioe voltage ue to the flow of the photocurrent. In the above equation V CC is bias voltage, R is loa resistance an I(t) is the total current flow. In orer to obtain the total current flow I(t) we shoul fin the carriers concentrations an carriers velocities, which play the role in rift currents, an iffusion current. The require concentrations for the case of Heavisie s time input excitation can be obtaine from the results for Dirac s time input excitation. For the case of Dirac s input time excitation mean values, in space, for electrons an holes concentrations are alreay obtaine in [2], given by: hni ¼ expð aþexp@ a Z t t m n ð UðtÞ V ðþ ÞtAA; ð2aþ hpi ¼ exp@ a Z t t m p ð UðtÞ V ÞtA expð aþa: ð2bþ I represents, here, the intensity of incient Dirac s excitation, μ n an μ p stan for electrons an holes mobilities respectively, α is the absorption coefficient, is the with of i region, t represents the time moment when Dirac s impulse appears, U(t) is the photoioe

4 J Infrare Milli Terahz Waves voltage, V is punch-through voltage, an integral of U(t)-V in the exponent stans for the total path which electrons (or holes) pass over the photoioe. In the above expressions miss the Heavesie s members for borer conitions, wherein the total path of the carriers must be less than with of the photoioe. In the case of constant velocities, or slowly changeable velocities, we can use (2a) an (2b) in a ifferent manner as: hi¼ n I ð exp a ð Þexp ð au nðtþðt t ÞÞÞ; ð3aþ Utilizing the property of Dirac s function: hpi ¼ I exp au pðtþðt t Þ expð aþ : ð3bþ ΦðtÞ ¼ Z Φðt Þðt t Þt ; ð4þ where Φ(t) is an arbitrary function in time, an uner the assumption that the velocities are slowly changeable, we can obtain the mean values of carriers concentrations as: hn Φ i ¼ I Z Φðt t Þhn D ðþ t it; ð5aþ hp Φ i ¼ I Z Φðt t Þhp D ðþ t it: ð5bþ where variable t stans for expression (t-t ). In (5a) an(5b) inexes n Φ an p Φ stan for the mean values of concentrations of electrons an holes, for the case of arbitrary function of time input excitation, an <n D > an <p D > are alreay obtaine the mean values of carriers concentrations for Dirac s input excitation, given in (3a) an(3b). The Eqs. 5a an 5b coul be solve analytically in many cases, i.e. for a large number of ifferent incient excitations functions Φ(t). If we replace the arbitrary function with Heavisie s function we can obtain the mean values of concentrations of electrons an holes in i region for Heavisie s input excitation as: hn h i ¼ Φ Zun ðht ð t t ÞÞð expð aþexpðau n tþþt; ð6aþ hp h i ¼ Φ Zup ðht ð t t ÞÞ exp au p t expð aþ t: ð6bþ In the above equations t is the time moments when the Heavisie s starts, an Φ stans for an intensity of Heavesie s function (in physical sense it is the input flux). Here the carriers

5 J Infrare Milli Terahz Waves velocities are marke with υ n an υ p, regarless of whether we work with constant or slowly changeable velocities. Solutions of the Eq. 6a are: For t >t hn h ðtþi ¼ : ð7aþ For t-t </(v n (t)) int hn h ðtþi ¼ t t au n ðtþ ð exp ð a þ au nðtþðt t ÞÞ expð aþþ : ð7bþ An for t-t >/(v n (t)) hn h ðtþi ¼ int u n ðtþ au n ðtþ ð exp a : ð7cþ Here n h marks the mean value (in space) for electrons concentration for the case of Heavisie s input function in time. Also we obtain the mean value for concentration of holes as: For t >t For t-t </(v p (t)) hp h ðtþi ¼ For t-t >/(v p (t)) int hp h ðtþi ¼ hp h ðtþi ¼ : ð8aþ au p ðtþ exp au pðtþðt t Þ expð aþ ð t t Þ : ð8bþ int exp a au p ðtþ ð ð Þ Þ exp ð a Þ : ð8cþ u p ðtþ Int stans for the intensity of Heavisie s function. All above equations are vali in the case of constant or slowly changeable velocities. For the case of Heavisie s square wave input excitation, with intensity int, given with the formula: ht; ð t ; t 2 Þ ¼ int ðht ð t Þ ht ð t 2 ÞÞ; ð9þ it is easy to obtain the carriers concentrations as: hn h ðtþi ¼ hn h ðt; t Þi hn h ðt; t 2 Þi; ðaþ hp h ðtþi ¼ hp h ðt; t Þi hp h ðt; t 2 Þi: ðbþ In this case we first obtain the carriers concentrations for the first Heavisie s signal an then for the secon signal. The rule of superposition can be use as long as the working regime is quasi-linear.

6 J Infrare Milli Terahz Waves In orer to obtain total rift current in i region, besies alreay given expressions for the carriers concentrations, we nee the equations for the carriers velocities. The electron s velocity is given in [9] Eq. 4, here presente with (a), an for the holes velocity we use (b): UðtÞ V u n ¼ m n r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðaþ ð þ UðtÞ V Þ E 2u ðð UðtÞ V Þ E Þ E c UðtÞ V u p ¼ m p : ðbþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E an E c are constants (E ¼ 2 E M þ EM 2 4E2 C, E C ¼ v sat m, E M is constant, v sat is n electron saturation velocity); u represents function which obtains value (for U(t)-V>E ) or (otherwise) epening on the value of expression in brackets; U(t) is the instantaneously photoioe voltage; V is the punchtrough voltage an is the with of i region. The total current flow, as it is alreay mentione, is the sum of total rift current in i region an iffusion current in N region. Using the continuity equation for N region, with borer conitions, we obtain the expression for iffusion current in the case of Heavisie s input excitation as: I pif ¼ int qsd p a 2 exp a g ð Þ ð exp ð g ð t t ÞÞÞ; ð2þ (see in Appenix) where: g ¼ L 2 p a 2 =tp. Here t p is the hole s life time in N region, L p is the hole s iffusion length, S is the photoioe cross section area, D p is hole s iffusion constant, an q is the electron s charge. In the total current flow there is one more term that represents isplacement current the last member in (3): IðtÞ ¼I n þ I p þ I pif þ "S UðtÞ : ð3þ t In the equation I n an I p are rift electrons an holes currents, an ε is ielectric constant. For the series of Heavisie s square waves given by expression: h ¼ int ðht ð t Þ ht ð t 2 ÞÞþint 2ðht ð t 3 Þ ht ð t 4 ÞÞþ...; ð4þ total electrons an hole s concentrations are: hi¼ n hnt; ð t Þi hnt; ð t 2 Þiþ hnt; ð t 3 Þi hnt; ð t 4 Þiþ... ð5aþ hpi ¼ hpt; ð t Þi hpt; ð t 2 Þiþhpt; ð t 3 Þi hpt; ð t 4 Þiþ... ð5bþ The total rift currents, for the series of Heavisie s square waves are: I n ¼ Squ n hni; ð6aþ I p ¼ Squ p hpi; ð6bþ

7 J Infrare Milli Terahz Waves wherein the mean values for concentrations are obtaine from (5a) an (5b), an iffusion current is: I if ¼ I if ðt; t Þ I if ðt; t 2 ÞþI if ðt; t 3 Þ I if ðt; t 4 Þþ... ð6cþ The last equation shows that the iffusion current, like rift currents, is obtaine by summing iffusion currents for each Heavisie s function. Now we have all the require epenencies to create the equivalent electric circuit of the P-i-N photoioe for the case of series of Heavisie s square waves (Fig. 2). The main circuit consists of the inepenent voltage generator V CC, resistance R, capacitance C t =εs/, parasitic capacitance C s, an three epenent currents sources (two for rift currents an one for iffusion current). The main circuit is obtaine from () an (3). One subcircuit solves epenence of intensity of electric fiel on the photoioe voltage. Two subcircuits solve the epenence of velocities on electric fiel, wherein functions f an f 2 are given with expressions (a) an (b). In the case of constant carriers velocities there is no nee for these subcircuits an velocities can be taken as the input parameters. Since we use them as slowly changeable there is a nee to recalculate them in the subcircuits. In accorance with (5a), (5b) an (6c) one by one subcircuit is if cc p n U(t)-V E(t) / f (E(t)) n(t) f 2 (E(t)) p(t) f 3 ( n (t),t ) n(t,t ) n(t,t 2 ) f 3 ( n (t),t 2 ) p(t,t ) f 4 ( p (t),t ) f 4 ( p (t),t 2 ) p(t,t 2 ) t s I if (t,t ) I if (t,t 2 ) f 5 (t,t ) f 5 (t,t 2 ) Fig. 2 The equivalent electric circuit of the P-i-N photoioe.

8 J Infrare Milli Terahz Waves inserte to solve carriers concentrations an iffusion current for each Heavisie s pulse (in the series of Heavisie s square waves). So we involve 2 subcircuits to resolve the electrons concentration for the case of Heavisie s square waves, together with 2 subcircuits for holes concentration, an 2 subcircuits for iffusion current. All these subcircuits use resistance of Ohm. These subcircuits solve the require carriers concentrations given within the formulas (7a) (7c) an (8a) (8c), an iffusion current given by (2). The circuit presente in Fig. 2 was ensue from the previous consierations. Completely the same circuit can be obtaine for ifferent time forms of input excitation, only the functions use in Eqs. 7a 7c an 8a 8c will iffer. The circuit is incorporate into SPICE program an the simulate results are obtaine. 3 Results an iscussion In the calculations we use next parameters of the GaAs photoioe: α= 4 cm, =5 μm, m n ¼ 75 cm2 Vs, m p ¼ 42 cm2 Vs, E M ¼ 3; 2 kv cm, v sat ¼ 8 6 cm s, R=5Ω, V =.6 V, an V CC = 5 V. Dielectric constant is ε r =2, photoioe area S=7 μm 2, an the wavelength of input excitation is l=.8 μm. It shoul be note that in all simulations holes velocity is calculate using the Eq. b. The first results presente in Fig. 3, obtaine for one Heavisie s square wave input excitation, show the ifferences in photoioe responses when saturation electrons velocity an slowly changeable electrons velocity υ sat an υ n (t) are use. The both results are obtaine using the SPICE program. In the first case, instea of the Eq. a, electrons velocity is given as the constant saturation velocity. In the secon case velocity is calculate in SPICE, in the subcircuit which solves Eq. a (subcircuit solves electron s velocity epenence on the instantaneously photoioe voltage). The ifferences obtaine in results are significant concerning the minimum of the photoioe voltage an time response. The error, occurre as a consequence of using the constant saturation electrons velocity, is about 22% comparing to the secon case. 5, photoioe voltage [V] 4, υ n = υ sat υ n (t) 3, time [ps] Fig. 3 The photoioe responses to the Heavisie s square wave, P=.5 W(h(t)-h(t- ps)), in the cases of: saturation electrons velocity an changeable electrons velocity. Both results are obtaine using SPICE program.

9 J Infrare Milli Terahz Waves (Here the time response of the photoioe is efine as a time in which photoioe voltage reaches the value of.99 V CC ). Generally the error epens on the parameters of input signal (power an time uration of the input signal). On Fig. 4 we compare the SPICE results with more exact FORTRAN results. Here FORTRAN results can be taken as exact because in FORTRAN we use Eqs. 2a an 2b with integral of photoioe voltage for overpasse path of carriers. The maximum obtaine error between SPICE an FORTRAN results is about 6%, an there is no error in the minimum of the photoioe response. So we can say that SPICE moel use here gives more accurate results than the moel with constant velocities. It is obvious that approximation use in this moel, where the overpasse path is calculate like υ(t) t, gives reliable results as long as the change of the photoioe voltage is not too high, an when the velocities can be taken as slowly changeable. It is necessary to efine the omain of input parameters where the approximation is vali. With the aim to establishing the relevant omain we analyze the maximum errors between the SPICE an FORTRAN results. The results are obtaine for one Heavisie s square wave, for ifferent input energies an ifferent time uration of the input signal, an they are presente in Fig. 5. Figure 5 inicates the increase of error (between SPICE an FORTRAN) with the increase of input energy. In the case of the same input energy, error increase for ecrease of time uration of input pulse. High input energy leas to bigger changes of velocities an, as the consequence, the obtaine error in overpasse paths is bigger. It means that higher the nonlinearity, of the working regime, the higher error is obtaine. If we efine quasi-linear working regime as a regime where the error between FORTRAN moel an SPICE moel is less than 5%, we can efine the omain of input parameters. For one pulse (one Heavisie s square wave) for the photoioe, linear working regime is obtaine for input energies less than.75 pj, an for the case of input energies between.75 pj an 2.2 pj linear working regime is obtaine if the time uration of the pulse is higher than 4 ps. The similar results, concerning the behavior of photoioe response, coul be obtaine using only SPICE program. In some segments, minimum of the photoioe response has linear epenence on the input energy as shown in Fig. 6. The photoioe response shows 5, photoioe voltage [V] 4, error=6% SPICE FORTRAN 3, time [ps] Fig. 4 The photoioe response to the Heavisie s square wave, P=.5 W(h(t)-h(t- ps)), when changeable electrons velocity is use (SPICE), an when integral of photoioe voltage is use for calculation of overpasse path of electrons (FORTRAN).

10 J Infrare Milli Terahz Waves max. error [%] E=2.2pJ 2pJ.85pJ.75pJ.5pJ pj.5pj time uration of pulse [ps] Fig. 5 The maximum error obtaine in SPICE compare to FORTRAN results. The error is the consequence of ifferences between the total path calculate in SPICE an FORTRAN. linear behavior up to the values of minimum photoioe voltage higher than 3.25 V. If the photoioe voltage minimum is lower than 3.25 V the borer of the linear epenence is exceee. So we have two ways of investigating the linearity of the case. First way is to check the values of input parameters, an secon is to check minimum of the photoioe response an its linearity. For more than one pulse, the situation is more complicate. The question is when it is possible to obtain response on the two separate light signals as a sum of two inepenent responses, an when the total response iffers from the sum of two inepenent responses. This means that now linear working regime epens on three parameters input energy, time uration of pulse an time perio between two pulses. As long as the photoioe response acts like a sum of two inepenent responses to the excitation of two inepenent light pulses we consier a working regime as the linear one. V CC -U min [V] 3, 2,5 2,,5,,5 time uration of the pulse [ps] ,,25,5,75,,25,5,75 2, 2,25 2,5 energy [pj] Fig. 6 The obtaine ifference between the bias voltage an minimum of the photoioe response for ifferent energies an time urations of pulse.

11 J Infrare Milli Terahz Waves First we obtain the total photoioe response to the two ientical input signals, an then we obtain the secon response, calculate as a sum of responses (obtaine inepenently for each pulse). Also, we have change the time perio between those two pulses. If the ifference between these two responses is less than 5% we consier the case as a linear one. For given energies, time uration of one pulse an time perio between two ientical pulses we get Fig. 7. The straight lines, in Fig. 7, inicate the minimum time, which has to pass between two pulses, for given energy an time uration of one pulse, to stay in linear regime within the error of 5%. From Fig. 7 it is obvious that minimum time between two pulses (when two ientical pulses come) has linear epenences on the uration of the pulse for a given energy. It shoul be note that the points on the straight lines in Fig. 7 are not obtaine for the same errors, but were chosen to fulfill the request that minimum error is less than 5%. The linear epenences shown in Fig. 7 coul be fitte by equation: where A epens on the input energy as: A ¼ 36:2 exp t ¼ A :35ðT 5psÞ; ð7þ E :78 :57 : ð8þ Here energy is given in [pj], an time an constant A in [ps]. So for given energy an time uration of the signal T, it is possible to fin minimum time t, between two ientical pulses, either using Fig. 7 or using formulas (7) an (8), to obtain linear working regime. If actual time between two pulses is greater than t the photoioe works in linear regime an vice versa. The example of the fact that linearity of the case epens on the time between two pulses is presente in Fig. 8. Here in the first case (lower curve) time between the pulses is ps an in the secon case it is 3 ps. Inepenently observe, both pulses put the photoioe in the linear working regime. When pulses are taken together it is possible that working regime exceees the linear one. time between two pulses [ps] E = pj E =.5pJ Numerical calculations τ = A-.35(T-5ps) E = 2pJ time uration of one pulse [ps] Fig. 7 The minimum time between two ientical pulses, in orer to achieve the linear working regime, is obtaine as a linear epenence on the time uration of one pulse. If the time between two pulses is above the corresponing straight line than it is obtaine linear working regime an vice versa.

12 J Infrare Milli Terahz Waves photoioe voltage [V] 5, 4, 3, 2,5 2,,5 sum of inepenent responses response on the sum of pulses error = % error = 9% 6, 5,5 5, 4, 3, 2,5, 2, time [ps] Fig. 8 Comparison of obtaine responses for two ientical Heavisie s square waves. The left axis refers to the upper curves, P=.2 W(h(t)-h(t-5 ps)+h(t-8 ps)-h(t-23 ps)), an right axis refers to the lower curves, P=.2 W(h(t)-h(t-5 ps)+h(t-5 ps)-h(t-2 ps)). The error in results epens on the time which passes between two input signals. In the case of two input pulses, with energies of pj an time urations of 5 ps, in Fig. 7, we fin that minimum time between pulses to stay in linear working regime is ps. So the upper curve in Fig. 8 is obtaine for linear working regime, whereas lower curve inicates that the working regime slightly excees the borers of linearity. Before explanation what happens in the case of more than two input signals, it is necessary to investigate the case of two input signals with ifferent characteristics. The results, from the observe behaviors of responses, show that Fig. 7 can be use in the similar way as it is alreay use in the case of ientical pulses. For the first input signal we fin τ. For the secon input signal we fin τ 2, an for the require time between the two signals, to enable the linear working regime, higher of these two values shoul be chosen. Here it shoul be note that it is also possible to obtain linear working regime for time less than chosen τ. But, to be sure that the selecte regime is linear, it is better to choose such more strict criteria, an it woul only result in the error less than 5%. Figure 9 is obtaine for two ifferent input signals, where the time between two signals was change. For the two signals it is calculate τ an τ 2 using the Eqs. 7 an 8. Higher value (between these two) is τ =2 ps. So the upper curves in Fig. 9 are obtaine for the time perio between two pulses equals 2 ps, an lower curves are obtaine for much shorter time perio, equals 5 ps. Consiering the error in obtaine response the upper curves belong to the linear regime whereas the lower curves inicate the nonlinear regime. So to enable linear working regime of the photoioe, the parameters of one input signal shoul be chosen in relevant omain, an the time perio between two or more signals must be appropriate. To fulfill the first eman, in case of the photoioe, the energy of input signal shoul be less than.75 pj, or for energies between.75 pj an 2.2 pj the time uration of the signal shoul be greater than 4 ps. For two signals, to stay within linear regime, time between two signals shoul be greater than calculate minimum τ (t>max(t, t 2 )). For more than two input signals, to fulfill the request for the linear working regime, it is necessary that time between each two pulses is higher than etermine τ (herewefinτ for each par of input pulses). The

13 J Infrare Milli Terahz Waves photoioe voltage [V] 5, 4, 3, 2,5 2,,5 sum of inepenent responses response on the sum of pulses error = 5% error = 2% 6, 5,5 5, 4, 3, 2,5, 2, time [ps] Fig. 9 The change of error in obtaine results epens on the time between two pulses. The left axis refers to the upper curves, P=.5 W(h(t)-h(t- ps))+. W(h(t-35 ps)-h(t-45 ps)), an right axis refers to the lower curves, P=.5 W(h(t)-h(t- ps))+. W(h(t-5 ps)-h(t-25 ps)). examples of the photoioe responses to ifferent input excitation are presente in Figs. an. Figure presents photoioe response to the series of 4 Heavisie s square waves. Each pulse inepenently observe gives linear working regime. The time perio between them is chosen to enable linear working regime. The error between presente response an the sum of inepenent responses (on each pulse) is equal to 7.5%, which is less than 5% an the request for linear regime is fulfille. Figure presents photoioe response to the series of the same Heavisie s square waves as in Fig.. Only the time perios between signals were change an it le to the error of 23%. 5, photoioe voltage [V] 4, 3, 2,5 sum of inepenent responses response on the sum of pulses error = 7.5% 2, time [ps] Fig. The response to the series of Heavisie s pulses, P=.5 W(h(t)-h(t- ps))+. W(h(t-3 ps)-h(t- 4 ps)+.5 W(h(t-45 ps)-h(t-85 ps))+. W(h(t-95 ps)-h(t- ps)).the maximum error is less than 5%. The photoioe response is linear.

14 J Infrare Milli Terahz Waves photoioe voltage [V] 5, 4, 3, 2,5 sum of inepenent responses response on the sum of pulses error = 23% 2, time [ps] Fig. The response to the series of Heavisie s pulses, P=.5 W(h(t)-h(t- ps))+. W(h(t-5 ps)-h(t- 25 ps)+.5 W(h(t-45 ps)-h(t-85 ps))+. W(h(t-9 ps)-h(t-5 ps)). The maximum error (23%) occurs in response to the first two pulses, because the time perio between them is ecrease to 5 ps instea of 2 ps (in the case of Fig. ). The photoioe working regime is nonlinear. 4 Conclusions The evelope moel, which takes into account the effect of change of carriers velocities, shows significant ifferences compare to the moels with constant velocities fin in literature, especially concerning minimum of the photoioe voltage response an time response. All results are presente as a voltage time epenences to enable the stuy of igital an signals an to show the ifferences in behavior of response. The SPICE moel is compare to erive FORTRAN moel. The ifferences in the programs results are ue to the way of calculating the overpasse path. SPICE an FORTRAN results iffer significantly for higher values of input energies, where the photoioe linear working regime is exceee. For smaller input energies, i.e. in linear an quasi-linear working regime, the results show very goo agreement. We assume that quasi-linear working regime lasts as long as the values of the instantaneous carriers velocities o not iffer significantly from the mean values. Also, concerning more than one incient pulse, linear working regime is efine as a regime where the sum of inepenent responses (to each pulse inepenently) is equal to the total response to the series of incient pulses. The total response in that case epens not only on energy an time uration of the signal pulses but also on the time between two signal pulses. All results inicate that it is practical an convenient to use this moel, an the corresponing SPICE program, whenever the input energy is not too high so we can say that photoioe is in some quasi-linear regime. Practically, whether the regime is linear or not epens on photoioe s characteristics an energy an time epenence of input excitation. For an actual photoioe, as long as minimum of the photoioe response has linear epenence on the input energy we can say that results are accurate. Results also show goo corresponence with FORTRAN results when we succee linear regime, but nonlinearity is ifficult to comment without taking into account the effect of space charge. Acknowlegments This work was supporte by the Serbian Ministry of Science an Technological Development with contract No. 6A.

15 J Infrare Milli Terahz Waves Appenix The iffusion current is obtaine using the continuity equation for holes in N region in the case of Dirac s time input D 2 p 2 þ p n p n t p ¼ ai expð axþðt t Þ: ða:þ In above equation p n is the holes concentration in N region, D p is iffusion constant for holes, τ p is the holes life time in N region, p n is the thermal-equilibrium hole s concentration in N region. Here we propose borer conitions in steay state as p n ðx ¼ W n Þ ¼ an p n ðx ¼/ Þ ¼ p no. This approximation is involve with the aim to simplify solving the case. Solution of the Eq. A. is p n ðx; tþ ¼ p n p n exp x L p þ ai expð ax Þexpð gðt t o ÞÞht ð t Þ; ða:2þ where γ=(-α 2 L 2 p )/t p, an x>. The holes iffusion current in N region can be obtaine n I pif ¼ qsd : ða:3þ x¼ Consiering that the last member in Eq. A.2 has the biggest value, we exclue the rest members an obtain I pif ¼ qsd p a 2 I expð aþexpð gðt t ÞÞ: ða:4þ Using the Eq. A.4 an parameter τ instea of (t-t ), we obtain I pif for the case of Heavisie s time input excitation as I pif ¼ qsd p a 2 ðint Þexpð aþ Z hðt t tþ expð gtþt; ða:5þ where int is the intensity of Heavisie s function. Solving the above integral we get Eq. 2. References. W. Chen, S. Liu, IEEE Journal of Quantum Electronics, vol. 32, no. 2, pp. 25, (996) 2. J. Jou, C. Liu, C. Hsiao, H. Lin an Hsiu-Chih Lee, IEEE Photonic Technology Letters, vol. 4, no. 4, pp. 525, (22) 3. Y. Batawy, J. Deen, Journal of Lightwave Technology, vol. 23, no., pp. 423, (25) 4. Y. Batawy, J. Deen, an N. Das, Journal of Lightwave Technology, vol. 2, no. 9, pp. 23, (23) 5. Y. Batawy, J. Deen, IEEE Transaction on Electron Devices, vol. 52, no. 3, pp. 325, (25) 6. M. Lu, J. Deng, C. Juang, M. Jou, an B. Lee, IEEE Journal of Quantum Electronics, vol. 3, no. 8, pp. 48, (995) 7. G. Rossi, R. Paoletti, an M. Meliga, IEEE/OSA Journal of Lightwave Technology, vol. 6, no. 8, pp. 59, (998) 8. B. Tsou, D. Pulfrey, IEEE Journal of Quantum Electronics, vol. 33, no. 2, pp. 246, (997) 9. C. Chang, H. Fetterman, Soli-state electronics, vol. 29 no. 2, pp. 295, (986). S. Malyshev, A. Chizh, Journal of Selecte Topics in Quantum Electronics, vol., no. 4, pp. 679, (24). P. Matavulj, D. Gvozić, J.Raunović, Journal of Lightwave Technology, vol. 5, no. 2, pp. 227, (997) 2. M. Lazović, P. Matavulj an J. Raunović, Microwave an Optical Technology Letters, vol. 4, no. 6, pp. 468, (24)