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Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption and Multiplication Regions

Written By

Viacheslav Kholodnov and Mikhail Nikitin

Submitted: 16 February 2012 Published: 19 December 2012

DOI: 10.5772/50778

From the Edited Volume

Photodiodes - From Fundamentals to Applications

Edited by Ilgu Yun

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1. Introduction

Minimal value of dark current in reverse biased pn junctions at avalanche breakdown is determined by interband tunneling. For example, tunnel component of dark current becomes dominant in reverse biased pn junctions formed in a number semiconductor materials with relatively wide gap Eg already at room temperature when bias Vb is close to avalanche breakdown voltage VBD (Sze, 1981), (Tsang, 1981). The above statement is applicable, for example, to pn junctions formed in semiconductor structures based on ternary alloy In0.53Ga0.47As which is one of the most important material for optical communication technology in wavelength range λ up to 1.7 μm (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981). Significant decreasing of tunnel current can be achieved in avalanche photodiode (APD) formed on multilayer heterostructure (Fig. 1) with built-in pn junction when metallurgical boundary of pn junction (x=0) lies in wide-gap layer of heterostructure (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011). Design and specification of heterostructure for creation high performance APD must be such that in operation mode the following two conditions are satisfied. First, space charge region (SCR) penetrates into narrow-gap light absorbing layer (absorber) and second, due to decrease of electric field E(x) into depth from x=0 (Fig. 1), process of avalanche multiplication of charge carriers could only develop in wide-gap layer. This concept is known as APD with separate absorption and multiplication regions (SAM-APD). Suppression of tunnel current is caused by the fact that higher value of E corresponds to wider gap Eg. Electric field in narrow-gap layer is not high enough to produce high tunnel current in this layer. Dark current component due to thermal generation of charge carriers in SCR (thermal generation current with density JG) is proportional to intrinsic concentration of charge carriers niexp(Eg/2kBT), here kB – Boltzmann constant, T – temperature (Sze, 1981), (Stillman, 1981). Tunnel current density JT grows considerably stronger with narrowing Eg than ni and depends weakly on T (Stillman, 1981), (Burstein & Lundqvist, 1969). Therefore, component JT will prevail over JG in semiconductor structures with reasonably narrow gap Eg even at room temperature. Another dark current component − diffusion-drift current caused by inflow of minority charge carriers into SCR from quasi-neutral regions of heterostructure is proportional to ni2×N1 (Sze, 1981), (Stillman, 1981) (where N is dopant concentration). To eliminate it one side of pn junction is doped heavily and narrow-gap layer is grown on wide-gap isotype heavily doped substrate (Tsang, 1981). Thus heterostructure like as pwg+nwgnngnwg+ is the most optimal, where subscript ‹wg› means wide-gap and ‹ng› − narrow-gap, properly. To ensure tunnel current’s density not exceeding preset value is important to know exactly allowable variation intervals of dopants concentrations and thicknesses of heterostructure’s layers. Thickness of narrow-gap layer W2 is defined mainly by light absorption coefficient γ and speed-of-response. But as it will be shown further tunnel current’s density depends strongly on thickness of wide-gap layer W1 and dopant concentrations in wide-gap N1 and narrow-gap N2 layers. Approach to optimize SAM-APD structure was proposed in articles (Kim et al, 1981), (Forrest et al, 1983) (see also (Tsang, 1981)). Authors have developed diagram for physical design of SAM-APD based on heterostructure including In0.53Ga0.47As layer. However, diagram is not enough informative, even incorrect significantly, and cannot be reliably used for determining allowable variation intervals of heterostructure’s parameters. The matter is that diagram was developed under assumption that when electric field E(x) (see Fig. 1b) at metallurgical boundary of pwg+nwg junction E(0)E1 is higher than 4.5×105 V/cm then avalanche multiplication of charge carriers occurs in InP layer where pwg+nwg junction lies at any dopants concentrations and thicknesses of heterostructure’s layers. However, electric field E1=E1BD at which avalanche breakdown of pn junction occurs depends on both doping and thicknesses of layers (Sze, 1981), (Tsang, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1988), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholodnov, 1998), (Kholodnov & Kurochkin, 1998). As a consequence, avalanche multiplication of charge carriers in considered heterostructure can either does not occur at electric field value E1=4.5×105 V/cm or occurs in narrow-gap layer (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989). Value of electric field required to initialize avalanche multiplication of charge carriers can even exceed E1BD (Sze, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholodnov, 1998), (Kholodnov & Kurochkin, 1998) that has physical meaning in the case of transient process only (Groves et al, 2005), (Kholodnov, 2009). Further, in development of diagram was assumed that maximal allowable value of electric field in absorber at hetero-interface with multiplication layer E2 (see Fig. 1b) is equal to 1.5×105 V/cm. But tunnel current density JT in narrow-gap absorber In0.53Ga0.47As (Osipov & Kholodnov, 1989) is much smaller at that value of electric field than density of thermal generation current JG which in the best samples of InPIn0.53Ga0.47AsInP heterostructures (Tsang, 1981), (Tarof et al, 1990), (Braer et al, 1990) can be up to 10-6 A/cm2. However, diagram does not take into account the fact that tunnel current in wide-gap multiplication layer can be much greater than in narrow-gap absorber (Osipov & Kholodnov, 1989). Therefore, total tunnel current can exceed thermal generation current.

In present chapter is done systematic analysis of interband tunnel current in avalanche heterophotodiode (AHPD) and its dependence on dopants concentrations N1 in nwgwide-gap and N2 in nngnarrow-gap layers of heterostructure and thicknesses W1 and W2, respectively (Fig. 1) and fundamental parameters of semiconductor materials also. Performance limits of AHPDs are analyzed (Kholodnov, 1996). Formula for quantum efficiency η of heterostructure is derived taking into account multiple internal reflections from hetero-interfaces. Concentration-thickness nomograms were developed to determine allowable variation intervals of dopants concentrations and thicknesses of heterostructure layers in order to match preset noise density and avalanche multiplication gain of photocurrent. It was found that maximal possible AHPD’s speed-of-response depends on photocurrent’s gain due to avalanche multiplication, as it is well known and permissible noise density for preset value of photocurrent’s gain also. Detailed calculations for heterostructure InPIn0.53Ga0.47AsInP are performed. The following values of fundamental parameters of InР (I, Fig. 1) and In0.53Ga0.47As (II, Fig. 1) materials (Tsang, 1981), (Stillman, 1981), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Braer et al, 1990), (Stillman et al, 1983), (Burkhard et al, 1982), (Casey & Panish, 1978) are used in calculations: band-gaps Eg1= 1.35 eV and Eg2= 0.73 eV; intrinsic charge carriers concentrations ni(1)=108 сm-3 and ni(2)=5.4×1011 сm-3; relative dielectric constants ε1 = 12.4 and ε2=13.9; light absorption coefficient in In0.53Ga0.47As γ=104 сm-1; specific effective masses m*=2mc×mv/(mc+mv) of light carriers m1= 0.06m0 and m2= 0.045m0, where m0 – free electron mass. The chapter material is presented in analytical form. For this purpose simple formulas for avalanche breakdown electric field EBD and voltage VBD of pn junction are derived taking into account finite thickness of layer. Analytical expression for exponent in well-known Miller’s relation was obtained (Sze, 1981), (Tsang, 1981), (Miller, 1955) which describes dependence of charge carriers’ avalanche multiplication factors on applied bias voltage Vb. It is shown in final section that Geiger mode (Groves et al, 2005) of APD operation can be described by elementary functions (Kholodnov, 2009).

Figure 1.

Energy diagram of heterostructure in operation mode (a) and electric field distribution in it (b). Ec and Ev − energy of conduction band bottom and valence band top. Solid lines − N2=N2(1), dashed − N2>N2(1)

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2. Formulation of the problem: Basic relations

Let’s consider pwg+nwgnngnwg+ heterostructure at reverse bias Vb sufficient to initialize avalanche multiplication of charge carries. This structure is basic for fabrication of AHPDs. From relations (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Grekhov & Serezhkin, 1980), (Artsis & Kholodnov, 1984)

Mn=M(Lp),Mp=M(Ln),M˜(Lp,Ln)=LpLng(x)M(x)dx/LpLng(x)dx,E1
M(x)=Y(x,Lp)/(1m),m(Lp,Ln)=LpLnα(x)Y(x,Wp)dx,Y(x,x0)=exp[x0x(βα)dx]E2

can be determined, in principal, dependences of multiplication factors M in pn structures on Vb, where Mn and Mp – multiplication factors of electrons and holes inflow into space charge region (SCR); value of multiplication factor of charge carriers generated in SCR M˜ lies between Mn and Mр; specific rate of charge carriers’ generation in SCR g=gd+gph consists of dark gd and photogenerated gph components; Lp and Ln – thicknesses of SCR in p and n sides of structure; α(E) and β(E)= K(E)×α(E) – impact ionization coefficients of electrons α(E) and holes β(E); Е(х) – electric field. Let’s denote by N1pt dopant concentration N1 so that for N1<N1pt “punch-through” (depletion) of nwg layer occurs that means penetration of non-equilibrium SCR into nng layer (Fig. 1). Optical radiation passing through wide-gap window is absorbed in nng layer and generates electron-holes pairs in it. When N1<N1pt then photo-holes appearing near nwg/nng heterojunction (х=W1) are heated in electric field of non-equilibrium SCR and, at moderate discontinuities in valence band top Ev at х=W1, photo-holes penetrate into nwg layer (layer I) due to emission and tunneling. If W1 is larger than some value W1min(N1,N2,W2) (Osipov & Kholodnov, 1989), which is calculated below, then avalanche multiplication of charge carriers occurs only in nwg layer, i.e. photo-holes fly through whole region of multiplication. In this case photocurrent’s gain (Tsang, 1981), (Artsis & Kholodnov, 1984) Mph=Mр. Let pwg+ layer is doped so heavy that avalanche multiplication of charge carriers in it can be neglected (Kholodnov, 1996-2), (Kholodnov & Kurochkin, 1998). Under these conditions thicknesses in relations (1) and (2) can be put Lp=0 and Ln=W1, i.e.

Mph=Y(W1,0)/[1m(0,W1)]E3

It is remarkable that responsivity SI(λ) (where λ – is wavelength) of heterostructure increases dramatically once SCR reaches absorber nng (layer II on Fig. 1) and then depends weakly on bias Vb till avalanche breakdown voltage value VBD (Stillman, 1981). This effect is caused by potential barrier for photo-holes on nwg/nng heterojunction and heating of photo-holes in electric field of non-equilibrium SCR. If losses due to recombination are negligible (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Forrest et al, 1983), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), for example, at punch-through of absorber, then SI(λ) in operation mode is determined by well-known expression (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2011):

SI(λ)=η(λ)×λ1.24×MphE4

where λ in µm and value of quantum efficiency η is considered below. Photocurrent gaining and large drift velocity of charge carriers in SCR allow creating high-speed high-performance photo-receivers with APDs as sensitive elements (Sze, 1981), (Tsang, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Woul, 1980). Reason is high noise density of external electronics circuit at high frequencies or large leakage currents that results in decrease in Noise Equivalent Power (NEP) of photo-receiver with increase of Мph despite of growth APD’s noise-to-signal ratio (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966). Decrease in NEP takes place until Мph becomes higher then certain value Мphopt above which noise of APD becomes dominant in photo-receiver (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980). Even at low leakage current and low noise density of external electronics circuit, avalanche multiplication of charge carriers may lead to degradation in NEP of photo-receiver due to decreasing tendency of signal-to-noise ratio dependence on APD’s Мph under certain conditions (Artsis & Kholodnov, 1984). Moreover, excess factor of avalanche noise (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966) may decrease with powering of avalanche process as, for example, in metal-dielectric-semiconductor avalanche structures, due to screening of electric field by free charge carriers (Kurochkin & Kholodnov 1999), (Kurochkin & Kholodnov 1999-2). Using results obtained in (Artsis & Kholodnov, 1984), (McIntyre, 1966), noise spectral density SN of pwg+nwgnngnwg+ heterostructure which performance is limited by tunnel current can be written as:

SN=2×q×AS×Mph2×i=12JT,i(V)Fef,i(Mph),E5

where q – electron charge; АS – cross-section area of APD’s structure; Fef,i(Mph) – effective noise factors (Artsis & Kholodnov, 1984) in wide-gap multiplication layer (i=1) and in absorber (i=2); JT,i(V) – densities of primary tunnel currents in those layers, i.e. tunnel currents which would exist in layers I and II in absence of multiplication of charge carriers due to avalanche impact generation. Comparison of two different APDs in order to determine which one is of better performance is reasonable only at same value of Мph. Expression (5) shows, that for preset gain of photocurrent, noise density is determined by values of primary tunnel currents IT1=JT1×AS and IT2=JT2×AS (total primary tunnel current IT=IT1+IT2). Distribution of electric field Е(х) that should be known to calculate parameters (4) and (5) of AHPD is obtained from Poisson equation and in layers I and II is determined by expressions:

E(x)=(E1qN1xε0ε1)×U_(l1x),E6
E(x)=[E2qN2ε0ε2(xW1)]×U_(W1+l2x),E7

Where

E2=(ε1ε2E1qN1W1ε0ε2)×U_(l1W1),E8
li=εiε0qNiEi×U+(Wili)+Wi×U_(liWi),E9

U(x) and U+(x) – asymmetric unit stepwise functions (Korn G. & Korn T., 2000), ε0 – dielectric constant of vacuum, ε1 and ε2 – relative dielectric permittivity of nwg and nng layers (Fig. 1).

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3. Avalanche multiplication factors of charge carriers in p-n structures

3.1. Preliminary remarks: Avalanche breakdown field

For successful development of semiconductor devices using effects of impact ionization and avalanche multiplication of charge carriers is necessary to know dependences of avalanche multiplication factors M(V) of charge carriers in pn structures on applied bias Vb. We need to know among them dependence of avalanche breakdown voltage VBD on parameters of pn structure and distribution of electric field E(x) related to VBD dependence. Usual way to compute required dependencies is based on numerical processing of integral relations (1) and (2) in each case. Impact ionization coefficients of electrons α(E) and holes β(E) depend drastically on electric field E. At the same time theoretical expressions for α(E) and β(E) include usually some adjustable parameters. Therefore, to avoid large errors in calculating of multiplication factors, in computation of (1) and (2) are commonly used experimental dependences for α(E) and β(E). Avalanche breakdown voltage VBD is defined as applied bias voltage at which multiplication factor of charge carriers tends to infinity (Sze, 1981), (Tsang, 1981), (Miller, 1955), (Grekhov & Serezhkin, 1980). Therefore, as seen from (2), breakdown condition is reduced to integral equation with m=1 where field distribution E(x) is determined by solving Poisson equation. Bias voltage at which breakdown condition V=VBD is satisfied can be calculated by method of successive approximations on computer. Thus, this method of determining VBD and, hence, E(x) at V=VBD requires time-consuming numerical calculations. The same applies to dependence M on V. Similar calculations were performed for a number of semiconductor structures for certain thicknesses of diode’s base by which is meant high-resistivity side of p+n homojunction or narrow-gap region of heterojunction (Kim et al, 1981), (Stillman et al, 1983), (Vanyushin et al, 2007). In addition to great complexity, there are other drawbacks of this method of M(V) and VBD determination – difficulties in application and lack of illustrative presentation of working results. Availability of analytical, more or less universal expressions would be very helpful to analyze different characteristics of devices with avalanche multiplication of charge carriers, for example, expression describing E(x), when we estimate tunnel currents in AHPDs. In this section are presented required analytical dependences (Osipov & Kholodnov, 1987), (Kholodnov, 1988), (Kholodnov, 1996-3). For quick estimate of breakdown voltage in abrupt p+n homojunction or heterojunction is often used well-known Sze-Gibbons approximate expression (Sze, 1981), (Sze & Gibbons, 1966):

VBD=AV×N(s2)/s, V,E10

where

s=8,AV=6×1013×(Eg/1.1)3/2,E11

Gap Eg of semiconductor material forming diode’s base and dopant concentration N in it are measured in eV and cm-3, properly. As follows from Poisson equation, voltage value given by (10) corresponds to value of electric field at metallurgical boundary (x=0, Fig. 2) of p+n junction:

E(0)=EBD=A×N1/s,E12

where at s=8

A=1.2×qεε0×(Egεε0)3/4×1010,E13

ε0 and ε − dielectric constant of vacuum and relative dielectric permittivity of base material; q − electron charge. Unless otherwise stated, in formulas (12) and (13) and below in sections 3.1-3.3 is used SI system of measurement units.

Figure 2.

Schematic drawing of diode based on p+nn+ heterostructure (a) and distribution of electric field in it at avalanche breakdown voltage (b); 1N1=N1(0), 2N1>N1(0); N1 − dopant concentration nwg in wide-gap layer I

Formulas (10) and (11) cannot be used for reliable estimates of VBD and EBD in semiconductor structures with thin enough base. Indeed, dependence of VBD on N is due to two factors. First, as follows from Poisson equation, the larger N the steeper the field E(x) decreases into the depth from x=0 comparing to value E1=E(0) (Fig. 1b). Second, value of electric field E1=E(0) at V=VBD falls with decreasing of N due to decreasing of |E| in SCR. Drop of E(x) becomes more weaker with decreasing of N (Fig. 1b), therefore, at preset base’s thickness W, initiation of avalanche process will require fewer and fewer field intensity E1. At sufficiently low concentration N, the lower the thicker W will be, variation of electric field E(x) on the length of base is so insignificant that probability of impact ionization becomes practically the same in any point of base. It means that breakdown voltage VBD and field EBD are independent on N and at the same time are dependent on W, moreover, the thinner W then, evidently, the higher EBD. So using of formulas (10) and (11) at any values of W, that done in many publications, contradicts with above conclusion. In next section 3.2 will be shown that value of breakdown field of stepwise p+n junction in a number of semiconductor structures can be estimated by following formula:

EBD(N,W)=EBD(0,W)×[1+NN˜(W)]1/s,E14

where

EBD(0,W)=A×(εε0×As×q×W)1/(s1),E15
N˜(W)=(εε×0As×q×W)s/(s1).E16

It seen from expression (14) that at N<N˜(W) electric field of avalanche breakdown EBD is practically independent on dopant concentration N in diode’s base.

3.2. Avalanche breakdown field

Consider pwg+nwgnngnwg+ heterostructure (Fig. 2). Symbols nwg and nng indicate to unequal, in general, doping of high-resistivity layers of structure. Denote as W1, W2 and N1, N2 thicknesses of nwg and nng layers and dopant concentrations in them, properly. Case W2=0 corresponds to diode formed on homogeneous p+nn+ structure. Let values N1 and W1 such that upon applying avalanche breakdown voltage VBD to structure, SCR penetrates into narrow-gap nng layer (Fig. 2). When W1 and N1, N2 are small enough and W2 is thick enough then avalanche process develops in nng layer. In other words, with increasing bias Vb applied to heterostructure, electric field E=E2 in narrow-gap layer on nwg/nng heterojunction (Fig. 2) reaches avalanche breakdown field E2BD in this layer earlier than electric field E1 on metallurgical boundary (х=0) of p+n  junction becomes equal to breakdown field E1BD in wide-gap nwg layer. This is due to the fact that at small values of W1 and N1 variation of field E(х) within wide-gap layer is insignificant and probability of impact ionization in narrow-gap layer is much higher than in wide-gap. If, however, W1 and N1, N2 are large enough and W2 thin enough, then avalanche process is developed in wide-gap nwg layer only. For these values of thicknesses and concentrations electric field E1 reaches value E1BD earlier than E2 – value E2BD. Because of significant decreasing of electric field E(х) in nwg layer with increasing distance from х=0, field E2 remains smaller E2BD despite the fact that band-gap Eg1 in nwg layer is wider than band-gap Eg2 in nng layer. Distribution of electric field E(x) in nwg and nng layers of considered heterostructure is obtained by solving Poisson equation as defined by (6)-(9). When avalanche breakdown voltage VBD is applied to structure, then either E1=E1ВD(N1,W1) or E2=E2ВD(N2,W2). In section 3.1 is noted that at low enough concentrations Ni avalanche breakdown fields EiBD(Ni,Wi) should not depend on Ni and have definite value depending on Wi, where i=1,2. To account for this effect, formula (12) should be modified so that when N0 then breakdown field EBD tends to some non-zero value. It would seem that it is enough to add some independent on N constant to right side of (12). It is easy to see that such modification of formula (12) leads to contradiction. To verify that let’s consider situation when avalanche multiplication of charge carriers occurs in nwg layer, i.e. E1 is close to E1ВD and multiplication factor of holes Мр (1) is fixed. Then, with increasing concentration N1, field EI(W1) (Fig. 2b) shall be monotonically falling function of N1. Indeed, with increasing N1, field E1ВD and |EI(x)| are increasing also. Increasing |EI| must be such that when x became larger some value x¯ then value EI(x) has decreased (Fig. 2b). Otherwise, field E(x) would increase throughout SCR that reasonably would lead to growth of Мр. This is evident from (1) and (2). On the other hand, adding constant to right side of expression (12) does not change E1BD/N1 and therefore results in, as follows from (6) and (9), non-monotonic dependence EI(W1) on N1. Equation (14) which can be rewritten for each of nwg and nng layers as:

EiBD(Ni,Wi)=Ai×[Ni+N˜i(Wi)]1/sE17

does not lead to that and other contradictions, From (17) follows that:

N˜i(Wi)=[EiBD(0,Wi)Ai]sE18

To determine dependences EiBD(0,Wi), let’s consider behavior of EI(W1) when parameters of heterostructure N1, N2 and W2 are varying. From (6)-(9), (17) and (18) we find that when value

Δ=N2+N˜2(W2)(ε1×A1ε2×A2)s×N˜1(W1)>0E19

then avalanche breakdown is controlled by nwg layer. It means that

EI(W1)=E1ВD(N1,W1)qN1×W1ε0ε1E20

If, however, Δ<0 then avalanche breakdown is controlled by nwg/nng heterojunction, i.e.

EI(W1)=ε2ε1E2ВD(N2,W2)E21

From (17)-(21) we obtain that

EI(WI)N1|N10={A1s×N˜1(1/s)1q×W1ε0ε1 , at Δ >00                               , at Δ <0E22

Formulas (15) and (16) follow from expressions (18), (19) and requirement (23)

limΔ0{EI(WI)N1|N10}=limΔ+0{EI(WI)N1|N10}E23

which means smoothness of field dependence E(x) in real heterostructures, where parameters are varying continuously. Particularly, in semiconductors for which relations (11) and (13) are valid, breakdown field at metallurgical boundary of p+n junction (or at heterojunction boundary, in narrow-gap layer of heterojunction, including isotype) can be described by formula

EBD(N,W)=EBD(0,W)×[1+NN˜(W)]1/8E24

where

EBD(0,W)=Xε3/7×Xg6/7×EВD(InP)(0,W);N˜(W)=Xε4/7×Xg6/7N˜InP(W)E25

And values for InP semiconductor widely used for manufacturing of AHPDs (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011) are as follows:

EВD(InP)(0,W)=4.3×105×W1/7, V/cm;N˜InP(W)=3.4×1015×W8/7, cm3E26

Xε=12.4/ε, Xg=1.35/Eg and gap Eg in diode’s base is measured in eV and its thickness W – in μm, respectively.

3.3. Avalanche breakdown voltage

It follows from expressions (6)-(9) and (14)-(16) that breakdown voltage VBD for p+nn+ structure is given by expressions

VBD=εε02qA2×[1+N˜(W)N]2/s×Ns2sAV×[1+N˜(W)N]2/s×Ns2s, ifN˜N<1θE27

i.e. when diode’s base is not punch-through and

VBD(N,W)=VBD(0,W)×{[1+NN˜(W)]1/sN2s×N˜(W)} , ifN˜N>1θE28

i.e. when diode base is punch-through. In expression (28)

VBD(0,W)=A×(εε0×As×q)1s1×Ws2s1E29

Value of parameter θ is defined from equation θ=s×(1+θ)1/s and with good degree of accuracy it equals to ss/(s1). Because θ>>1, therefore expression (27) practically coincides with formula (10), i.e. VВD of diode with thick base is independent on its thickness W. For diodes with thin base formed on semiconductors with parameters satisfying relations (11) and (14), namely when

WW˜(N)=9×Xε1/2×Xg3/4×(3×1015N)7/8E30

breakdown voltage of diode depends on W and N as follows

VBD(N,W)=VBD(0,W)××[(1+Xε4/7×Xg6/7×W8/7×N2.65×1015)1/8Xε4/7×Xg6/7×W8/7×N4.24×1018]E31

where

VBD(0,W)=43.1×Xε3/7×Xg6/7×W6/7E32

In expressions (30)-(32) Xε=12.4/ε, Xg=1.35/Eg and gap Eg in base, dopant concentration in it N and thickness W is measured in eV, cm-3 and μm, respectively.

Avalanche breakdown voltage of double heterostructure discussed in Section 4 (Fig. 1) depends on relations between fundamental parameters of materials of nwg and nng layers, their thicknesses and doping, and is determined, as follows from (6)-(9) and (14)-(16), by different combinations (with slight modification) of expressions (27)-(29) for these layers of heterostructure.

3.4. About correlation between impact ionization coefficients of electrons and holes

One of main goals of many experimental and theoretical studies of impact ionization phenomenon in semiconductors is to determine impact ionization coefficients of electrons α(E) and holes β(E) as functions of electric field E (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Parameters of some semiconductor devices, for example, APDs (Sze, 1981), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (Stillman & Wolf, 1977) depend significantly on ratio K(E)=β(E)/α(E). Performance of APD can be calculated on computer if α(E) and β(E) are known (Sze, 1981), (Tsang, 1985), (Filachev et al, 2011), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Dependences α(E) and β(E) are known, with greater or lesser degree of accuracy, for a number of semiconductors (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). However in works concerned determination of impact ionization coefficients the problem of interrelation between α(E) and β(E) has never been put. Even so, laws of conservation of energy and quasi-momentum in the act of impact ionization are maintained mainly by electron-hole subsystem of semiconductor (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Dmitriev et al, 1987). Therefore, there is a reason to hypothesize some correlation between α(E) and β(E), although perhaps not quite unique, for example, owing to big role of phonons in formation of distribution functions. It is shown in this section that for number of semiconductors the following approximate relation is satisfied (Kholodnov, 1988)

Z(E,α(E),β(E))9×102×(105E)7×α(E)β(E)ln[α(E)β(E)]=C(E)×Z0Z0ε3Eg6,E33

Where: ε – relative dielectric permittivity, and gap Eg, electric field E, α and β are measured in eV, V/cm and 1/cm, properly.

To derive relation (33) let’s consider thin p+nn+ structure in which thickness of high-resistivity base layer W satisfies to inequality

W<W0=Aεε0qs×N1ssE34

where ε0 – dielectric constant of vacuum; ε – relative dielectric permittivity of base material; q – electron charge; s and A – constants defining dependence of electric field EBDA×N1/s at metallurgical boundary (x=0) of abrupt p+n junction on dopant concentration N in base for avalanche breakdown in thick p+nn+ structure (Sections 3.1-3.3, (Sze, 1981), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966)). When condition (34) is satisfied then avalanche breakdown field can be written as

EBD(W)A×(Aεε0sqW)1s1E35

And, under these conditions, variation of electric field Е(x) along length of base W is so insignificant that probability of impact ionization is practically the same in any point of base of considered structure. For many semiconductors including Ge, Si, GaAs, InP, GaP relations given below are valid (Sze, 1981), (Kholodnov, 1988-2), (Kholodnov, 1996), (Sze & Gibbons, 1966)

s=8,A=1.2qεε0×(Eg11q)3/4×1010,E36

In this case as it follows from (34) and (35)

W0=14×ε×Eg3/4×(3×1015N)7/8,E37

And avalanche breakdown electric field for thin p+nn+ structure is defined by approximate universal formula

EBD(W)(Eg2ε)3/7×106W1/7,E38

In expressions (37) and (38) and below in this Section 3.4 concentration is measured in cm-3, energy – in eV, length – in µm, electric field – in V/cm. On the other hand condition of avalanche breakdown of p+nn+ structure (Sections 2, 3.1 and (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977))

m(0,W)=0Wα(E(x))×exp{0x[β(E(x))α(E(x))]dx}dx=1,E39

takes the form

W×[α(EBD)β(EBD)]=ln[α(EBD)β(EBD)],E40

That means the same probability of impact ionization in any point of diode’s base. And relation (33) follows from expressions (38) and (40). Let’s estimate applicable electric field interval for this relation. Expression (38) will be valid when inequality (41) is satisfied both for electrons and for holes

EBD(W)×W>(WλR×ER+Eion)×104E41

where λR,Еion,ЕR – mean free path for charge carriers scattered by optical phonons, threshold ionization energy of electrons or holes and energy of Raman phonon, respectively (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Taking into account that for many semiconductors

5×103×εEgEion7/6Wmin<<EionERλR<<Wmax1014×λR7Eg6ER7ε3,E42

From (38) and (41) we find desired interval of electric field:

104×ERλREBD(Wmax)Emin<E<EmaxEBD(Wmin)2×106×Egε×Eion6.E43

Interval of electric field (43) is most often realized in experimental studies (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Ratio Wmin/λR is usually not more than a few units. Therefore, when W<Wmin then EBDEionW×104 and hence when E>Emax instead of (33) must be valid relation

EionE×α(E)β(E)ln[α(E)β(E)]=c(E)1E44

where Еion to be understood by largest in value threshold ionization energy of electrons and holes. On basis of relations (33) (or its upgraded version, if parameters s and A differ from values of (36)) and (44)) can be obtained although approximate but relatively simple and universal analytical dependences of charge carriers multiplication factors and excess noise factors (Tsang, 1985), (Stillman et al, 1983), (Artsis & Kholodnov, 1984), (Woul, 1980), (McIntyre 1966), (Stillman & Wolf, 1977) on voltage as well as analytical expressions for avalanche breakdown voltage at different spatial distributions of dopant concentration in pn structures.

3.5. Miller’s relation for multiplication factors of charge carriers in p-n structures

Usual way to calculate dependences of avalanche multiplication factors of charge carriers M (Section 2) in pn structures on applied voltage Vb is based on numerical processing of integral relations (1) and (2) in each case. Distribution of specific rate of charge carriers’ generation g(x) in space charge region (SCR), i.e. when Lp<x<Ln (see inset in Fig. 3), is accepted in this Section 3.5 as exponential (and as special case − uniform). It is valuable for practical applications to have analytical, more or less universal, dependences M on Vb. In article (Sze & Gibbons, 1966) was proposed analytical expressions for avalanche breakdown voltage VBD, i.e. applied voltage value at which M=, in asymmetric abrupt and linear pn junctions. Expression for VBD (Sze & Gibbons, 1966) in the case of asymmetric abrupt p+n junction was generalized in (Osipov & Kholodnov, 1987) for the case of thin p+n(p)n+ structure (like as pin) as discussed in Section 3.3. Using as model abrupt (stepwise) pn junction under assumption that K(E)=β/α=const (Kholodnov, 1988-2) has been shown that from (1), (2) and approximate relation (33), which is valid for number of semiconductors including Ge, Si, GaAs, InP, GaP, can be obtained analytical dependences of multiplication factors of charge carriers on voltage.

Figure 3.

Dependences of exponents in Miller’s relation for electron nn and holes np for "thick" abrupt рn junction on applied voltage V at different values K=β/α equal to 1, 2, 3, and 4

Rewrite (33) in the form

α(E)K(E)1lnK(E)=56×(εε06×108×q)3×(1.1Еg)6×(E105)7E45

In (45) and below in this Section 3.5 is accepted (unless otherwise specified) the following, convenient for this study, system of symbols and units (Sze, 1981): gap Eg and threshold ionization energy Еion in eV; electric field E in V/cm; bias Vb in V; multiplication factors α and β in cm-1, electron charge q in C; dielectric constant of vacuum ε0 in F/m; concentration including shallow donors ND and acceptors NA in cm-3; concentration gradient a in cm-4; width of SCR Lр and Ln in p and n layers and thicknesses of these layers (inset in Figure 3) in μm, light absorption coefficient γ in cm-1. In this section, analytical dependences M(V) in pn structures have been calculated under no K(E)=const condition. Such calculations are possible because ratio K(E)1lnK(E) varies, typically, much slighter than E7. In some cases it allows using relation (45) to integrate analytically (in some cases – approximately) expressions (1) and (2) and, thus, get analytical, more or less universal, relatively simple dependences M(V). The most typical cases are considered: abrupt (stepwise) and gradual (linear) pn junctions like as in model given in (Sze, 1987), (Sze & Gibbons, 1966) and thin p+n(p)n+ structure (like as pin) with stepwise doping profile as in model presented in (Osipov & Kholodnov, 1987). For purposes of discussion and comparison of obtained results with numerical calculations and experimental data, multiplication factors will be written in traditional common form

Mn=11vnn,Mp=11vnp,M˜=11vn˜,E46

where v=V/VBD. This form was first proposed by Miller in 1955 (Miller, 1955) and then, despite lack of analytical expressions for exponents nn, np, n˜, has been widely used as "Miller’s relation" (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986). It was found that values of these exponents depend on many factors including, in general, voltage as well (Kholodnov, 1988-2), (Grekhov & Serezhkin, 1980), Fig. 3. Form of writing (46) clearly shows that M(V) when VVBD.

3.5.1. Stepwise pn junction

In this case from relations (1), (2) and (45) and Poisson equation (SI units)

dEdx={qNAεε0,       x<0qNDεε0,     x>0E47

follow that

Mn=(K01)/(K0K0v4),Mp=K0v4×MnE48
VBD=6×1013×(Еg1.1)3/2×Neff3/4,Neff=NA×NDNA+NDE49

where K0 – value K(х) when Е(х)=Е(0)=Е0, i.e. value of K at metallurgical boundary of pn junction (see inset in Fig. 3). Formula (49) for VBD at ND<<NAor NA<<ND becomes well-known Sze-Gibbons relation (Sze, 1981), (Sze & Gibbons, 1966). If charge carriers are generated uniformly in SCR then computations lead to following expressions:

M˜Mn=NA×exp[ξA×e(K01)+ξD/g]+ND×exp[ξD×e(1K0)+ξA/g]NA+ND,E50

when

ξA,D(v)×(Neff/NA,D)×v4×|lnK0|1;E51
M˜=(1Keff8v4×1Keff1v4Keff1)1×Mn,E52

when

Keffv4×NeffNA,D>>1,E53

e(x) – unity function (Zeldovich & Myshkis, 1972), Keff=K0+K01. Expression (50) is obtained by expanding the function Y(x,Lp) as a power series in

Lpx(βα)dx,

and expression (52) was derived by standard method of integrating fast-changing functions (Zeldovich & Myshkis, 1972).

3.5.2. Gradual (linear) pn junction

In this case Poisson equation can be written as (SI units):

dEdx=q×σεε0×xE54

where σ- slope of linear concentration profile

and therefore

Mn=(K01)/(K0K0v5),Mp=K0v4×MnE55
VBD=60×(3×1020σ)2/5×(Eg1.1)6/5×(17.7ε)1/5.E56

In derivation of relations (55) and (56) was used known expression for voltage distribution on linear pn junction (Sze, 1981) and was also taken into account that (Gradstein & Ryzhyk, 1963)

0y(yx)7xdx=40966435y15/2E57

Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear pn junction (Sze, 1981), (Sze & Gibbons, 1966) by last multiplicand, which for typical values of ε10 (Sze, 1981), (Casey & Panish, 1978) is close to unity.

3.5.3. Thin p+n(p)n+ structure (pin)

When thickness of high-resistivity region (base) of considered structure

W>W˜=6εε05q×(Еg1.1)3/4×1010N7/82ε×Еg3/4×(3×1015N)7/8,E58

where N – dopant concentration (for example, donor) in base, and when Vb=VBD then SCR does not extend to entire thickness of base ((Osipov & Kholodnov, 1987), Sections 3.1-3.3, inset in Fig. 4). In this case, expressions (48)-(53) remain apparently valid. In opposite case, base is depleted by free charge carriers when Vb<VBD that gives in the result substantially other expressions for avalanche multiplication factors of charge carriers and avalanche breakdown voltage. When W<W˜ then from relations (1), (2) and (45) and Poisson equation

dEdx=qNεε0E59

we find that

Mn=(K01)/(K0K0v˜8),Mp=K0v˜8×Mn,E60

where

v˜8=(V+V1)8(VV1)8V28,E61
V1=qNW22εε0×106,E62
V28=(6εε05×108qW2)4×(1.1Eg)6×1N.E63

Figure 4.

Dependences of analytical (solid lines) and numerical (dashed lines) (Leguerre & Urgell, 1976) limiting values of exponent nB=limVVBDn(V) in Miller’s relation (46) on concentration of donor dopant ND in "thick" high-resistivity layer of stepwise p+nn+ structure, 1 − Si, 2 − Ge, 3 − GaAs, 4 − GaP. Values K(E), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966). In inset − scheme of "thick" p+nn+ structure

In deriving expressions (60)-(63), multiplication of charge carriers in p+ and n+ layers and voltage drop on them is considered negligible. This is justified because of significant decreasing of electric field E(х) deep into high-doped layers of the structure (Sze, 1981), (Kholodnov 1996-1), (Kholodnov 1998), (Leguerre & Urgell, 1976). Admissibility of such neglect is confirmed also by formula (49) when NA<<ND or when ND<<NA. Avalanche breakdown voltage is determined by equation v˜ = 1 which has no exact analytical solution. However, till W surpasses W˜/8, then value of field at x= W is much less than value of field at x=0. In this case, using smallness parameter

(12×V1V2)8<<1E64

we find that in zeroth-order approximation with respect to this parameter

VBD=V2V1.E65

In the case of very thin base when

WW0=18W˜,E66

electric field varies so slightly along base that probability of impact ionization is practically the same in any point of it ((Osipov & Kholodnov, 1987), (Kholodnov 1988-1), Sections 3.2 and 3.4). As a result

Mn=(K1)/(KKv7),Mp=Kv7×Mn,E67
M˜=γWγW+v7×lnK×Kv7×exp(γW)1exp(γW)1×Mn,E68
VBD=7×325×(3q50εε0)3×(Еg1.1W)6×10698×(W×Еgε)6/7,E69

where γ< 0, if structure is illuminated through p+ region (front-side illuminated) and γ> 0 if structure is illuminated through n+ region (back-side illuminated).

3.6. Discussion of the results. Comparison with computed and experimental data

3.6.1. To formulas for avalanche breakdown electric field and voltage for abrupt p+ - n junction

In sections 3.1-3.3 were derived approximate universal formulas for avalanche breakdown field EBD and voltage VBD for abrupt p+n junction taking into account finite thickness of high-resistivity layer W. Comparative values of breakdown field EBD(0,W) for Si, Ge and InP most often used for fabrication of APDs computed by formulas (25) and (26) and found from numerical solution of breakdown integral equation m=1, where m is defined by (2) are shown on Fig. 5 (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Groves et al, 2005), (Stillman et al, 1983), (Trommer, 1984), (Woul, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986), (Gasanov et al, 1988), (Brain, 1981), (Tager & Vald-Perlov, 1968). It is seen that in the most practically interesting range W(0.2÷10) μm for all a.m. semiconductors analytical EBD(a)(0,W) and calculated EBD(c)(0,W) values of breakdown field differ by less than 20 %. Relatively drastic fall of ratio EBD(a)(0,W)/EBD(с)(0,W) in comparison to unity with decrease of W (for thin enough W) is due to the fact that, as shown in Sec. 3.4, if

W<Wmin5×103×εЕg×Еion7/6, µm,E70

then formulas (25) and (26) are not true. To estimate breakdown field EBD(0,W) at values W defined by (70) can be used the following formula

EBD(0,W)=104×EionW, V/cmE71

Figure 5.

Dependence of ratio between analytical value of breakdown field EBD(а)(0,W) obtained by formulas (25) and (26) and numerical value EBD(c)(0,W). Dashed curve − analytical value of effective avalanche breakdown field EBD*(0,W)=Xε3/7×Xg6/7×EBD(0,W)EВD(InP)(0,W). Curves 1 and 1' − Si, 2 − InP, 3 − Ge. Values α(E) and β(E) are taken: for curves 1 and 3 − from Table 1 of monograph (Grekhov & Serezhkin, 1980), for curve 1' − from (Kuzmin et al, 1975), for curve 2 − from (Cook et al, 1982)

If assume that in Si threshold energy of impact ionization Еion of holes is higher than electrons, and it equals to 5 eV (Sze, 1981), then from (70) we find for Si Wmin0.1 μm. Estimates based on data from studies (Sze, 1981), (Tsang, 1985), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977) show that for Ge and InP value Wmin is 2-3 times smaller.

Therefore curve 1 in Fig. 5 starts to fall significantly below unity at larger values W than curves 2 and 3. Analytical and computed dependences EBD on N for InP used in high-performance APDs for wavelength range λ=(1÷1,7) µm as wide-gap layers in double heterostructures (Fig. 1, 2) are shown on Fig. 6 (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Forrest et al, 1983), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984). It is seen that EBD(a)(N,W) and EBD(c)(N,W) differ from each other by less than 10 %. In Fig. 7 and 8 are shown universal dependences of breakdown voltage VBD(a) on N and W calculated by formulas (11), (27)-(29). It is seen from Fig. 7 that Sze-Gibbons relations (10) and (11) (Sze, 1981), (Sze & Gibbons, 1966) can be used to determine VBD when N>Nmin10×N˜(W) only. Value of this minimal concentration, for example, for classic semiconductors Si, Ge, GaAs, GaP and InP at W= (1-2) μm equals to (1÷5)×1016cm-3. As shown on lower inset in Fig. 7, dependence VBD on N is in the strict sense non-monotonic. Such kind of dependence VBD on N is due to the fact that for small enough N breakdown field EBD is growing faster with increasing N than |E|N in diode’s base. Maximum VBD is reached, as it follows from (28), at

N=Nmax=(2ss11)×N˜(W)E72

and expressed as

VBD(max)=[(s1)×2ss1+1]×(2s)1×VBD(0,W)E73

when s=8, value Nmax1.2×N˜, ΔVmax(rel)2.86×102<<1 and absolute value ΔVmax can reach tens Volts, and even more (see Fig. 7). The analytical dependences VBD(a)(N,W) (Fig. 7 and 8) for a number of semiconductors are in good agreement with VBD(c)(N,W) computed on the basis of integral equations (1) and (2) (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976). Note that results of comparison VBD(a)(N,W) with VBD(c)(N,W) and ЕBD(a)(N,W) with ЕBD(c)(N,W) depend on accuracy of determination of impact ionization coefficients of electrons α(Е) and holes β(Е) which are sharp functions of electric field Е. As a rule, different authors obtain different results (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) (see, for example, curves 1 and 1' in Fig. 5). In addition, deducing of relations (1) and (2) is based on local relation between α and β (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2011), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) which is not always valid (McIntyre, 1972), (Gribnikov et al, 1981), (Okuto & Crowell, 1974), (McIntyre, 1999).

Figure 6.

Dependence of field EBD on N for InP: 1 − W=0.5 μm, 2 − W=2 μm, 3 − W=8 μm. Solid lines – formulas (24)-(26), dashed curves − numerical calculation. Values α(E) and β(E) are taken from (Cook et al, 1982). In inset is shown dependence of effective concentration N˜*=Xε4/7×Xg6/7×N˜N˜InP on W. Concentration is measured in cm-3, field − in V/cm and thickness W − in μm.

Figure 7.

Dependence of avalanche breakdown voltage VBD of homogeneous p+nn+ structure on dopant concentration N in base: solid line − (31) and (32), dotted line − expressions (10) and (11). In lower inset: dependence of relative voltage ΔV(rel)=[VBD/VBD(0,W)]1 normalized to concentration N˜(W) at N4×N˜(W). In upper inset: dependence of effective ΔVmax*=Xε3/7×Xg6/7×[VBDVBD(0,W)]maxΔVmax(InP) on base thickness W. Voltage is measured in V, thickness W − in μm.

Figure 8.

Dependence of effective avalanche breakdown voltage VBD*=Xε3/7×Xg6/7× VBDVBD(InP) of homogeneous p+nn+ structure on thickness of its base W for three values of effective concentration N*=Xε4/7×Xg6/7×NNInP: 1 − N*=3×1016cm-3, 2 − N*=3×1014cm-3, 3 − N*=3×1012cm-3. In inset is shown dependence W˜ on N*. Concentration is measured in cm-3, voltage − in V, thickness W − in μm

3.6.2. To сorrelation between values of impact ionization coefficients of electrons and holes

In Section 3.4 is shown that there is reason to suppose existence of some correlation between values of impact ionization coefficients of electrons α(Е) and holes β(Е), and form of required relation (expression (33) and (45)) is proposed. It is obvious from Fig. 9 that values Z0ε3/Eg6 may differ by many orders of magnitude in different semiconductors. At the same time, for presented in Fig. 9 Ge, Si and GaP, function c(Е) (see relations (33) and (45)) in range of fields where α(Е) and β(Е) vary in several orders of magnitude (Okuto & Crowell, 1975), remains, as it follows from (33) and (45), of the order of unity. Calculations based on experimental dependences α(Е) and β(Е) (Cook et al, 1982) show that in InP value c(Е) is some more closely to 1. It is evident from Fig. 10 that for GaAs, regardless of orientation of crystal with respect to electric field, function c(Е) depends weakly on Е in comparison with impact ionization coefficients of charge carriers (which values are taken from (Lee & Sze, 1980)), and differs from unity by no more than 2-3 times. A similar situation takes place in Ge (Fig. 11, according to (Mikawa et al, 1980)). As shown in (Kobajashi et al, 1969) dependences α(Е) and β(Е) measured in (Miller, 1955), (McKay & McAfee, 1953) in the range of fields Е=(1.5÷2.7)×105 V/cm can be described in Ge by formulas α(Е)=7.81×1034×Е7, β(Е)2α(Е). This result agrees well with expression (33). Note that, c(Е) differs from unity approximately by the same factor, as values α(Е) and β(Е) for the same material obtained by different authors differ, respectively, from each other (Sze, 1981), (Tsang, 1985), (Forrest et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (Cook et al, 1982), (Okuto & Crowell, 1974), (Okuto & Crowell, 1975), (Lee & Sze, 1980), (Mikawa et al, 1980), (Kuzmin et al, 1975). Using procedure described in Section 3.4, we can also determine relation between α(Е) and β(E)=K(E)×α(E) in the case when relations (11) and (13) are not satisfied (Grekhov & Serezhkin, 1980). It seems, relation required for such case, i.e. under assumption of power dependence α on Е and K(E)=const, was obtained for the first time in (Shotov, 1958).

Figure 9.

Dependence Z(E) [relation (33)] in Ge, Si, and GaP for α(E) and β(E) from (Okuto & Crowell, 1975)

3.6.3. To Miller’s relation

From (48), (55) and (67) follow that, exponents in Miller’s relation (46) for multiplication factors of electrons and holes are given by

nn×lnv=ln[(K0vξ1)/(K01)],E74
np×lnv=ln[K0K01(1K0vξ)],E75

where ξ= 4, 5 and 7 for stepwise pn junction, linear pn junction and very thin (66) p+nn+ structure (situation 1, 2 and 3, respectively). If thickness of base in p+nn+ structure is not very small, i.e., W0<W<W˜ (situation 4) then as it follows from formula (60), exponents nn and nр are also expressed by (74) and (75) but in right side of those expressions v˜ substitutes v and ξ=8. Value of exponent n˜ lies between values nn and nр. From (1) and (2) apparent that when α=β then factors Mn, Mp and M˜ coincide with each other, i.e., nn=nр=n˜=n, and, as it follows from expressions (74) and (75), regardless of bias voltage applied, n= 4, 5 and 7 for situations 1, 2 and 3, respectively. Exponents in Miller’s relation have the same values when V<<VBD, more exactly, when |lnK0/lnv|<<ξ, regardless of ratio K0=β(E0)/α(E0). When VVBD or more exactly, if

Δv=1v<<min{1ξ|lnK0|;1ξ},M>>1

Then for these situations

nn=nnBξ×K0×lnK0(K01),np=npBξ×lnK0(K01).E76

Graphs in Fig. 4 allow comparing numerical values of exponents nnB and nрB calculated in (Leguerre & Urgell, 1976) nB(c) and analytical nB(a) computed by formulas (76) for asymmetrical stepwise pn junction. Like as in (Leguerre & Urgell, 1976), experimentally determined functional dependencies α(E0) and β(E0) (Sze & Gibbons, 1966) were used in calculations of dependences nB(a). As follows from (46), when M>>1, then ratio of analytical value of multiplication factor M(a) to calculated M(c) equals to ratio nB(c) to nB(a) (Fig. 11-13). It obviously from Fig. 11-13 that for all considered semiconductors (with curves α(Е) and β(Е) taken from (Sze & Gibbons, 1966)), dependences M(a)(V) and M(c)(V) do not differ by more than 50 %. Dependences of exponents nn(a) and np(a) on voltage and nnB(a) and npB(a) on ratio K=β/α are illustrated in Fig. 3 and 14, respectively. It should be noted that numerical values of exponent in Miller’s relation, as well as, value VBD depend, obviously, on what functions α(Е) and β(Е) are used in (1) and (2) in calculations. Let’s take the simplest case when α(E)=β(E) and pn junction is stepwise. Varying expressions (1) and (2), we find that under considered conditions

nB=εε0500×q×Neff×α(EBD)×EBD,E77

where EBD=E(0) at V=VBD is determined from condition

0EBDα(E)dE=100εε0×NeffE78

Figure 10.

Dependence С(E) at different orientations of GaAs crystal with respect to electric field for values α(E) and β(E) from (Lee & Sze, 1980)

Figure 11.

Dependence of ratio between analytical values of avalanche multiplication factors М(а) of electrons and holes and numerical values М(с) (Leguerre & Urgell, 1976) in stepwise asymmetric Ge pn junction on value of multiplication factor М=М(а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of pn junction N, cm-3: 1 − 1015, 2 − 3×1015, 3 − 1016, 4 − 3×1016, 5 − 6×1016. Values K(E), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

Figure 12.

Dependence of ratio between analytical values of avalanche multiplication factors М(а) of electrons and holes and numerical values М(с) (Leguerre & Urgell, 1976) in stepwise asymmetric Si pn junction on value of multiplication factor М=М(а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of pn junction N, cm-3: 1 − 1015, 2 − 3×1015, 3 − 1016, 4 − 3×1016, 5 − 6×1016. Values K(E), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

Figure 13.

Dependence of ratio between analytical values of avalanche multiplication factors М(а) of electrons and holes and numerical values М(с) (Leguerre & Urgell, 1976) in stepwise asymmetric GaAs (solid lines) and GaP (dashed lines) pn junctions on value of multiplication factor М=М(а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of pn junction N, cm-3: 1 − 1015, 2 − 3×1015, 3 − 1016, 4 − 3×1016, 5 − 6×1016. Values K(E), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

Figure 14.

Dependence of limiting values nB=limVVBDn(V) of exponents in Miller’s relation for electron nn and holes np for "thick" abrupt рn junction on K=β/α

In Fig. 15a are shown dependences nB(Neff) calculated from relations (77) and (78) for four values α(E)=β(E) obtained for GaAs by different authors (Grekhov & Serezhkin, 1980), (Okuto & Crowell, 1975), (Kressel & Kupsky, 1966), (Nuttall & Nield, 1974). It is seen that analytical value nnB=npB=4 calculated by formulas (76) approximately equals to mean value with respect to curves 1-4 in Fig. 15a. According to obtained above results expressions (48)-(53) are not valid when concentration

Neff>(Neff)max2×1017×(Еg)2×Еion4/3E79

which for many semiconductors is of the order of 1017 cm-3. At such high concentrations, as it follows from Section 3.4 and (Kholodnov, 1988-1) and relations (1) and (2), for stepwise pn junction

nn=ln((K0v1)/(K01))lnv,np=nn+1vlnvlnK0,E80

moreover

nnB=K0ln(K0/(K01))=K0×npB.E81

Figure 15.

Dependences nB(Neff) in GaAs calculated on the base of different dependences α(E)=β(E), taken from: 1 − (Shabde & Yeh, 1970), 2 − (Grekhov & Serezhkin, 1980), 3 − (Okuto & Crowell, 1975), 4 − (Kressel & Kupsky, 1966), 5 − (Sze & Gibbons, 1966). Dashed lines − analytical values

For comparison, in Fig. 15b are presented dependences of nB(c)(Neff) and nB(a)(Neff)=1 for the case α=β, when nnВ=nрВ=nВ. It is seen that value nB(a)(Neff)= 1 is approximately equal to mean value with reference to curves 2, 3 and 5 in Fig. 15b plotted on the base of numerical data. Note that starting from Neff(Neff)max breakdown voltage VBD dependence on Neff becomes, with growth Neff, more and more weaker than that described by equation (49), and in limit tends to value VBD=Eion/q. This conclusion accords with results of studies (Grekhov & Serezhkin, 1980), (Nuttall & Nield, 1974). Obtained results agree well with experimental results for a number of pn structures, including based on Ge, Si, GaAs, GaP (Sze, 1981), (Tsang, 1985), (Stillman et al, 1983), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Bogdanov et al, 1986), (Cook et al, 1982), (Shotov, 1958). We present here three cases of studies. In experimental study (Miller, 1955) of breakdown in Ge stepwise pn junction was found that measured values of exponents in Miller’s relation were lying in range from 3 to 6.6. The same values of exponents are obtained from expressions (74) and (75) with ξ=4 if we take into account that in Ge with doping levels used in (Miller, 1955) К02÷3 (Sze, 1981), (Tsang, 1985), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Shotov, 1958). In experimental study (Bogdanov et al, 1986) of APD based on MIS structure (metal-insulator-semiconductor APD) multiplication of charge carriers occurs in thick pSi substrate. From point of view of avalanche process this structure is similar to asymmetric stepwise n+p junction. Therefore, avalanche process in MIS APD can be described by expressions (74)-(76) with ξ=4. Concentration of shallow acceptors in substrate of investigated structure was 1015 cm-3. At this doping avalanche breakdown in Si occurs when electric field near insulator-semiconductor interface reaches value EBD3×105 V/cm (Sections 3.1 and 3.2, (Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966)), and therefore К0102 (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Kuzmin et al, 1975). Measured in (Bogdanov et al, 1986) value nn at VBDV<<VBD was found equal to 0.2. From formulas (76) with К0102 follows that nnВ=0.186. In Tables 1 and 2 are presented experimental (Shotov, 1958) and calculated by formulas (48) and (55) values of multiplication factors of electrons Mn(V) and holes Mp(V) in Ge stepwise and linear pn junctions. Obviously, for these pn junctions, experimental and analytical values of multiplication factors differ from each other by less than 20 % in whole voltage V range used in measurements.

V/VBD
Mp
Experiment(Shotov, 1958) Theory
0.65 1.35 1.30
0.70 1.50 1.44
0.75 1.75 1.65
0.80 2.10 1.98
0.85 2.65 2.55
0.90 3.70 3.71
0.95 7.00 7.30

Table 1.

Experimental (Shotov, 1958) and computed [from Equation (48)] hole avalanche multiplication factor Mp in step-wise pn junction in pGe for different ratios of applied voltage to avalanche breakdown voltage V/VBD. It is assumed that K0=2 (Shotov, 1958)

V/VBD
Mp Mn K0 (*)
Experiment (*) Theory Experiment (*) Theory
0.65 1.25 1.19 1.12 1.09 2.10
0.70 1.40 1.28 1.20 1.14 2.00
0.75 1.60 1.44 1.30 1.22 2.00
0.80 1.85 1.70 1.40 1.33 2.10
0.85 2.40 2.13 1.70 1.56 2.00
0.90 3.50 3.10 2.20 2.00 2.10
0.95 6.80 5.89 3.90 3.45 2.00
0.975 13.00 11.64 7.00 6.32 2.00
0.98 - 14.52 - 7.76 2.00
0.985 - 19.33 - 10.16 2.00
0.99 30.00 28.90 - 14.97 2.00

Table 2.

Experimental (*) (Shotov, 1958) and computed [from Equation (55)] avalanche multiplication factors Mp and Mn for holes and electrons in Ge linear pn junction for different ratios of applied voltage to avalanche breakdown voltage V/VBD (Shotov, 1958)

Finally, it is interesting to analyze application of expressions (45) and (76) to describe avalanche process in InSb. The fact is that dependence α(Е) in InSb was quite well known already in 1967 (Baertsch, 1967), but no one could obtain information about dependence β(Е) (Dmitriev et al, 1987), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968). Substituting in (45) dependence α(Е) for InSb (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968), we find that ratio K=β(E)/α(E) is vanishingly small up to electric field E4×104 V/cm resulting in extremely high value nрВ when at the same time value nnВ is extremely small. It means that Mn(V) becomes much larger than unity, even at voltages Vb noticeably lower avalanche breakdown voltage VBD, and value Mp(V) remains equal to unity up to values Vb very close to VBD. Effect obtained from application of relations (45) and (76) accords very well with experimental data (Baertsch, 1967), (Dmitriev et al, 1983) and explains why multiplication of holes in InSb is extremely hard to observe (Dmitriev et al, 1987), (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968).

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4. Tunnel currents in avalanche heterophotodiodes

4.1. Calculation of tunnel currents in approximation of quasi-uniform electric field and conditions of its applicability

In act of interband tunneling electron from valence band overcomes potential barrier ABC (Fig. 16a). The length of tunneling lT, i.e. length on which energy of bottom of conduction band Ес(x) changes by value equal to Еg is found by solving integral equation

Eg=q×xx+lT(x)E(x')dx'E82

If variation of electric field within length of tunneling ΔЕ<<Е, i.e. specific length of variation of field lE>>lT, then expanding function Е(x) in Taylor series around point x=x, we find that in the first order of parameter of smallness lT/lE equation (82) takes the form

lT=EgqE(x)×[1(lT/2E)×|E/x|]E83

When N(x)=const then equation (83) is exact. As can be seen from Fig. 16a, if

|CC|ΔlT<<lT,|CB|ΔEc<<Eg,E84

then true ABC barrier coincides to high degree of accuracy with triangle ABC′ to which corresponds uniform field Е(х) (Fig. 16b).

It follows from (83) and Poisson equation that inequalities (84) are satisfied if

δ(x)N(x)×Eg2εε0×E2(x)<<1,E85

at that

lT(Eg,E)=Egq×E(x)E86

Figure 16.

Physical meaning of quasi-uniform field approximation: a − band diagram, b − field distribution on length of tunneling. ABC − true potential barrier, ABC' − potential barrier used de facto. Dashed lines − Е(х)=const

As shown below, due to large values of field Е at avalanche breakdown of pn structures, inequality (85) is valid for almost all materials up to concentration N=1017 cm-3 and even high.

Under these conditions specific rates of charge carriers’ tunnel generation gTi(x) in layers I and II of structure can be described by expression

gTi(x)1q×JTix=ATi×E2(x)×exp[aiE(x)],E87

obtained in (Kane, 1960) (see also (Burstein & Lundqvist, 1969)) for Е(х)=const, in which

ATi=q2(2π)3×2×2mi*Egi,ai=π4q××2mi*×Egi3.E88

Here , Еgi and mi*=2mc×mv/(mc+mv) – crossed Plank constant, gaps and specific effective masses of light charge carriers in proper layers. Approximation of quasi-uniform field (87) and expressions (6)-(9) result in convenient formula for analysis of primary interband tunnel current density

JT=i=12JTi=2×q3(2π)3×2×i=12mi*Egi×LTi×Ei2×exp(aiEi),E89

where characteristic dimensions of areas of charge carriers’ tunnel generation in layers I and II

LTi(Ei,Wi)=min{WTiε0εi×Ei2q×ai×Ni,Wi}.E90

Equation (89) is valid under conditions

δiNi×Egi2εε0×Ei2<Eiai<<1,E91
lTilT(Egi,Ei)=Egiq×Ei<<li.E92

These conditions mean the following. If inequalities (91) for gTi(E) are satisfied then expression (87) is valid, at least in the neighborhood of field value E=Ei. When right side of inequalities (91) is satisfied then tunnel generation drops sharply with decreasing E, and therefore ITi at WTi<Wi is mainly determined by tunneling in areas 0xWT1 and W1xW1+WT2.

Fulfillment of conditions (92) is necessary at punch-through of proper layers of structure for neglecting tunneling through its hetero-interfaces which is not accounted for by formula (89). We show further, that at avalanche breakdown, inequalities (91) and (92) are valid for almost all real values of material parameters, concentrations Ni and layers’ thicknesses Wi of heterostructure. Avalanche breakdown occurs when one of fields Ei becomes close to breakdown field EiBD of proper layer of structure ((Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), Sections 3.1-3.3).

Breakdown fields EiBD can be obtained by formula (14) (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989), i.e.,

EiBD(Ni,Wi)=EiBD(0,Wi)×[1+NiN˜i(Wi)]1/s,E93

where

EiBD(0,W)=Ai×(Ai×εiε0sqWi)1/(s1),N˜i(Wi)=(Ai×εiε0sqWi)s/(s1)E94

(s and Ai – some constants).

For many semiconductors including InxGa1xASyP1y alloy which is one of the main materials for avalanche heterophotodiodes fabrication (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), (Woul, 1980)

s=8,Ai=1.2×qεiε0×(Egi11q)3/4×1010.E95

From expressions (93) and (94) when relations (95) are satisfied we find the following.

  1. When

NiNi(1)=8.9×1019Xmi4×Xεi4×Xgi6, cm-3,WiWi(1)=Χmi3.5×Χεi3×Χgi6×1.4×104, µm,E96

then ratio Еi to ai is less than 0.1, where Χmi=0.06/mi0*,Χεi=12.4/εi,Χgi=1.35/Egi (for InP which is often used for growing of wide-gap layers of heterostructure (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), (Woul, 1980)), Xmi=Xεi=Xgi=1, mi0*=mi*/m0 (m0– free-electron mass)

  1. When

NiNi(2)=Χmi0.2×Χεi1.6×Χgi0.4×3.3×1017,cm3,WiWi(2)=Xgi0.4×1.8×102Xmi0.7×Xεi1.9,μmE97

then under avalanche breakdown of proper layer of structure ratio δi to ЕiBD/ai is not exceed unity, moreover, even when Ni=Ni(2)

δi<Χmi0.6×Χεi1.2×Χgi0.8×101.E98
  1. When

Wi>>1.8×102Xεi×Xgi6, µm,E99

then length of tunneling lTi at Ei=EiBD is much shorter than thickness Wi of this layer.

In expressions (96)-(99) Еgi is measured in eV. Analysis shows that under avalanche breakdown of heterostructure inequities (91) and (92) are satisfied for real values of Ni and Wi and Ei<EiBD, i.e. in layer which does not control avalanche breakdown also. As can be seen from Fig. 17, when punch-through of layer nwg stops then, obviously, conditions (91) and (92) become no longer valid. Note that calculations of tunnel currents in approximation of quasi-uniform field lead to some overestimation of actually available. In fact, due to high doping of рwg+ layer, tunnel current in it can be ignored; this is situation similar to MIS structures (Anderson, 1977). In n type layers electric field decreases with increasing distance from metallurgical boundary of p+n junction (Fig. 1b), and because gradient of potential is expressed as dφ/dx=E then slope of zones Еc(x) and Еv(x) decreases with increasing x. It is shown from Figure 16a that use of quasi-uniform field approximation means underestimating of thickness of actual barrier ABC. As expected, numerical calculations in WKB approximation (Anderson, 1977) give a somewhat smaller value of tunnel currents than formula (89). Since tunnel currents are strongly dependent on parameters of material, which in real samples, usually, more or less different from those used in calculations (moreover, exact dopant’s distribution profile Ni(x) and hence shape of barrier ABC are usually unknown), then slight overestimation of tunnel currents values provides some technological margin that is needed for development of devices with required specifications.

Figure 17.

Dependence of generalized parameters of smallness δ2* and lT2* in quasi-uniform field approximation on concentration N1, at Мph=100, in case, when charge carriers multiplication occurs in nwg:InP layer. Solid lines − δ2*, dashed − lT2*. Values W1, μm: 1 − 0.5, 2 – 2, 3 − 8. N1pt − maximal concentration N1 at which punch-through of nwg layer is possible; δ2=(N2/1016)×(ε2/ε1)×Eg2×δ2*; lT2=(ε2/ε1)×Eg2×lT2*; Eg2- eV, concentration − cm-3.

4.2. Features of interband tunnel currents in p+n heterostructures under avalanche breakdown

Analysis of expression (89) under avalanche breakdown of p+n heterostructure, i.e., when either E1=E1BD or E2=E2BD, shows that in contrast to homogeneous pn junction (Stillman, 1981), (Ando et al, 1980) density of initial tunnel current JT, as a rule, is not a monotonic function N1. An increase in N2 cause, for some values of N1 and Wi, the rise of tunnel current and vice versa – decrease of tunnel current when N1 and Wi have different values. Depending on gap Egi of heterostructure’s layers and their thicknesses Wi the following situations are possible.

4.2.1. Independent doping levels of wide-gap and narrow-gap n type layers

I.

W1W2W1/2W1/2*=(ε1×A1ε2×A2)s×[N˜2(W2)N2+N˜2(W2)](s1)/s.E100

In this case, at any concentration N1, field E1=E1BD(N1,W1), and E2<E2BD, i.e., avalanche breakdown is controlled by nwg layer.

As follows from (6)-(9), (89) and (93), if

exp[a1E1BD(0,W1)×(1ε2×а2ε1×а1)]<<1,E101

which is fulfilled with large margin at a2ε2<a1ε1 due to large ratio of a1 to E1BD(0,W1) (1-2 orders of magnitude) while

N1<N˜1(T)s×(2s1×ε1ε2×a2×E1BD(0,W1))1/2×N˜1(W1)W1(s+0.5)/(s1)E102

then tunnel current is almost independent on N1.

If s sufficiently large ((Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966), Sections 3.1-3.3), then with further increase of N1 tunnel current is monotonically falling. However, in most real cases, for example, when relations (95) is valid, tunnel current at N1>N˜1(T) first decreases and then increases.

One can see that at minimum of tunnel current, as a rule, the following inequality is valid

ξE1BD(0,W1)a1<κ(s2)/(s1)s1/(s1)×yf2(y),E103

where

f(y)=(y+r1)1/s,r=(κ×s)s/(s1),κ=1a2×ε2a1×ε1,y=N1r×N˜1

When (103) is fulfilled then WT1<W1.

Therefore, as it follows from (6)-(9), (89), (90) and (93), concentration N1=N1min(T) at which JT reaches minimum is defined by equation

yf(y)+ξr1(2/s)×[s×f(y)r1(1/s)×y]×ln[Λ(y;ξ)]=1,E104

where

Λ(y;ξ)=B×f3s(y)y×[f(y)κ×y]2×[κ×sf1s(y)]×1ξ×r1/s×f(y)×(s4+sr×y)1+ξ×4r(1κ)×κ×s×[f(y)κ×y],

B=(m1*m2*)3/2×(Eg1Eg2)5/2×N2N˜1×(1κ)2r.E105

Expression (105) is valid when inequality WT2<W2 is fulfilled. This inequality and inequality (103) also are fulfilled at minimum of tunnel current in the most practically interesting cases. Below is explained difference between situations WT2>W2 and WT2<W2 at N1=N1min(T). Equation (104) can be solved by successive approximations using parameters of smallness ξ and 1/s.

As a result we find

N1min(T)=[ε1ε0×A1q×W1×(1ε2×a2ε1×a1)]ss1×y0×{1ξ×1κκ×r1/s×ln[Λ(y0;0)]×y0×(κ×s×y0+1)(s1)×κ×y0+1+0(ξ)},E106

where

y0=1+1κ×s2+0(1s2).E107

It is shown from (105) and (106) that N1min(T) is decreased with growth W1 and, also, although weakly, with increase N2.

When N1=N1min(T)then density of tunnel current

JT(N1)=JTmin=C0×ε1ε0×q32π4××Eg12×E1BD4(0,W1)N˜1(W1)×Λn1(y0;0)×exp[C1×a1E1BD(0,W1)]×[1+0(1)],E108

Where

C0=y03×y0×κ×(s1)+1(s×κ1)×y0+1×(κ×s)(4s)/(s1),C1=[y0×(κ×s)1/(s1)]1,

n1=y0×(1κ)(s1)×κ×y0+1.

From (94), (105) and (108) follow that JTmin decreases sharply with increasing W1. Value JTmin decreases also, although weakly, with increasing N2. Ratio

JTminJT(N1)|N1N˜1(T)[N2N˜1(W1)]n2×exp[(C1+κ1)×a1E1BD(0,W1)],E109

Where

n2=y0×(κ×s1)+1(s1)×κ×y0+1,

drops sharply, same as JTmin, with increase W1, but it increases with increasing N2. Value of this ratio is usually several orders of magnitude less than unity. For example, for combination of layers nwg:InP/nng:In0.53Ga0.47As, differential of currents, as can be shown, does not exceed values (N2/1018)0.9×2×104, where N2 is measured in cm-3.

When concentrations

N2<ε2×a2ε1×a1ε2×a2×W1W2×N1min(T)E110

then in minimum of JT(N1) takes place punch-through of narrow-gap layer, i.e. non-equilibrium SCR reaches nwg+ layer. When N1>N1min(T) then tunnel current increases with increasing N1, and at the same time, non-equilibrium SCR will penetrate into narrow-gap layer until concentration N1 reaches value

N1=N1pt=(A1×ε1ε0qW1)s/(s1)×[1+0(1)]>N1min(T)E111

Nature of above dependence JT on N1 is competition between tunnel currents in wide-gap and narrow-gap layers of heterostructure (Fig. 1a). When N1N˜1(T) then field E=EI(W1) in nwg layer at its heterojunction (Fig. 1b) coincide with very high accuracy with E1BD. Due to relatively large field E2=(ε1/ε2)×E1BD, current density JT is determined by tunneling of charge carriers in narrow-gap layer, i.e. JTJT2 (Fig. 1a). With increasing N1, field E2 and therefore current JT2 decrease due to fall EI(W1) (Fig. 18). Decrease EI(W1) with increase N1 is caused by requirement (1) of constancy of photocurrent gain Mph=Mр. Indeed, increase N1 for given Mph should lead to growth E1. Otherwise, due to growth |Е(х)| with increasing N1, field would be reduced everywhere in SCR, which in turn would lead to a decrease Mph. However, increase E1 should not be too large, and it should be such that Е(х) at х greater than some value in interval 0<x<W1 is decreased. In other words, Е(х) anywhere in SCR would increase, that, evidently, would increase Mph. It can be seen directly from (1) and (2). Note that for sufficiently large values of multiplication factors Mph, field E1 is practically independent on Mph and very close to breakdown field E1BD(N1,W1) when value of integral m (2) is equal to unity. This allows to use value E1=E1BD(N1,W1) (93) instead of true value Е1(N1,W1,Mph). When N1>N˜1(T), then variation of field Е(х) at distance W1 in nwg layer is still very insignificant, but it is enough to affect value JT2. Due to decrease E2 with growth N1 (especially when N1>N˜1), current is more and more determined by tunneling of charge carriers in nwg layer, therefore when N1>N1min(T), current density JTJT1 increases with increase N1 because E1BD grows with increase N1. Initial plateau (Fig. 18a) on the graph JT(N1) is caused by extremely weak dependences E1BD on N1 (93) and Е on х in nwg layer when N1<N˜1(T). Reducing of value JTmin (108) with growth N2 is due to increasing length of tunneling in narrow-gap nng layer (Fig. 1). Indeed, in this layer Е~N2<0, and E2 under these conditions does not depend on N2. It means, that Е(х) everywhere in nng layer, except of point x=W1, falls with increase N2 (1b). Since dEcdx=dEvdx=dφdx=E<0, then slopes of Ec(x) and Ev(x) everywhere in nng layer, except of point x=W1, decrease also with increasing N2, that leads to increase length of tunneling. Reducing of JT is more significant with growth N2 when N1<N1min(T) (Fig.18b), because current density JT2 increases with decrease N1 while JT1 decreases. When N1<N˜1(T) then current density JT1JT2, and if N1=N1min(T) it exceeds JT2. Therefore, ratio of JTmin to JT|N1<N˜1(T) (109) increases with increasing N2. Because at N1=N1min(T) value JT1>JT2, then, naturally, concentration N1min(T) (106) slightly decreases with increasing N2 (Fig. 18b). For small values N2, when WT2>W2, Е(х) in nng layer coincides with E2 with high accuracy. Therefore, length of tunneling in this layer, and hence JT also, do not depend on N2. Reducing of values N1min(T) (106) and JTmin (108) with increasing W1 (Figure 18a) is due to the fact that the more is W1 then the less is E1BD and the greater is fall of field Е(х) in depth of nwglayer.

Figure 18.

Dependence of tunnel current density on concentration N1 in case of independent doping levels of nwg:InP and nng:In0.53Ga0.47As layers at W2=2 μm. aN2=1014cm-3; W1, μm: 1 − 0.1, 2 − 0.2, 3 − 0.5, 4 − 1. b − neighborhood of value N1=N1min(T); W2=2 μm; N2, cm-3: 1 − 1014, 2 − 1015, 3 − 1016, 4 − 1017

II.

Condition (100) is not satisfied. For example, for combination of layers nwg:InP/nng:In0.53Ga0.47As such situation takes place when

W1W2×(1+N22.2×1015×W28/7)7/8<21.5,E112

where N2 and Wi are measured in cm-3 and μm, respectively. Under this condition, when N1<N¯1, where N¯1 satisfies equation

ε2ε1×A2×[N2+N˜2(W2)]1/s+q×N¯1×W1ε1ε0=A1×[N1+N˜1(W1)]1/s,E113

avalanche breakdown is controlled by nng layer, i.e. E2=E2BD(N2,W2), and E1<E1BD and it increases linearly with N1. Therefore, strictly speaking, when N1<N¯1 then tunnel current increases with increasing N1. At the same time, JT2 does not depend on N1 under following conditions.

  1. If

(W1/2W1/2*)1/(s1)>1s12s2,E114

then at N1<N¯1,JT2>>JT1 with margin of several orders of magnitude, and therefore with very high accuracy JT(N1)=const. If N1>N¯1 then due to decrease E2 and hence JT2 also, density of tunnel current JT(N1) begins drop sharply and, reaching minimum value (108) at concentration (106), then starts to grow again due to growth JT1(N1).

  1. If

(W1/2W1/2*)1/(s1)<<1s12s2,E115

then after initial plateau JT(N1) grows monotonically. It is due to monotonic increase in component of tunnel current density JT(N1), which at N1N¯1 is considerably superior to JT2.

  1. If

(W1/2W1/2*)1/(s1)1s12s2,E116

then for small enough thicknesses W1 of layer nwg dependence JT(N1) has distinct maximum at N1=N¯1, however, at least in this case minimum is not deep. This is due to the fact that components of tunnel current density JT1 and JT2 are equal to each other in order of magnitude at small enough W1. Characteristics of tunnel currents in heterostructure with independent doping of nwg and nng layers are illustrated in Fig. 18. Note that if in case I increase N2 leads to decrease JT at all values N1, then in case II, increase N2, when N1 is small enough, leads to increase of tunnel current, but at sufficiently large N1 tunnel current decreases, particularly, in the vicinity of concentration N1=N1min(T).

4.2.2. Equal doping levels of wide-gap and narrow-gap n type layers

Under this condition density of tunnel current is given by expression (89), where N1=N2=N

i.

W1W2(ε1×A1ε2×A2)sE117

At this relation of parameters avalanche breakdown is controlled by nwg layer, i.e. E1=E1BD(N1,W1), and E2<E2BD(N2,W2) regardless of doping. Dependence JT on N has identical character with JT(N1)|N2=const in the case of 4.2.1. I, and is caused by the same physical grounds. The only difference is that when N<N2, then curves JT(N) lie higher on plotting area, and when N>N2 – lower, than curves JT(N1)|N2=const in the case of 4.2.1. I.

This occurs because at given value E2 length of tunneling in narrow-gap layer is the greater the higher is level of doping of this layer.

ii. Condition (117) is not satisfied.

Then, till N<N¯, (where N¯ is determined by equation (113), where N¯1=N2=N¯) avalanche breakdown is controlled by nng layer, i.e. E2=E2BD(N,W2), and E1<E1BD(N,W1) and increases linearly with N. Dependence JT(N) has, in contrast to situation 4.2.1, not only deep minimum, but high maximum also (Fig. 19a). This is due to the fact that when N<N¯ then E1 grows and E2 grows also reaching at N=N¯ maximal value (Fig. 19b). As a result, when N<N¯ then JT1 grows with increase N and JT2 grows also. Note that when doping of nwg and nng layers are equal then concentration N=Nmin(T), at which tunnel current density JT has minimal value, is determined by formula (106) with accuracy up to small corrections of order ξ=E1BD(0,W1)/a1<<1, as in the case of independent doping of nwg and nng layers. Formula for JTminmay be obtained from expression (108), if we replace N2 by Nmin(T) in it.

Figure 19.

Dependences of tunnel current density (a) and fields Е (b) on dopant concentration N in case of equal doping levels of nwg:InP and nng:In0.53Ga0.47As layers, at W2=2 μm. W1, μm: 1 – 10, 2 – 1, 3 – 0.1, Curves 1', 2', 3' – Е2(N); curve 4 − Е1(N), weakly dependent on W1

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5. Basic performance of avalanche heterophotodiode

5.1. Responsivity

In punch-through conditions of absorber nng, current responsivity SI(λ) of heterostructure under study can be described by relation (4). In calculating quantum efficiency η of heterostructure, we take into account that optical radiation is not absorbed in its wide-gap layers. Let’s assume that light beam falls perpendicularly to front surface of heterostructure (Fig. 1), and absorption coefficient in narrow-gap layer γ(λ) does not depend on electric field. Quantum efficiency is ratio of number of electron-hole pairs generated in sample by absorbed photons per unit time to incident flux of photons.

Therefore, (Fig. 20a)

η=(1R1)×(1R2)1R1×R2×η1,E118

where reflection coefficient of light from illuminated surface R1= (εexε1)2/(εex+ε1)2 and from interfaces of heterostructure R2= (ε2ε1)2/(ε2+ε1)2; εex– relative dielectric constant of environment; and quantum efficiency η1 with respect to light ray which has penetrated into narrow-gap layer is written

η1=1ζ+η2×ζ;E119

quantum efficiency η2 with respect to light ray which has reached to second interface of heterostructure,

η2=R2(1ζ)+R22ζ(1ζ)+η2(ζR2)2+R1R2(1R2)2ζ1R1R2(1ζ+η2ζ)+(1R2)2R31R2R3××[(1ζ)(1+R2ζ)+η2R2ζ2+R1(1R2)2ζ1R1R2(1ζ+η2ζ)],E120

ζ=exp(γW2), R3 – reflection coefficient of light from not illuminated (backside) surface. From expressions (118)-(120) follow, that

η(γW2)=η()×[1exp(γW2)]×1+R23exp(γW2)1R12R23exp(2γW2)E121

where

η()=(1R1)(1R2)1R1R2,E122
Rij=Ri(1Rj)+Rj(1Ri)1RiRj,i,j=1,2,3.E123

Particularly,

η(γW2)={η()1exp(γW2)1R12exp(γW2),atR3=R1,η()1exp(2γW2)1R12exp(2γW2),atR3=1.E124

Dependence η on W2 for heterostructure InP/In0.53Ga0.47As/InP is shown in Fig. 20b. It should be noted that since in operation, electric field is high even in absorption layer, then, due to Franz-Keldysh effect, quantum efficiency is slightly higher than given in Fig. 20b. This is especially true when absorbing layer W2 is very thin.

Figure 20.

Layout view of multiple internal reflections and absorptions of light beam in heterostructure (a) and dependence of quantum efficiency η of structure InP/In0.53Ga0.47As/InP on absorption layer thickness W2, μm (b): 1 − R3=R1 2 − R3=1. It is assumed that relative dielectric permittivity of environment εex=1

5.2. Noise

It was noted above that in order to achieve the best performance of SAM-APD special doping profile is formed in heterostructure which facilitates penetration of photogenerated charge carriers with higher impact ionization coefficient into multiplication layer. In this case, at given voltage bias on heterostructure, current responsivity SI(λ) is maximal, and effective noise factor Fef,i(Mph) is minimal (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (McIntyre 1966), and hence, as it is evident from expression (5), noise spectral density SN is also minimal. If α=β, then (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (McIntyre 1966) Fef(Mph)=Mph, and therefore

SN=2q×A×JT×Mph3.E125

In InP ratio K(E)=β/α in interval of fields of interest E=(3.3÷7.7)×105 V / cm varies from 2.3 to 1.4 (Tsang, 1985), (Filachev et al, 2011), (Cook et al, 1982). Therefore, noise spectral density of heterostructure with InP multiplication layer and optimal doping is slightly less than value given by formula (125). When N1>N¯1, (where N¯1 satisfies equation (113) (see Fig. 21), in which N˜i(Wi) is defined by formula (94) for i=1,2) then avalanche multiplication of charge carriers in narrow-gap layer does not occur. Under these conditions, field value at metallurgical boundary of p+n junction (х=0, Fig. 1) equals to E1=E1B(N1,W1) (see (93) and (94)). For many semiconductors (see Sections 3.1-3.2) including InхGa1хAsyP1y, values s and Ai are defined by relations (95). In the case of heterostructure InP/In0.53Ga0.47As/InP, in first approximation in parameters of smallness

δ1E1BD(0,W1)a1=2.786×102W11/7,δ2=1s2=164E126

we find that value of concentration N1=N1min(T) at which function JT(N1) reaches its minimum

JTmin(W1,N2)=2.19×108×W10,49N20,07×exp(27.88×W11/7), A/cm2,E127

is given by

N1min(T)(W1,N2)=2.33×1016W18/7×[12.52×102W11/7×(lnN2×W18/73.69×10151.41)], cm-3E128

Formulas (127) and (128) are valid when WT2W2, i.e., as follows from Section 4.2.1, when

N2×W2Q(W1)=5W12/7×1014,E129

where concentration and thicknesses, as in (127) and (128), are measured in cm-3 and µm, respectively.

If inequality (129) is not satisfied, then values N1min(T) and JTmin will be again determined by (127) and (128), in which N2 is replaced by Q(W1)/W2. It is shown from (127) and (128) that N1min(T) and JTmin are decreasing, moreover JTmin sharply, with increase W1 (see Fig. 21, 22), and, also, although weakly, with increase N2. Decrease of values N1min(T) and JTmin with increase W1 is caused by situation when the thicker W1 the less E1BD and the greater fall of field E(x) on nwg layer thickness. Slight decrease N1min(T) and JTmin with growth N2 is due to increasing of length of interband tunneling lTng in narrow-gap nng layer with increase N2 and the fact that at minimum JT1>JT2. For small values either N2 or W2, field E(x) is so weakly dependent on x in nng layer, that value lTng in it is almost constant. Therefore, when N2W2<Q(W1) then values N1min(T) and JTmin do no longer depend on N2 and slightly decrease with increase W2 due to reducing the length of tunneling generation region in narrow-gap material. In high performance diode, absorber should be punched-through when voltage bias Vb on heterostructure is less than voltage of avalanche breakdown VBD. This eliminates dark diffusion current from narrow-gap layer and increases operational speed. Condition of punch-through of absorber, as follows from 4.1 and 4.2 is given by:

N1×W1+N2×W2<ε0ε1q×E1BD(N1,W1).E130

Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when R1=R3 quantum efficiency reaches almost its maximal value when W2=2 μm. Therefore, for development of concentration – thickness nomogram in Fig. 21, namely this value W2 was selected. Note that decrease in dispersion in N2 results in increase in dispersion N1 and W1, while increase gives the opposite result. Value of noise current density IN1012A/Hz1/2 corresponds to JT1.8×105А/сm2, and value IN1013 А/ Hz1/2 corresponds to JT1.8×107А/сm2.

Figure 21.

Concentration-thickness nomogram for avalanche InP/In0.53Ga0.47As/InP heterophotodiode when N2=(1÷5)×1015cm-3, W2=2 μm, Мph=15, cross-section area A=5×103 μm2. When noise current IN=SN1013 A/Hz1/2, then allowable set of points in space (N1,W1) lies inside figure a-b-c-d; when IN=SN1012 A/Hz1/2 − inside figure a-e-f-g. Dashed and dash-dot curves − dependences N1min(T)(W1) and N¯1(W1), respectively: 1 − N2=1015cm-3, 2 − N2=5×1015cm-3. N1 is measured in units of 1016cm-3, W1 − in μm

Figure 22.

Dependences of minimal tunnel current JTmin, A/cm2 of avalanche heterophotodiode InP/In0.53Ga0.47As/InP on multiplication layer thickness W1, μm: 1 − N2=1013cm-3, 2 − N2=1015cm-3, 3 − N2=1017cm-3

5.3. Operational speed

Minimal possible time-of-response of this class of devices

τ=2×(τtr1×f(Mph)+τtr2)E131

is determined by time-of-flight of charge carriers through multiplication layer τtr1 and absorber τtr2, and also by value of function f(Mph), which is close to 1 when K>>1, and is equal to Mph when K=1 (Tsang, 1985), (Filachev et al, 2011), (Emmons, 1967), (Kurochkin & Kholodnov, 1996). It was noted above that in InP 1<K2.3. Therefore, in InP/InxGa1xAsyP1y/InP SAM-APD

τ2×(τtr1×Mph+τtr2).E132

As is evident from Fig. 20b, in InP/In0.53Ga0.47As/InP heterostructure quantum efficiency value η lies in interval 0.5η0.686 when R3=1 and W20.5 μm. It means that, because of not so much loss in quantum efficiency η compared to maximal possible (only 27 % less), time-of-response value τtr2=5 ps can be achieved by forming absorber with thickness W2=0.5 μm and fully reflecting backside surface. Minimal value τtr1 is determined by maximum allowable minimal value W1min. When JT106 A/cm2, then as follows from Fig. 22, W1min2 μm, and therefore τmin(4Mph+1)×102 ns.

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6. Analytical model of avalanche photodiodes operation in Geiger mode

We consider possibility to describe transient phenomena in pin APDs by elementary functions, first of all, when initially applied voltage V0 is greater than avalanche breakdown voltage VBD. Formulation of the problem is caused by need to know specific conditions of APDs operation in Geiger mode. Simple expression describing dynamics of avalanche Geiger process is derived. Formula for total time of Geiger process is obtained. Explicit analytical expression for realization of Geiger mode is presented. Applicability of obtained results is defined. APDs in Geiger mode (pulsed photoelectric signals) make possible detection of single photons (Groves et al, 2005), (Spinelli & Lacaita, 1997), (Zheleznykh et al, 2011), (Stoppa et al, 2005), (Gulakov et al, 2007). It is worked at reverse bias voltages Vb>VBD. Different types of devices are realized on APDs in Geiger mode (Groves et al, 2005), (Spinelli & Lacaita, 1997), (Zheleznykh et al, 2011), (Stoppa et al, 2005), (Gulakov et al, 2007). At the same time, review of publications shows that theoretical studies have tendency to carry out increasingly sophisticated numerical simulations. In (Vanyushin et al, 2007) was proposed discrete model of Geiger avalanche process in pin structure. Obtained iterative relations allow to determine, although fairly easy, but only by numerical method, options for realization of Geiger mode when ratio Kβ/α differs very much from unity, where α(EiGE) and β(EiGE) – impact ionization coefficients of electrons and holes and EiGE – electric field in i– layer (base 0<x<WiGE, Fig. 23). "Continuous" model (Kholodnov, 2009) developed in this section admits value К=1. Considered below approach allows also to describe conditions of realization of Geiger mode and its characteristics by mathematically simple, graphically illustrative relations. It is adopted that photogeneration (PhG) is uniform over sample cross-section area S transverse to axis x (Fig. 23). Then, in the most important single-photon process, area S, according to uncertainty principle, shall not exceed in the order of magnitude, square of wavelength of light λ. Under these conditions, it is allowably to consider problem as one-dimensional (axis x, Fig. 23). There are grounds to suppose that go beyond one-dimensional model at local illumination make no sense. Single-photon case arises itself when S>>S1π×λ2. The matter is that charge, during Geiger avalanche process, as show estimates below, has no time to spread significantly over cross section area. Consider serial circuit: pin diode – load resistance R – power supply source providing bias Vb>VBD. Let p and n regions are heavily doped, so that prevailing share of bias falls across base i. Then after charging process voltage on it can be considered equals to V0=Vb. When electron-hole pairs appear in the base then occurs their multiplication that results in decrease Vi due to screening of field EiGE in base by major charge carriers inflowing into p and n regions (Fig. 23) in quantity Nn and Pp and voltage drop across load resistor VR and, hence, current in external circuit arise

IR=VRRVbViR.E133

In present structure charge is mainly concentrated in thin near border ni and pi layers (let's call them plates, Fig. 23). Therefore, as in (Vanyushin et al, 2007), field EiGE will be assumed uniform. Numerical value EiGE=EBD when Vi=VBD for a number of materials can be quickly determined by formulas given in Section 3. As in (Vanyushin et al, 2007), we restrict consideration by PhG in base only, we neglect recombination in it, and we assume that currents of electrons IN and holes IP are determined by their drift in electric field with velocity of saturation vs, i.e.,

IN(x,t)=q×vs×N(x,t),IP(x,t)I(x,t)IN(x,t)=q×vs×P(x,t),E134

where N and P – linear density (per unit length) of electrons and holes, I – full conductive current, q – absolute value of electron charge, t - time.

Figure 23.

Avalanche process inpin structure: “-“ − acceptors charge in boundary ip layer (cathode plate − Cathode); “+” − donors charge in boundary in layer (anode plate − Anode); ⊖ and ⊕ − generated in i − region avalanche photoelectrons and photoholes; Nn and Pp − inflowing in n − and p − regions avalanche photoelectrons and photoholes; Ес and Еv – energy of conduction band bottom and valence band top; hν − photon energy

Substituting volume charge density from Poisson equation in continuity equation for I and integrating over depletion layer (DL) we obtain that, in approximation of zero-bias current, in quasi-neutral parts of structure

IR=Cd×Vdt+<Id>,<Id>=1WiGE×0WdI(x,t)dxE135

where Vd – voltage on DL, Cd=εε0×S/Wd and Wd – DL capacity and thickness, ε0

dielectric constant of vacuum, ε – dielectric permittivity, <Id> let’s call avalanche current Iav.

Relation (135) generalizes well-known theorem of Rameau (Spinelli & Lacaita, 1997), it takes into account key feature of Geiger mode – variation over time of voltage across DL, and it is valid for any distribution profile of dopant. In our formulation of the problem (in pin structure) i – layer can be considered as DL, i.e., d in (135) and below should be replaced by i. By integrating continuity equation for IN and IP with respect to x from 0 to WiGE and marking linear density of photogeneration rate as G(x,t) we obtain equations

q×Ni(t)t=α×I˜N(t)+β×I˜P(t)+IN(WiGE,t)IN(0,t)+q×G˜(t),E136
q×Pi(t)t=α×I˜N(t)+β×I˜P(t)IP(WiGE,t)+IP(0,t)+q×G˜(t),E137
Ni=0WiGEN(x,t)dx,Pi=0WiGEP(x,t)dx,I˜N,P(t)=0WiGEIN,P(x,t)dx,G˜(t)=0WiGEG(x,t)dxE138

Because plates are very thin, then generation and recombination in them can be neglected. Now by integrating same equations with respect to thickness of plates, we find that in approximation of absence of minority carriers in p and n regions

IN(0,t)=IR+q×Nnt=IRCi×Vit=IP(WiGE,t),IP(0,t)=IN(WiGE,t)=0E139

Strictly speaking, equations (139) are valid when r1Pp/Nn=1, from which r2|PiNi|/Nn=0. Therefore, let’s assume uniform PhG along x. Then, at К=1, symmetry requires r1=1. Equations (139) are correct in concern of the order of magnitude both when К is not too big and when small also. This follows from quasi-discrete computer iterations in uniform static field. Computer iterations are performed in several evenly spaced points of PhG xg succeeded by averaging with respect to xg and take into account much more number acts of impact ionization by holes than similar iterations in (Vanyushin et al, 2007). Iteration procedure performed in interval equals to several time-of-flight of charge carriers through base ttr gives 0.6 <r1< 1, and r2< 0.4 (Fig. 24a), which corresponds to approximation of uniform field. Note that smallness r2 does not mean smallness Pi+Ni (curve 3 in Fig. 24a).

Figure 24.

Evaluation of applicability of quasi-uniform field approximation. (a) − Results of quasi-discrete computer iterative procedure rj(K): r1=Pp/Nn, r2=|PiNi|/Nn, r3=(Pp+Nn)/(Pi+Ni). (b) − Dependence of error ER during determination of breakdown field on K=β/α; accepted (Tsang, 1985), (Grekhov & Serezhkin, 1980) α(E)=AGE×exp(B/E), where AGE, 1/μm: 1 − 200, 2 − 400, 3 − 800, 4 − 2000, 5 − 5000

Relations (133)-(139) allow obtaining equations

F[VR;(1/τi)]2VRt2+{1τivs×Y[EiGE(VR)]}×VRtvsτi×Y[EiGE(VR)]×VR=q×G˜(t)×2×vsCi×WiGE,E140

with initial conditions

VR(0)=0,VRt|t=0=2vsCi×WiGE×limt0q×ttG˜(t')dt'E141

where

Y(EiGE)=Χ(EiGE)(2/WiGE),X=α(EiGE)+β(EiGE),EiGE=(VbVR)/WiGE,τi=RCi,Ci=εε0×(S/WiGE)E142

At delta-shaped time-evolving illumination G˜i(t)=Nph×δ(t) relations (140) and (141) are converted into

F[VR;(1/τi)]=0,VR(0)=0,VRt|t=0=AGEq×2vs×Nphεε0×SE143

where Nph – number of absorbed photons.

If we take R=0 and limtG˜i(t)=const0 then we find that breakdown is determined by condition WiGE×Χ(EiGE)=2, which at К1 gives another value for breakdown field EiGE=Eav than EiGE=EBD obtained directly from solving of stationary problem in Section 3. However, discrepancy between Eav and EBD is no more than 20 %, if К is different from 1 by no more than two orders of magnitude (Fig. 24b). Equation (140) admits only numerical solution. However, Geiger mode can be described without solving this equation, by using physical grounds and limit R, when

Iav=Ci×ΔVit,F[ΔV;0]=0,EiGE=Vi/WiGE[VbΔVi(t)]/WiGE,ΔVi(0)=0,ΔVit|t=0=AGEE144

and problem is solved in quadratures. To solve in elementary functions let’s approximate exact dependence Y[EiGE(ΔVi)] by piecewise-linear function passing through principal point ΔVi=DavVbVav=VbEav×WiGE (Fig. 25 and 26), where Y=0, and Iav reaches its peak during tav.

Figure 25.

Form of approximation of function Y(ΔV). Dependences (1 - exact, 2 - approximate) are plotted for Ge with orientation <100> (Tsang, 1985) taken WiGE= 1 μm, Dav=4 V

Figure 26.

Ratio of approximate dependence Y˜(ΔV) to exact Y(ΔV) for Ge with orientation <100>; δ=ΔV/(ΔV)max; WiGE=1 μm, WiGE=2 μm; Dav, V: 1 − 0.25, 2 − 0.5; 3 − 1

Suppose, for simplicity X(Eb)4/WiGE, where Eb=Vb/WiGE. Then ΔVimaxlimtΔVi(t) is not more than value of break point ΔVk of piecewise-linear approximation (Fig. 25). Under these conditions

ΔVi(t)=ΔVimax×Zt/tav1Zt/tav+Z,ΔVimax=2Dav,tav=lnZvs×Yb,Zεε0×S×Yb×Davq×Nph>>40,E145

where Yb=Yb(Eb). Geiger mode occurs when during time R×Ci of inverse recharge of avalanche diode, avalanche is able to develop and cancel itself in full. As seen from (145) it is happened when RRmintav/Ci. Maximal voltage drop on load equals to VRmax=ΔVimax. Since ttr<<tav, then results of computer evaluation of uniform field approximation applicability can be considered reasonable. To evaluate transverse charge spreading let’s use expression (21) from (Pospelov et al, 1974). It determines dependence χ(t)r/r0, where r(t) and r0 – current and initial radii of charge "drop" of parabolic type. Implying under capacity in (Pospelov et al, 1974) value Ci and putting WiGE=1 μm, r0=λ=1 μm, in the case of single-photon process we get χ(tav)<˜21/41.2. This justifies our assumption that charge spreading over sample cross-section during avalanche Geiger process is not intensive.

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7. Conclusions

The above analysis shows that to create high performance SAM-APD (in particular, based on widely used InP/InxGa1xAsyP1y/InP heterostructures) it is necessary to maintain close tolerances on dopants concentration in wide-gap multiplication layer I – N1 and in narrow-gap absorption layer II – N2, and also on thickness W1 of wide-gap multiplication layer (Fig.1). This is due to strong dependence of interband tunnel current in such heterostructures on N1,N2 and W1. Allowable variation intervals of values N1,N2 and W1, and, optimal thickness of absorber also, can be determined using results obtained in Sections 4 and 5. Value of minimal possible time-of-response τmin depends not only on photocurrent’s gain Мph but on allowable noise density at preset value of photocurrent’s gain also. The lower noise density, the larger is value τmin. For example, for heterostructure InP/In0.53Ga0.47As/InP minimal time-of-response equals to τmin0.6 ns, when noise current equals to 3.3×1011 А/Hz1/2 and current responsivity 10.3 A/W. Analysis shows that operational speed can be slightly increased by means of inhomogeneous doping of wide-gap multiplication layer. To ensure operational speed in picosecond range it is necessary to use as multiplication layer semiconductor layer with low tunnel current and impact ionization coefficients of electrons and holes much different from each other, for example, indirect-gap semiconductor silicon. As has long been known maximal operational speed is achieved by APD if light is absorbed in space-charge region. In this case, as it was shown in Section 6, when bias voltage Vb exceeds breakdown voltage VBD of no more than a few volts, then, for Kβ/α values lying in interval from a few hundredths to a few tens, elementary relations (145) can be used for approximate description of Geiger mode in pin APD. Moreover if cross-section area S>S1π×λ2, then we can expect that in single-photon case under S in (145) should imply value of order S1. This is due to finite size of single-photon spot S1 and not intensive spreading of charge during time of avalanche Geiger process tav when photogeneration of charge carriers occurs in i – region of pin structure depleted by charge carriers. Proposed approach allows describing Geiger mode by elementary functions at voltages higher Vb as well. Note that equation (140) and physical grownds allow to expect three possible process modes at pulse illumination under Vb>VBD. When RC<<tav then generated photocurrent will tend to reach some constant and flow indefinitely (unless, of course, ignore energy losses). When RC=tav then generated photocurrent will be of infinitely long oscillatory character. When RC>>tav then Geiger mode is realized.

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Written By

Viacheslav Kholodnov and Mikhail Nikitin

Submitted: 16 February 2012 Published: 19 December 2012