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

Minimal value of dark current in reverse biased p −n junctions at avalanche breakdown is determined by interband tunneling. For example, tunnel component of dark current be‐ comes dominant in reverse biased p −n junctions formed in a number semiconductor ma‐ terials with relatively wide gap Eg already at room temperature when bias V b is close to avalanche breakdown voltage V BD (Sze, 1981), (Tsang, 1981). The above statement is ap‐ plicable, for example, to p −n junctions formed in semiconductor structures based on ter‐ nary alloy I n0.53Ga0.47As 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 photo‐ diode (APD) formed on multilayer heterostructure (Fig. 1) with built-in p −n junction when metallurgical boundary of p −n junction (x =0) lies in wide-gap layer of heterostruc‐ ture (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-

(where N is dopant concentration). To eliminate it one side of p − n junction is doped heavily and narrow-gap layer is grown on wide-gap isotype heavily doped substrate (Tsang, 1981). Thus heterostructure like as p wg + − n wg − n ng − n wg + 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 W 2 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 W 1 and dopant concentrations in wide-gap N 1 and narrow-gap N 2 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 I n 0.53 Ga 0. 47 As 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 p wg + − n wg junction E (0) ≡ E 1 is higher than 4.5×10 5 V/cm then avalanche multiplication of charge carriers occurs in InP layer where p wg + − n wg junction lies at any dopants concentrations and thicknesses of heterostructure's layers. However, electric field E 1 = E 1BD at which avalanche breakdown of p − n junction occurs depends on both doping and thicknesses of layers (Sze, 1981), (Tsang, 1981), (Osipov & Kholodnov, 1987), , (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 E 1 =4.5×10 5 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 E 1BD (Sze, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholod-nov, 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 heterointerface with multiplication layer E 2 (see Fig. 1b) is equal to 1.5×10 5 V/cm. But tunnel current density J T in narrow-gap absorber I n 0.53 Ga 0. 47 As (Osipov & Kholodnov, 1989) is much smaller at that value of electric field than density of thermal generation current J G which in the best samples of InP − I n 0.53 Ga 0. 47 As − InP heterostructures (Tsang, 1981), (Tarof et al, 1990), (Braer et al, 1990) can be up to 10 -6 A/cm 2 . 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 N 1 in n wg wide-gap and N 2 in n ng narrow-gap layers of heterostructure and thicknesses W 1 and W 2 , respectively ( Fig. 1) and fundamental parameters of semiconductor materials also. Performance limits of AHPDs are analyzed . 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 InP − I n 0.53 Ga 0. 47 As − InP are performed. The following values of fundamental parameters of InР (I, Fig. 1) and I n 0.53 Ga 0.47 As (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 E g 1 = 1.35 eV and E g 2 = 0.73 eV; intrinsic charge carriers concentrations n i (1) =10 8 сm -3 and n i (2) =5.4×10 11 (Sze, 1981), (Tsang, 1981), (Miller, 1955) which describes dependence of charge carriers' avalanche multiplication factors on applied bias voltage V b . It is shown in final section that Geiger mode (Groves et al, 2005) of APD operation can be described by elementary functions (Kholodnov, 2009).

Formulation of the problem: Basic relations
Let's consider p wg + − n wg − n ng − n wg + heterostructure at reverse bias V b 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 can be determined, in principal, dependences of multiplication factors M in p − n structures on V b , where M n and M p -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 M n and M р ; specific rate of charge carriers' generation in SCR g = g d + g ph con- sists of dark g d and photogenerated g ph components; L p and L n -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 N 1 pt dopant concentration N 1 so that for N 1 < N 1 pt "punch-through" (depletion) of n wg layer occurs that means penetration of non-equilibrium SCR into n ng layer (Fig. 1). Optical radiation passing through wide-gap window is absorbed in n ng layer and generates electron-holes pairs in it. When N 1 < N 1 pt then photo-holes appearing near n wg /n ng heterojunction (х = W 1 ) are heated in electric field of non-equilibrium SCR and, at moderate discontinuities in valence band top E v at х = W 1 , photo-holes penetrate into n wg layer (layer I) due to emission and tunneling. If W 1 is larger than some value W 1min (N 1 , N 2 , W 2 ) (Osipov & Kholodnov, 1989), which is calculated below, then avalanche multiplication of charge carriers occurs only in n wg layer, i.e. photo-holes fly through whole region of multiplication. In this case photocurrent's gain (Tsang, 1981), (Artsis & Kholodnov, 1984) M ph =M р . Let p wg + 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 L p = 0 and L n = W 1 , i.e. 1 1 It is remarkable that responsivity S I (λ) (where λ -is wavelength) of heterostructure increases dramatically once SCR reaches absorber n ng (layer II on Fig. 1) and then depends weakly on bias V b till avalanche breakdown voltage value V BD (Stillman, 1981). This effect is caused by potential barrier for photo-holes on n wg /n ng 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 S I (λ) in operation mode is determined by well-known expression (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2011): (4) 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 М ph 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 М ph becomes higher then certain value М ph opt 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 М ph 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 1999-2). Using results obtained in (Artsis & Kholodnov, 1984), (McIntyre, 1966), noise spectral density S N of p wg + − n wg − n ng − n wg + heterostructure which performance is limited by tunnel current can be written as: where q -electron charge; А S -cross-section area of APD's structure; F ef ,i (M ph ) -effective noise factors (Artsis & Kholodnov, 1984) in wide-gap multiplication layer (i = 1) and in absorber (i = 2); J T ,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 М ph . Expression (5) shows, that for preset gain of photocurrent, noise density is determined by values of pri-mary tunnel currents I T 1 = J T 1 × A S and I T 2 = J T 2 × A S (total primary tunnel current I T = I T 1 + I T 2 ). 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: U − (x) and U + (x) -asymmetric unit stepwise functions (Korn G. & Korn T., 2000), ε 0 -dielectric constant of vacuum, ε 1 and ε 2 -relative dielectric permittivity of n wg and n ng layers (Fig. 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 p − n structures on applied bias V b . We need to know among them dependence of avalanche breakdown voltage V BD on parameters of p − n structure and distribution of electric field E (x) related to V BD 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 V BD 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 = V BD is satisfied can be calculated by method of successive approximations on computer.
Thus, this method of determining V BD and, hence, E (x) at V = V BD 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 V BD 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, 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): Gap E g 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 ε 0 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. ; N 1 − dopant concentration n wg in wide-gap layer I Formulas (10) and (11) cannot be used for reliable estimates of V BD and E BD in semiconductor structures with thin enough base. Indeed, dependence of V BD 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 E 1 = E (0) (Fig. 1b). Second, value of electric field E 1 = E (0) at V = V BD 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 E 1 . 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 V BD and field E BD are independent on N and at the same time are dependent on W , moreover, the thinner W then, evidently, the higher E BD . 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: It seen from expression (14) that at N < Ñ (W ) electric field of avalanche breakdown E BD is practically independent on dopant concentration N in diode's base.

Avalanche breakdown field
Consider p wg + − n wg − n ng − n wg + heterostructure (Fig. 2). Symbols n wg and n ng indicate to unequal, in general, doping of high-resistivity layers of structure. Denote as W 1 , W 2 and N 1 , N 2 thicknesses of n wg and n ng layers and dopant concentrations in them, properly. Case W 2 = 0 corresponds to diode formed on homogeneous p + − n − n + structure. Let values N 1 and W 1 such that upon applying avalanche breakdown voltage V BD to structure, SCR penetrates into narrow-gap n ng layer (Fig. 2). When W 1 and N 1 , N 2 are small enough and W 2 is thick enough then avalanche process develops in n ng layer. In other words, with increasing bias V b applied to heterostructure, electric field E = E 2 in narrow-gap layer on n wg /n ng heterojunction ( Fig. 2) reaches avalanche breakdown field E 2BD in this layer earlier than electric field E 1 on metallurgical boundary (х = 0) of p + − n junction becomes equal to breakdown field E 1BD in wide-gap n wg layer. This is due to the fact that at small values of W 1 and N 1 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, W 1 and N 1 , N 2 are large enough and W 2 thin enough, then avalanche process is developed in wide-gap n wg layer only. For these values of thicknesses and concentrations electric field E 1 reaches value E 1BD earlier than E 2 -value E 2BD . Because of significant decreasing of electric field E (х) in n wg layer with increasing distance from х = 0, field E 2 remains smaller E 2BD despite the fact that band-gap E g 1 in n wg layer is wider than band-gap E g 2 in n ng layer. Distribution of electric field E (x) in n wg and n ng layers of considered heterostructure is obtained by solving Poisson equation as defined by (6) (12) should be modified so that when N → 0 then breakdown field E BD 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 n wg layer, i.e. E 1 is close to E 1ВD and multiplication factor of holes М р (1) is fixed.
Then, with increasing concentration N 1 , field E I (W 1 ) (Fig. 2b) shall be monotonically falling function of N 1 . Indeed, with increasing N 1 , field E 1ВD and | ∇ E I (x) | are increasing also. Increasing | ∇ E I | must be such that when x became larger some value x then value E I (x) has decreased (Fig. 2b). Otherwise, field E (x) would increase throughout SCR that reasonably would lead to growth of М р . This is evident from (1) and (2). On the other hand, adding constant to right side of expression (12) does not change ∂ E 1BD / ∂ N 1 and therefore results in, as follows from (6) and (9), non-monotonic dependence E I (W 1 ) on N 1 . Equation (14) which can be rewritten for each of n wg and n ng layers as: does not lead to that and other contradictions, From (17) follows that: To determine dependences E iBD (0, W i ), let's consider behavior of E I (W 1 ) when parameters of heterostructure N 1 , N 2 and W 2 are varying. From (6)- (9), (17) and (18) we find that when then avalanche breakdown is controlled by n wg layer. It means that If, however, Δ < 0 then avalanche breakdown is controlled by n wg /n ng heterojunction, i.e.
From (17) , at 0 Formulas (15) and (16) follow from expressions (18), (19) and requirement (23) 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 where 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: X ε = 12.4 / ε, X g = 1.35 / E g and gap E g in diode's base is measured in eV and its thickness W -in μm, respectively.

Avalanche breakdown voltage
It follows from expressions (6)- (9) and (14)-(16) that breakdown voltage V BD for p + − n − n + structure is given by expressions i.e. when diode's base is not punch-through and i.e. when diode base is punch-through. In expression (28) Value of parameter θ is defined from equation θ = s × (1 + θ) 1/s and with good degree of accuracy it equals to s s/(s−1) . Because θ > > 1, therefore expression (27) practically coincides with formula (10), i.e. V ВD 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 In expressions (30)-(32) X ε = 12.4 / ε, X g = 1.35 / E g and gap E g 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 n wg and n ng 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.
To derive relation (33) let's consider thin p + − n − n + structure in which thickness of high-resistivity base layer W satisfies to inequality where ε 0 -dielectric constant of vacuum; ε -relative dielectric permittivity of base material; q -electron charge; s and A -constants defining dependence of electric field E BD ≈ A × N 1/s at metallurgical boundary (x = 0) of abrupt p + − n junction on dopant concentration N in base for avalanche breakdown in thick p + − n − n + 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 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), , ( In this case as it follows from (34) and (35) Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 And avalanche breakdown electric field for thin p + − n − n + structure is defined by approximate universal formula 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 + − n − n + structure (Sections 2, 3.1 and (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977)) takes the form 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 4 ( ) 10 where λ R , Е ion , Е R -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 7 6 3 7 /6 14 min max 7 3 5 10 10 , From (38) and (41) we find desired interval of electric field: 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 W min / λ R is usually not more than a few units. Therefore, when W < W min then E BD ≈ E ion W × 10 4 and hence when E > E max instead of (33) must be valid relation where Е ion 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 p − n structures.

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 p − n structures on applied voltage V b 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 − L p < x < L n (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 V b . In article (Sze & Gibbons, 1966) was proposed analytical expressions for avalanche breakdown voltage V BD , i.e. applied voltage value at which M = ∞, in asymmetric abrupt and linear p − n junctions. Expression for V BD (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 p − i − n) as discussed in Section 3.3. Using as model abrupt (stepwise) p − n 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. Rewrite (33) in the form 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 E g and threshold ionization energy Е ion in eV; electric field E in V/cm; bias V b in V; multiplication factors α and β in cm -1 , electron charge q in C; dielectric constant of vacuum ε 0 in F/m; concentration including shallow donors N D and acceptors N A in cm -3 ; concentration gradient a in cm -4 ; width of SCR L р and L n 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 p − n structures have been calculated under no K (E) = const condition. Such calculations are varies, typically, much slighter than E 7 . 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) p − n junctions like as in model given in (Sze, 1987), (Sze & Gibbons, 1966) and thin p + − n(p) − n + structure (like as p − i − n) 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 where v = V / V BD . This form was first proposed by Miller in 1955 (Miller, 1955) and then, despite lack of analytical expressions for exponents n n , n p , ñ , 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

Gradual (linear) p − n junction
In this case Poisson equation can be written as (SI units): where σ-slope of linear concentration profile and therefore 5 4 In derivation of relations (55) and (56) was used known expression for voltage distribution on linear p − n junction (Sze, 1981) and was also taken into account that (Gradstein & Ryzhyk, 1963) Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear p − n junction (Sze, 1981), (Sze & Gibbons, 1966) by last multiplicand, which for typical values of ε ≈ 10 (Sze, 1981), (Casey & Panish, 1978) is close to unity.

10
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 N A < < N D or when N D < < N A . Avalanche breakdown voltage is determined by equation ṽ = 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 we find that in zeroth-order approximation with respect to this parameter 2 1 .
In the case of very thin base when 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 7 7 ( 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).

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 l T , i.e. length on which energy of bottom of conduction band Е с (x) changes by value equal to Е g is found by solving integral equation If variation of electric field within length of tunneling ΔЕ < < Е, i.e. specific length of variation of field l E > > l T , then expanding function Е(x ′ ) in Taylor series around point x ′ = x, we find that in the first order of parameter of smallness l T / l E equation (82) takes the form When N (x) = const then equation (83) is exact. As can be seen from Fig. 16a, if , , 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
at that As shown below, due to large values of field Е at avalanche breakdown of p − n structures, inequality (85) is valid for almost all materials up to concentration N = 10 17 cm -3 and even high.
Under these conditions specific rates of charge carriers' tunnel generation g Ti (x) in layers I and II of structure can be described by expression obtained in (Kane, 1960) where characteristic dimensions of areas of charge carriers' tunnel generation in layers I and II Equation (89) is valid under conditions 2 0 1, 2 These conditions mean the following. If inequalities (91) for g Ti (E) are satisfied then expression (87) is valid, at least in the neighborhood of field value E = E i . When right side of inequalities (91) is satisfied then tunnel generation drops sharply with decreasing E , and therefore I Ti at W Ti < W i is mainly determined by tunneling in areas 0 ≤ x ≤ W T 1 and 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 N i and layers' thicknesses W i of heterostructure. Avalanche breakdown occurs when one of fields E i becomes close to breakdown field E iBD of proper layer of structure ( (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), Sections 3.1-3.3).

When
then under avalanche breakdown of proper layer of structure ratio δ i to Е iBD / a i is not exceed unity, moreover, even when N i = N i (2) δ i < Χ mi 0.6 × Χ εi 1.2 × Χ gi 0.8 × 10 −1 .

When
then length of tunneling l Ti at E i = E iBD is much shorter than thickness W i of this layer.
In expressions (96)-(99) Е gi is measured in eV. Analysis shows that under avalanche breakdown of heterostructure inequities (91) and (92) are satisfied for real values of N i and W i and E i < E iBD , i.e. in layer which does not control avalanche breakdown also. As can be seen from Fig. 17, when punch-through of layer n wg 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 р wg + 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 Е c (x) and Е v (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 N i (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.

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 E 1 = E 1BD or E 2 = E 2BD , shows that in contrast to homogeneous p − n junction (Stillman, 1981), (Ando et al, 1980) density of initial tunnel current J T , as a rule, is not a monotonic function N 1 . An increase in N 2 cause, for some values of N 1 and W i , the rise of tunnel current and vice versa -decrease of tunnel current when N 1 and W i have different values. Depending on gap E gi of heterostructure's layers and their thicknesses W i the following situations are possible.
then tunnel current is almost independent on N 1 .
One can see that at minimum of tunnel current, as a rule, the following inequality is valid where Therefore, as it follows from (6) Expression (105) is valid when inequality W T 2 < W 2 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 W T 2 > W 2 and W T 2 < W 2 at N 1 = N 1min (T ) . Equation (104) can be solved by successive approximations using parameters of smallness ξ and 1 / s.
As a result we find It is shown from (105) and (106) that N 1min (T ) is decreased with growth W 1 and, also, although weakly, with increase N 2 .
When concentrations then in minimum of J T (N 1 ) takes place punch-through of narrow-gap layer, i.e. non-equilibrium SCR reaches n wg + layer. When N 1 > N 1min (T ) then tunnel current increases with increasing N 1 , and at the same time, non-equilibrium SCR will penetrate into narrow-gap layer until concentration N 1 reaches value Nature of above dependence J T on N 1 is competition between tunnel currents in wide-gap and narrow-gap layers of heterostructure (Fig. 1a). When N 1 ≤ Ñ 1 (T ) then field E = E I (W 1 ) in n wg layer at its heterojunction (Fig. 1b) coincide with very high accuracy with E 1BD . Due to relatively large field E 2 = (ε 1 / ε 2 )×E 1BD , current density J T is determined by tunneling of charge carriers in narrow-gap layer, i.e. J T ≈ J T 2 (Fig. 1a). With increasing N 1 , field E 2 and therefore current J T 2 decrease due to fall E I (W 1 ) (Fig. 18). Decrease E I (W 1 ) with increase N 1 is caused by requirement (1) of constancy of photocurrent gain M ph = M р . Indeed, increase N 1 for given M ph should lead to growth E 1 . Otherwise, due to growth | ∇ Е(х) | with increasing N 1 , field would be reduced everywhere in SCR, which in turn would lead to a decrease M ph . However, increase E 1 should not be too large, and it should be such that Е(х) at х greater than some value in interval 0 < x < W 1 is decreased. In other words, Е(х) anywhere in SCR would increase, that, evidently, would increase M ph . It can be seen directly from (1) and (2). Note that for sufficiently large values of multiplication factors M ph , field E 1 is practically independent on M ph and very close to breakdown field E 1BD (N 1 , W 1 ) when value of integral m (2) is equal to unity. This allows to use value E 1 = E 1BD (N 1 , W 1 ) (93) instead of true value Е 1 (N 1 , W 1 , M ph ). When N 1 > Ñ 1 (T ) , then variation of field Е(х) at distance W 1 in n wg layer is still very insignificant, but it is enough to affect value J T 2 . Due to decrease E 2 with growth N 1 (especially when N 1 > Ñ 1 ), current is more and more determined by tunneling of charge carriers in n wg layer, therefore when N 1 > N 1min (T ) , current density J T ≈ J T 1 in-creases with increase N 1 because E 1BD grows with increase N 1 . Initial plateau (Fig. 18a) on the graph J T (N 1 ) is caused by extremely weak dependences E 1BD on N 1 (93) and Е on х in n wg layer when N 1 < Ñ 1 (T ) . Reducing of value J T min (108) with growth N 2 is due to increasing length of tunneling in narrow-gap n ng layer (Fig. 1). Indeed, in this layer ∇ Е~− N 2 < 0, and E 2 under these conditions does not depend on N 2 . It means, that Е(х) everywhere in n ng layer, except of point x = W 1 , falls with increase N 2 (1b). Since slopes of E c (x) and E v (x) everywhere in n ng layer, except of point x = W 1 , decrease also with increasing N 2 , that leads to increase length of tunneling. Reducing of J T is more significant with growth N 2 when N 1 < N 1min (T ) (Fig.18b), because current density J T 2 increases with decrease N 1 while J T 1 decreases. When N 1 < Ñ 1 (T ) then current density J T 1 ≤ J T 2 , and if (T ) it exceeds J T 2 . Therefore, ratio of J T min to J T | N 1 <Ñ 1 (T ) (109) increases with increasing N 2 . Because at N 1 = N 1min (T ) value J T 1 > J T 2 , then, naturally, concentration N 1min (T ) (106) slightly decreases with increasing N 2 (Fig. 18b). For small values N 2 , when W T 2 > W 2 , Е(х) in n ng layer coincides with E 2 with high accuracy. Therefore, length of tunneling in this layer, and hence J T also, do not depend on N 2 . Reducing of values N 1min (T ) (106) and J T min (108) with increasing W 1 (Figure 18a) is due to the fact that the more is W 1 then the less is E 1BD and the greater is fall of field Е(х) in depth of n wg layer.

II.
Condition (100) is not satisfied. For example, for combination of layers n wg :InP / where N 2 and W i are measured in cm -3 and μm, respectively. Under this condition, when avalanche breakdown is controlled by n ng layer, i.e. E 2 = E 2BD (N 2 , W 2 ), and E 1 <E 1BD and it increases linearly with N 1 . Therefore, strictly speaking, when N 1 < N 1 then tunnel current increases with increasing N 1 . At the same time, J T 2 does not depend on N 1 under following conditions.  then at N 1 < N 1 ,J T 2 > > J T 1 with margin of several orders of magnitude, and therefore with very high accuracy J T (N 1 ) = const. If N 1 > N 1 then due to decrease E 2 and hence J T 2 also, density of tunnel current J T (N 1 ) begins drop sharply and, reaching minimum value (108) at concentration (106), then starts to grow again due to growth J T 1 (N 1 ).

If
then after initial plateau J T (N 1 ) grows monotonically. It is due to monotonic increase in component of tunnel current density J T (N 1 ), which at N 1 ≥ N 1 is considerably superior to J T 2 .

If
then for small enough thicknesses W 1 of layer n wg dependence J T (N 1 ) has distinct maximum at N 1 = N 1 , however, at least in this case minimum is not deep. This is due to the fact that components of tunnel current density J T 1 and J T 2 are equal to each other in order of magnitude at small enough W 1 . Characteristics of tunnel currents in heterostructure with independent doping of n wg and n ng layers are illustrated in Fig. 18. Note that if in case I increase N 2 leads to decrease J T at all values N 1 , then in case II, increase N 2 , when N 1 is small enough, leads to increase of tunnel current, but at sufficiently large N 1 tunnel current decreases, particularly, in the vicinity of concentration N 1 = N 1min (T ) .

Equal doping levels of wide-gap and narrow-gap n type layers
Under this condition density of tunnel current is given by expression (89) (Fig. 19a). This is due to the fact that when N < N then E 1 grows and E 2 grows also reaching at N = N maximal value (Fig. 19b). As a result, when N < N then J T 1 grows with increase N and J T 2 grows also. Note that when doping of n wg and n ng layers are equal then concentration N = N min (T ) , at which tunnel current density J T has minimal value, is determined by formula (106) with accuracy up to small corrections of order ξ = E 1BD (0, W 1 ) / a 1 < < 1, as in the case of independent doping of n wg and n ng layers. Formula for J T min may be obtained from expression (108), if we replace N 2 by N min (T ) in it.

Responsivity
In punch-through conditions of absorber n ng , current responsivity S I (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) where reflection coefficient of light from illuminated surface R 1 = ( ε ex − ε 1 ) quantum efficiency η 2 with respect to light ray which has reached to second interface of heterostructure, From expressions (118)-(120) follow, that (1 ) (1 ) , , 1,2,3. Particularly, Dependence η on W 2 for heterostructure InP / I n 0.53 Ga 0.47 As / 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 W 2 is very thin.

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 S I (λ) is maximal, and effective noise factor F ef ,i (M ph ) 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 S N is also minimal. If α = β, then (Tsang, 1985), (Filachev et (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 N 1 > N 1 , (where N 1 satisfies equation (113) (see  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 N 1min (T ) and J T min will be again determined by (127) and (128), in which N 2 is replaced by Q(W 1 ) / W 2 . It is shown from (127) and (128) that N 1min (T ) and J T min are decreasing, moreover J T min sharply, with increase W 1 (see Fig. 21, 22), and, also, although weakly, with increase N 2 . Decrease of values N 1min (T ) and J T min with increase W 1 is caused by situation when the thicker W 1 the less E 1BD and the greater fall of field E (x) on n wg layer thickness. Slight decrease N 1min (T ) and J T min with growth N 2 is due to increasing of length of interband tunneling l Tng in narrow-gap n ng layer with increase N 2 and the fact that at minimum J T 1 > J T 2 . For small values either N 2 or W 2 , field E (x) is so weakly dependent on x in n ng layer, that value l Tng in it is almost constant. Therefore, when Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when R 1 = R 3 quantum efficiency reaches almost its maximal value when W 2 = 2 μm. Therefore, for development of concentration -thickness nomogram in Fig. 21, namely this value W 2 was selected. Note that decrease in dispersion in N 2 results in increase in dispersion N 1 and W 1 , while increase gives the opposite result. Value of noise current density I N ≤ 10 −12 A/Hz 1/2 corresponds to J T ≤ 1.8 × 10 −5 А/сm 2 , and value I N ≤ 10 −13 А/ Hz 1/2 corresponds to J T ≤ 1.8 × 10 −7 А/сm 2 .

Operational speed
Minimal possible time-of-response of this class of devices is determined by time-of-flight of charge carriers through multiplication layer τ tr 1 and absorber τ tr 2 , and also by value of function f (M ph ), which is close to 1 when K > > 1, and is  equal to M ph when K = 1 (Tsang, 1985), (Filachev et al, 2011), (Emmons, 1967), (Kurochkin & Kholodnov, 1996). It was noted above that in InP 1 < K ≤ 2.3. Therefore, in InP / I n x Ga 1−x As y P 1− y / InP SAM-APD ( ) As is evident from Fig. 20b, in InP / I n 0.53 Ga 0.47 As / InP heterostructure quantum efficiency value η lies in interval 0.5 ≤ η ≤ 0.686 when R 3 = 1 and W 2 ≥ 0.5 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 τ tr 2 = 5 ps can be achieved by forming absorber with thickness W 2 = 0.5 μm and fully reflecting backside surface. Minimal value τ tr 1 is determined by maximum allowable minimal value W 1min . When J T ≤ 10 −6 A/cm2, then as follows from Fig. 22, W 1min ≅ 2 μm, and therefore τ min ≅ (4M ph + 1) × 10 −2 ns.

Analytical model of avalanche photodiodes operation in Geiger mode
We  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 crosssection 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 > > S 1 ≈ π × λ 2 . 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: p − i − n diode -load resistance R -power supply source providing bias V b > V BD . 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 V 0 = V b . When electron-hole pairs appear in the base then occurs their multiplication that results in decrease V i due to screening of field E i GE in base by major charge carriers inflowing into p and n regions (Fig.  In present structure charge is mainly concentrated in thin near border n − i and p − i layers (let's call them plates, Fig. 23). Therefore, as in (Vanyushin et al, 2007), field E i GE will be as- 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. where V d -voltage on DL, C d = εε 0 × S / W d and W d -DL capacity and thickness, ε 0dielectric constant of vacuum, ε -dielectric permittivity, < I d > let's call avalanche current I av .
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 p − i − n structure) i -layer can be considered as DL, i.e., d in (135)  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 Strictly speaking, equations (139) are valid when r 1 ≡ P p / N n = 1, from which r 2 ≡ | P i − N i | / N n = 0. Therefore, let's assume uniform PhG along x. Then, at К = 1, symmetry requires r 1 = 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 x g succeeded by averaging with respect to x g 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 t tr gives 0.6 <r 1 < 1, and r 2 < 0.4 (Fig. 24a), which corresponds to approximation of uniform field. Note that smallness r 2 does not mean smallness P i + N i (curve 3 in Fig. 24a).
Relations (133) with initial conditions 0 0 where N ph -number of absorbed photons. If we take R = 0 and lim t→∞ G i (t) = const ≠ 0 then we find that breakdown is determined by condition W i GE × Χ(E i GE ) = 2, which at К ≠ 1 gives another value for breakdown field E i GE = E av than E i GE = E BD obtained directly from solving of stationary problem in Section 3. However, discrepancy between E av and E BD 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 0 , [ ; 0] 0, and problem is solved in quadratures. To solve in elementary functions let's approximate exact dependence Y E i GE (ΔV i ) by piecewise-linear function passing through principal Fig. 25 and 26), where Y =0, and I av reaches its peak during t av .  (21) from (Pospelov et al, 1974). It determines dependence χ(t) ≡ r / r 0 , where r(t) and r 0 -current and initial radii of charge "drop" of parabolic type. Implying under capacity in (Pospelov et al, 1974) value C i and putting W i GE = 1 μm, r 0 = λ = 1 μm, in the case of singlephoton process we get χ(t av )< 2 1/4 ≅ 1.2. This justifies our assumption that charge spreading over sample cross-section during avalanche Geiger process is not intensive.

Conclusions
The above analysis shows that to create high performance SAM-APD (in particular, based on widely used InP / I n x Ga 1−x As y P 1− y / InP heterostructures) it is necessary to maintain close tolerances on dopants concentration in wide-gap multiplication layer I -N 1 and in narrowgap absorption layer II -N 2 , and also on thickness W 1 of wide-gap multiplication layer ( Fig.   1). This is due to strong dependence of interband tunnel current in such heterostructures on N 1 ,N 2 and W 1 . Allowable variation intervals of values N 1 ,N 2 and W 1 , and, optimal thickness of absorber also, can be determined using results obtained in Sections 4 and 5. Value of minimal possible time-of-response τ min depends not only on photocurrent's gain М ph but on allowable noise density at preset value of photocurrent's gain also. The lower noise density, the larger is value τ min . For example, for heterostructure InP / I n 0.53 Ga 0.47 As / InP minimal time-of-response equals to τ min ≈ 0.6 ns, when noise current equals to 3.3×10 −11 А/Hz 1/2 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 V b exceeds breakdown voltage V BD of no more than a few volts, then, for K ≡ β / 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 p − i − n APD. Moreover if cross-section area S > S 1 ≈ π × λ 2 , then we can expect that in single-photon case under S in (145) should imply value of order S 1 . This is due to finite size of single-photon spot S 1 and not intensive spreading of charge during time of avalanche Geiger process t av when photogeneration of charge carriers occurs in i -region of p − i − n structure depleted by charge carriers. Proposed approach allows describing Geiger mode by elementary functions at voltages higher V b as well. Note that equation (140) and physical grownds allow to expect three possible process modes at pulse illumination under V b > V BD . When RC < < t av then generated photocurrent will tend to reach some constant and flow indefinitely (unless, of course, ignore energy losses). When RC = t av then generated photocurrent will be of infinitely long oscillatory character. When RC > > t av then Geiger mode is realized.