Experimental (Shotov, 1958) and computed [from Equation (48)] hole avalanche multiplication factor in step-wise junction in for different ratios of applied voltage to avalanche breakdown voltage . It is assumed that (Shotov, 1958)
Minimal value of dark current in reverse biased junctions at avalanche breakdown is determined by interband tunneling. For example, tunnel component of dark current becomes dominant in reverse biased junctions formed in a number semiconductor materials with relatively wide gap already at room temperature when bias is close to avalanche breakdown voltage (Sze, 1981), (Tsang, 1981). The above statement is applicable, for example, to junctions formed in semiconductor structures based on ternary alloy which is one of the most important material for optical communication technology in wavelength range 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 junction when metallurgical boundary of junction 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 into depth from (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 corresponds to wider gap . 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 ) is proportional to intrinsic concentration of charge carriers , here – Boltzmann constant, – temperature (Sze, 1981), (Stillman, 1981). Tunnel current density grows considerably stronger with narrowing than and depends weakly on (Stillman, 1981), (Burstein & Lundqvist, 1969). Therefore, component will prevail over in semiconductor structures with reasonably narrow gap 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 (Sze, 1981), (Stillman, 1981) (where is dopant concentration). To eliminate it one side of junction is doped heavily and narrow-gap layer is grown on wide-gap isotype heavily doped substrate (Tsang, 1981). Thus heterostructure like as is the most optimal, where subscript ‹› means wide-gap and ‹› − 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 is defined mainly by light absorption coefficient and speed-of-response. But as it will be shown further tunnel current’s density depends strongly on thickness of wide-gap layer and dopant concentrations in wide-gap and narrow-gap 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 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 (see Fig. 1b) at metallurgical boundary of junction is higher than 4.5×105 V/cm then avalanche multiplication of charge carriers occurs in layer where junction lies at any dopants concentrations and thicknesses of heterostructure’s layers. However, electric field at which avalanche breakdown of 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 =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 (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 (see Fig. 1b) is equal to 1.5×105 V/cm. But tunnel current density in narrow-gap absorber (Osipov & Kholodnov, 1989) is much smaller at that value of electric field than density of thermal generation current which in the best samples of 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 in wide-gap and in narrow-gap layers of heterostructure and thicknesses and , respectively (Fig. 1) and fundamental parameters of semiconductor materials also. Performance limits of AHPDs are analyzed (Kholodnov, 1996). Formula for quantum efficiency 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 are performed. The following values of fundamental parameters of (I, Fig. 1) and (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 = 1.35 eV and = 0.73 eV; intrinsic charge carriers concentrations =108 сm-3 and =5.4×1011 сm-3; relative dielectric constants = 12.4 and =13.9; light absorption coefficient in =104 сm-1; specific effective masses of light carriers = 0.06and = 0.045, where – free electron mass. The chapter material is presented in analytical form. For this purpose simple formulas for avalanche breakdown electric field and voltage of 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 . It is shown in final section that Geiger mode (Groves et al, 2005) of APD operation can be described by elementary functions (Kholodnov, 2009).
2. Formulation of the problem: Basic relations
Let’s consider heterostructure at reverse bias 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 in structures on , where and – multiplication factors of electrons and holes inflow into space charge region (SCR); value of multiplication factor of charge carriers generated in SCR lies between and ; specific rate of charge carriers’ generation in SCR consists of dark and photogenerated components; and – thicknesses of SCR in and sides of structure; and = – impact ionization coefficients of electrons and holes ; – electric field. Let’s denote by dopant concentration so that for “punch-through” (depletion) of layer occurs that means penetration of non-equilibrium SCR into layer (Fig. 1). Optical radiation passing through wide-gap window is absorbed in layer and generates electron-holes pairs in it. When then photo-holes appearing near /heterojunction () are heated in electric field of non-equilibrium SCR and, at moderate discontinuities in valence band top at , photo-holes penetrate into layer (layer I) due to emission and tunneling. If is larger than some value (Osipov & Kholodnov, 1989), which is calculated below, then avalanche multiplication of charge carriers occurs only in layer, i.e. photo-holes fly through whole region of multiplication. In this case photocurrent’s gain (Tsang, 1981), (Artsis & Kholodnov, 1984) =. Let 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 and , i.e.
It is remarkable that responsivity (where – is wavelength) of heterostructure increases dramatically once SCR reaches absorber (layer II on Fig. 1) and then depends weakly on bias till avalanche breakdown voltage value (Stillman, 1981). This effect is caused by potential barrier for photo-holes on /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 in operation mode is determined by well-known expression (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2011):
where in µm and value of quantum efficiency 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 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 becomes higher then certain value 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 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 of heterostructure which performance is limited by tunnel current can be written as:
where – electron charge; – cross-section area of APD’s structure; – effective noise factors (Artsis & Kholodnov, 1984) in wide-gap multiplication layer () and in absorber (); – 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 . Expression (5) shows, that for preset gain of photocurrent, noise density is determined by values of primary tunnel currents and (total primary tunnel current +). Distribution of electric field 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:
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 of charge carriers in structures on applied bias . We need to know among them dependence of avalanche breakdown voltage on parameters of structure and distribution of electric field related to 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 and holes depend drastically on electric field . At the same time theoretical expressions for and 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 and . Avalanche breakdown voltage 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 where field distribution is determined by solving Poisson equation. Bias voltage at which breakdown condition is satisfied can be calculated by method of successive approximations on computer. Thus, this method of determining and, hence, at requires time-consuming numerical calculations. The same applies to dependence on . 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 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 and 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 , 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 homojunction or heterojunction is often used well-known Sze-Gibbons approximate expression (Sze, 1981), (Sze & Gibbons, 1966):
Gap of semiconductor material forming diode’s base and dopant concentration 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 Fig. 2) of junction:
− dielectric constant of vacuum and relative dielectric permittivity of base material; − 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.
Formulas (10) and (11) cannot be used for reliable estimates of and in semiconductor structures with thin enough base. Indeed, dependence of on is due to two factors. First, as follows from Poisson equation, the larger the steeper the field decreases into the depth from comparing to value (Fig. 1b). Second, value of electric field at falls with decreasing of due to decreasing of in SCR. Drop of becomes more weaker with decreasing of (Fig. 1b), therefore, at preset base’s thickness , initiation of avalanche process will require fewer and fewer field intensity . At sufficiently low concentration , the lower the thicker will be, variation of electric field 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 and field are independent on and at the same time are dependent on , moreover, the thinner then, evidently, the higher . So using of formulas (10) and (11) at any values of , that done in many publications, contradicts with above conclusion. In next section 3.2 will be shown that value of breakdown field of stepwise junction in a number of semiconductor structures can be estimated by following formula:
It seen from expression (14) that at electric field of avalanche breakdown is practically independent on dopant concentration in diode’s base.
3.2. Avalanche breakdown field
Consider heterostructure (Fig. 2). Symbols and indicate to unequal, in general, doping of high-resistivity layers of structure. Denote as , and , thicknesses of and layers and dopant concentrations in them, properly. Case corresponds to diode formed on homogeneous structure. Let values and such that upon applying avalanche breakdown voltage to structure, SCR penetrates into narrow-gap layer (Fig. 2). When and , are small enough and is thick enough then avalanche process develops in layer. In other words, with increasing bias applied to heterostructure, electric field in narrow-gap layer on /heterojunction (Fig. 2) reaches avalanche breakdown field in this layer earlier than electric field on metallurgical boundary () of junction becomes equal to breakdown field in wide-gap layer. This is due to the fact that at small values of and variation of field within wide-gap layer is insignificant and probability of impact ionization in narrow-gap layer is much higher than in wide-gap. If, however, and , are large enough and thin enough, then avalanche process is developed in wide-gap layer only. For these values of thicknesses and concentrations electric field reaches value earlier than – value . Because of significant decreasing of electric field in layer with increasing distance from , field remains smaller despite the fact that band-gap in layer is wider than band-gap in layer. Distribution of electric field in and layers of considered heterostructure is obtained by solving Poisson equation as defined by (6)-(9). When avalanche breakdown voltage is applied to structure, then either or . In section 3.1 is noted that at low enough concentrations avalanche breakdown fields should not depend on and have definite value depending on , where . To account for this effect, formula (12) should be modified so that when then breakdown field tends to some non-zero value. It would seem that it is enough to add some independent on 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 layer, i.e. is close to and multiplication factor of holes (1) is fixed. Then, with increasing concentration , field (Fig. 2b) shall be monotonically falling function of . Indeed, with increasing , field and are increasing also. Increasing must be such that when became larger some value then value has decreased (Fig. 2b). Otherwise, field would increase throughout SCR that reasonably would lead to growth of . This is evident from (1) and (2). On the other hand, adding constant to right side of expression (12) does not change and therefore results in, as follows from (6) and (9), non-monotonic dependence on . Equation (14) which can be rewritten for each of and layers as:
does not lead to that and other contradictions, From (17) follows that:
To determine dependences , let’s consider behavior of when parameters of heterostructure , and are varying. From (6)-(9), (17) and (18) we find that when value
then avalanche breakdown is controlled by layer. It means that
If, however, then avalanche breakdown is controlled by /heterojunction, i.e.
From (17)-(21) we obtain that
Formulas (15) and (16) follow from expressions (18), (19) and requirement (23)
which means smoothness of field dependence 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 junction (or at heterojunction boundary, in narrow-gap layer of heterojunction, including isotype) can be described by formula
, and gap in diode’s base is measured in eV and its thickness – in μm, respectively.
3.3. Avalanche breakdown voltage
It follows from expressions (6)-(9) and (14)-(16) that breakdown voltage for 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 is defined from equation and with good degree of accuracy it equals to . Because , therefore expression (27) practically coincides with formula (10), i.e. of diode with thick base is independent on its thickness . For diodes with thin base formed on semiconductors with parameters satisfying relations (11) and (14), namely when
breakdown voltage of diode depends on and as follows
In expressions (30)-(32) , and gap in base, dopant concentration in it and thickness 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 and 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 and holes as functions of electric field (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 . Performance of APD can be calculated on computer if and are known (Sze, 1981), (Tsang, 1985), (Filachev et al, 2011), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Dependences and 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 and 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 and , 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)
Where: – relative dielectric permittivity, and gap , electric field , and are measured in eV, V/cm and 1/cm, properly.
To derive relation (33) let’s consider thin structure in which thickness of high-resistivity base layer satisfies to inequality
where – dielectric constant of vacuum; – relative dielectric permittivity of base material; – electron charge; and – constants defining dependence of electric field at metallurgical boundary of abrupt junction on dopant concentration in base for avalanche breakdown in thick 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 along length of base is so insignificant that probability of impact ionization is practically the same in any point of base of considered structure. For many semiconductors including relations given below are valid (Sze, 1981), (Kholodnov, 1988-2), (Kholodnov, 1996), (Sze & Gibbons, 1966)
In this case as it follows from (34) and (35)
And avalanche breakdown electric field for thin 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 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
where – 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
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 is usually not more than a few units. Therefore, when then and hence when instead of (33) must be valid relation
where 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 and 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 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 (Section 2) in structures on applied voltage is based on numerical processing of integral relations (1) and (2) in each case. Distribution of specific rate of charge carriers’ generation in space charge region (SCR), i.e. when (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 on . In article (Sze & Gibbons, 1966) was proposed analytical expressions for avalanche breakdown voltage , i.e. applied voltage value at which , in asymmetric abrupt and linear junctions. Expression for (Sze & Gibbons, 1966) in the case of asymmetric abrupt junction was generalized in (Osipov & Kholodnov, 1987) for the case of thin structure (like as ) as discussed in Section 3.3. Using as model abrupt (stepwise) junction under assumption that (Kholodnov, 1988-2) has been shown that from (1), (2) and approximate relation (33), which is valid for number of semiconductors including , 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 and threshold ionization energy in eV; electric field in V/cm; bias in V; multiplication factors and in cm-1, electron charge in C; dielectric constant of vacuum in F/m; concentration including shallow donors and acceptors in cm-3; concentration gradient in cm-4; width of SCR and in and layers and thicknesses of these layers (inset in Figure 3) in μm, light absorption coefficient in cm-1. In this section, analytical dependences in structures have been calculated under no condition. Such calculations are possible because ratio varies, typically, much slighter than . 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 . The most typical cases are considered: abrupt (stepwise) and gradual (linear) junctions like as in model given in (Sze, 1987), (Sze & Gibbons, 1966) and thin structure (like as ) 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 . This form was first proposed by Miller in 1955 (Miller, 1955) and then, despite lack of analytical expressions for exponents , , , 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 when .
3.5.1. Stepwise junction
In this case from relations (1), (2) and (45) and Poisson equation (SI units)
where – value when ==, i.e. value of at metallurgical boundary of junction (see inset in Fig. 3). Formula (49) for at or 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:
– unity function (Zeldovich & Myshkis, 1972), . Expression (50) is obtained by expanding the function as a power series in
and expression (52) was derived by standard method of integrating fast-changing functions (Zeldovich & Myshkis, 1972).
3.5.2. Gradual (linear) junction
In this case Poisson equation can be written as (SI units):
where - slope of linear concentration profile
Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear junction (Sze, 1981), (Sze & Gibbons, 1966) by last multiplicand, which for typical values of (Sze, 1981), (Casey & Panish, 1978) is close to unity.
3.5.3. Thin structure ()
When thickness of high-resistivity region (base) of considered structure
where – dopant concentration (for example, donor) in base, and when 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 that gives in the result substantially other expressions for avalanche multiplication factors of charge carriers and avalanche breakdown voltage. When then from relations (1), (2) and (45) and Poisson equation
we find that
In deriving expressions (60)-(63), multiplication of charge carriers in and layers and voltage drop on them is considered negligible. This is justified because of significant decreasing of electric field 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 or when . Avalanche breakdown voltage is determined by equation = 1 which has no exact analytical solution. However, till surpasses , then value of field at is much less than value of field at . In this case, using smallness parameter
we find that in zeroth-order approximation with respect to this parameter
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
where < 0, if structure is illuminated through region (front-side illuminated) and > 0 if structure is illuminated through 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 and voltage for abrupt junction taking into account finite thickness of high-resistivity layer . Comparative values of breakdown field for , and most often used for fabrication of APDs computed by formulas (25) and (26) and found from numerical solution of breakdown integral equation , where 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 μm for all a.m. semiconductors analytical and calculated values of breakdown field differ by less than 20 %. Relatively drastic fall of ratio /in comparison to unity with decrease of (for thin enough ) is due to the fact that, as shown in Sec. 3.4, if
then formulas (25) and (26) are not true. To estimate breakdown field at values defined by (70) can be used the following formula
If assume that in threshold energy of impact ionization of holes is higher than electrons, and it equals to 5 eV (Sze, 1981), then from (70) we find for μm. Estimates based on data from studies (Sze, 1981), (Tsang, 1985), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977) show that for and value is 2-3 times smaller.
Therefore curve 1 in Fig. 5 starts to fall significantly below unity at larger values than curves 2 and 3. Analytical and computed dependences on for used in high-performance APDs for wavelength range 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 and differ from each other by less than 10 %. In Fig. 7 and 8 are shown universal dependences of breakdown voltage on and 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 when only. Value of this minimal concentration, for example, for classic semiconductors , , , and at = (1-2) μm equals to cm-3. As shown on lower inset in Fig. 7, dependence on is in the strict sense non-monotonic. Such kind of dependence on is due to the fact that for small enough breakdown field is growing faster with increasing than ||in diode’s base. Maximum is reached, as it follows from (28), at
and expressed as
when , value , and absolute value can reach tens Volts, and even more (see Fig. 7). The analytical dependences (Fig. 7 and 8) for a number of semiconductors are in good agreement with 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 with and with depend on accuracy of determination of impact ionization coefficients of electrons and holes which are sharp functions of electric field . 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 and (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).
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 and holes , and form of required relation (expression (33) and (45)) is proposed. It is obvious from Fig. 9 that values may differ by many orders of magnitude in different semiconductors. At the same time, for presented in Fig. 9 and , function (see relations (33) and (45)) in range of fields where and 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 and (Cook et al, 1982) show that in value is some more closely to 1. It is evident from Fig. 10 that for , regardless of orientation of crystal with respect to electric field, function depends weakly on 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 (Fig. 11, according to (Mikawa et al, 1980)). As shown in (Kobajashi et al, 1969) dependences and measured in (Miller, 1955), (McKay & McAfee, 1953) in the range of fields V/cm can be described in by formulas =, . This result agrees well with expression (33). Note that, differs from unity approximately by the same factor, as values and 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 and 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 on and , was obtained for the first time in (Shotov, 1958).
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
where = 4, 5 and 7 for stepwise junction, linear junction and very thin (66) structure (situation 1, 2 and 3, respectively). If thickness of base in structure is not very small, i.e., (situation 4) then as it follows from formula (60), exponents and are also expressed by (74) and (75) but in right side of those expressions substitutes and . Value of exponent lies between values and . From (1) and (2) apparent that when then factors , and coincide with each other, i.e., ===, and, as it follows from expressions (74) and (75), regardless of bias voltage applied, = 4, 5 and 7 for situations 1, 2 and 3, respectively. Exponents in Miller’s relation have the same values when , more exactly, when ||, regardless of ratio . When or more exactly, if
Then for these situations
Graphs in Fig. 4 allow comparing numerical values of exponents and calculated in (Leguerre & Urgell, 1976) and analytical computed by formulas (76) for asymmetrical stepwise junction. Like as in (Leguerre & Urgell, 1976), experimentally determined functional dependencies and (Sze & Gibbons, 1966) were used in calculations of dependences . As follows from (46), when , then ratio of analytical value of multiplication factor to calculated equals to ratio to (Fig. 11-13). It obviously from Fig. 11-13 that for all considered semiconductors (with curves and taken from (Sze & Gibbons, 1966)), dependences and do not differ by more than 50 %. Dependences of exponents and on voltage and and on ratio 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 depend, obviously, on what functions and are used in (1) and (2) in calculations. Let’s take the simplest case when and junction is stepwise. Varying expressions (1) and (2), we find that under considered conditions
where at is determined from condition
In Fig. 15a are shown dependences calculated from relations (77) and (78) for four values obtained for by different authors (Grekhov & Serezhkin, 1980), (Okuto & Crowell, 1975), (Kressel & Kupsky, 1966), (Nuttall & Nield, 1974). It is seen that analytical value 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
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 junction
For comparison, in Fig. 15b are presented dependences of and =1 for the case , when . It is seen that value = 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 breakdown voltage dependence on becomes, with growth , more and more weaker than that described by equation (49), and in limit tends to value . This conclusion accords with results of studies (Grekhov & Serezhkin, 1980), (Nuttall & Nield, 1974). Obtained results agree well with experimental results for a number of structures, including based on (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 stepwise 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 if we take into account that in with doping levels used in (Miller, 1955) (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 substrate. From point of view of avalanche process this structure is similar to asymmetric stepwise junction. Therefore, avalanche process in MIS APD can be described by expressions (74)-(76) with . Concentration of shallow acceptors in substrate of investigated structure was 1015 cm-3. At this doping avalanche breakdown in occurs when electric field near insulator-semiconductor interface reaches value V/cm (Sections 3.1 and 3.2, (Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966)), and therefore (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Kuzmin et al, 1975). Measured in (Bogdanov et al, 1986) value at was found equal to 0.2. From formulas (76) with follows that . In Tables 1 and 2 are presented experimental (Shotov, 1958) and calculated by formulas (48) and (55) values of multiplication factors of electrons and holes in stepwise and linear junctions. Obviously, for these junctions, experimental and analytical values of multiplication factors differ from each other by less than 20 % in whole voltage range used in measurements.
|Experiment (*)||Theory||Experiment (*)||Theory|
Finally, it is interesting to analyze application of expressions (45) and (76) to describe avalanche process in . The fact is that dependence in was quite well known already in 1967 (Baertsch, 1967), but no one could obtain information about dependence (Dmitriev et al, 1987), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968). Substituting in (45) dependence for (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968), we find that ratio is vanishingly small up to electric field V/cm resulting in extremely high value when at the same time value is extremely small. It means that becomes much larger than unity, even at voltages noticeably lower avalanche breakdown voltage , and value remains equal to unity up to values very close to . 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 is extremely hard to observe (Dmitriev et al, 1987), (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968).
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 , i.e. length on which energy of bottom of conduction band changes by value equal to is found by solving integral equation
If variation of electric field within length of tunneling , i.e. specific length of variation of field , then expanding function in Taylor series around point , we find that in the first order of parameter of smallness equation (82) takes the form
then true ABC barrier coincides to high degree of accuracy with triangle ABC′ to which corresponds uniform field (Fig. 16b).
It follows from (83) and Poisson equation that inequalities (84) are satisfied if
As shown below, due to large values of field at avalanche breakdown of structures, inequality (85) is valid for almost all materials up to concentration cm-3 and even high.
Under these conditions specific rates of charge carriers’ tunnel generation in layers I and II of structure can be described by expression
Here , and – 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
where characteristic dimensions of areas of charge carriers’ tunnel generation in layers I and II
Equation (89) is valid under conditions
These conditions mean the following. If inequalities (91) for are satisfied then expression (87) is valid, at least in the neighborhood of field value . When right side of inequalities (91) is satisfied then tunnel generation drops sharply with decreasing , and therefore at is mainly determined by tunneling in areas 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 and layers’ thicknesses of heterostructure. Avalanche breakdown occurs when one of fields becomes close to breakdown field of proper layer of structure ((Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), Sections 3.1-3.3).
Breakdown fields can be obtained by formula (14) (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989), i.e.,
(and – some constants).
For many semiconductors including 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)
From expressions (93) and (94) when relations (95) are satisfied we find the following.
then ratio to is less than 0.1, where (for 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)), , (– free-electron mass)
then under avalanche breakdown of proper layer of structure ratio to is not exceed unity, moreover, even when
then length of tunneling at is much shorter than thickness of this layer.
In expressions (96)-(99) is measured in eV. Analysis shows that under avalanche breakdown of heterostructure inequities (91) and (92) are satisfied for real values of and and , i.e. in layer which does not control avalanche breakdown also. As can be seen from Fig. 17, when punch-through of layer 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 layer, tunnel current in it can be ignored; this is situation similar to MIS structures (Anderson, 1977). In type layers electric field decreases with increasing distance from metallurgical boundary of junction (Fig. 1b), and because gradient of potential is expressed as then slope of zones and decreases with increasing . 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 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.
4.2. Features of interband tunnel currents in heterostructures under avalanche breakdown
Analysis of expression (89) under avalanche breakdown of heterostructure, i.e., when either or , shows that in contrast to homogeneous junction (Stillman, 1981), (Ando et al, 1980) density of initial tunnel current , as a rule, is not a monotonic function . An increase in cause, for some values of and , the rise of tunnel current and vice versa – decrease of tunnel current when and have different values. Depending on gap of heterostructure’s layers and their thicknesses the following situations are possible.
4.2.1. Independent doping levels of wide-gap and narrow-gap type layers
In this case, at any concentration , field (), and , i.e., avalanche breakdown is controlled by layer.
As follows from (6)-(9), (89) and (93), if
which is fulfilled with large margin at due to large ratio of to () (1-2 orders of magnitude) while
then tunnel current is almost independent on .
If sufficiently large ((Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966), Sections 3.1-3.3), then with further increase of tunnel current is monotonically falling. However, in most real cases, for example, when relations (95) is valid, tunnel current at first decreases and then increases.
One can see that at minimum of tunnel current, as a rule, the following inequality is valid
When (103) is fulfilled then .
Therefore, as it follows from (6)-(9), (89), (90) and (93), concentration at which reaches minimum is defined by equation
Expression (105) is valid when inequality 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 and at . Equation (104) can be solved by successive approximations using parameters of smallness and .
As a result we find
It is shown from (105) and (106) that is decreased with growth and, also, although weakly, with increase .
When then density of tunnel current
From (94), (105) and (108) follow that decreases sharply with increasing . Value decreases also, although weakly, with increasing . Ratio
drops sharply, same as , with increase , but it increases with increasing . Value of this ratio is usually several orders of magnitude less than unity. For example, for combination of layers , differential of currents, as can be shown, does not exceed values , where is measured in cm-3.
then in minimum of takes place punch-through of narrow-gap layer, i.e. non-equilibrium SCR reaches layer. When then tunnel current increases with increasing , and at the same time, non-equilibrium SCR will penetrate into narrow-gap layer until concentration reaches value
Nature of above dependence on is competition between tunnel currents in wide-gap and narrow-gap layers of heterostructure (Fig. 1a). When then field in layer at its heterojunction (Fig. 1b) coincide with very high accuracy with . Due to relatively large field ×, current density is determined by tunneling of charge carriers in narrow-gap layer, i.e. (Fig. 1a). With increasing , field and therefore current decrease due to fall (Fig. 18). Decrease with increase is caused by requirement (1) of constancy of photocurrent gain . Indeed, increase for given should lead to growth . Otherwise, due to growth with increasing , field would be reduced everywhere in SCR, which in turn would lead to a decrease . However, increase should not be too large, and it should be such that at greater than some value in interval is decreased. In other words, anywhere in SCR would increase, that, evidently, would increase . It can be seen directly from (1) and (2). Note that for sufficiently large values of multiplication factors , field is practically independent on and very close to breakdown field when value of integral (2) is equal to unity. This allows to use value (93) instead of true value . When , then variation of field at distance in layer is still very insignificant, but it is enough to affect value . Due to decrease with growth (especially when ), current is more and more determined by tunneling of charge carriers in layer, therefore when , current density increases with increase because grows with increase . Initial plateau (Fig. 18a) on the graph is caused by extremely weak dependences on (93) and on in layer when . Reducing of value (108) with growth is due to increasing length of tunneling in narrow-gap layer (Fig. 1). Indeed, in this layer , and under these conditions does not depend on . It means, that everywhere in layer, except of point , falls with increase (1b). Since , then slopes of and everywhere in layer, except of point , decrease also with increasing , that leads to increase length of tunneling. Reducing of is more significant with growth when (Fig.18b), because current density increases with decrease while decreases. When then current density , and if it exceeds . Therefore, ratio of to (109) increases with increasing . Because at value , then, naturally, concentration (106) slightly decreases with increasing (Fig. 18b). For small values , when , in layer coincides with with high accuracy. Therefore, length of tunneling in this layer, and hence also, do not depend on . Reducing of values (106) and (108) with increasing (Figure 18a) is due to the fact that the more is then the less is and the greater is fall of field in depth of layer.
Condition (100) is not satisfied. For example, for combination of layers :such situation takes place when
where and are measured in cm-3 and μm, respectively. Under this condition, when , where satisfies equation
avalanche breakdown is controlled by layer, i.e. , and <and it increases linearly with . Therefore, strictly speaking, when then tunnel current increases with increasing . At the same time, does not depend on under following conditions.
then at ,with margin of several orders of magnitude, and therefore with very high accuracy . If then due to decrease and hence also, density of tunnel current begins drop sharply and, reaching minimum value (108) at concentration (106), then starts to grow again due to growth .
then after initial plateau grows monotonically. It is due to monotonic increase in component of tunnel current density , which at is considerably superior to .
then for small enough thicknesses of layer dependence has distinct maximum at , however, at least in this case minimum is not deep. This is due to the fact that components of tunnel current density and are equal to each other in order of magnitude at small enough . Characteristics of tunnel currents in heterostructure with independent doping of and layers are illustrated in Fig. 18. Note that if in case I increase leads to decrease at all values , then in case II, increase , when is small enough, leads to increase of tunnel current, but at sufficiently large tunnel current decreases, particularly, in the vicinity of concentration .
4.2.2. Equal doping levels of wide-gap and narrow-gap type layers
Under this condition density of tunnel current is given by expression (89), where ==
At this relation of parameters avalanche breakdown is controlled by layer, i.e. =, and <regardless of doping. Dependence on has identical character with in the case of 4.2.1. I, and is caused by the same physical grounds. The only difference is that when , then curves lie higher on plotting area, and when – lower, than curves in the case of 4.2.1. I.
This occurs because at given value 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 , (where is determined by equation (113), where ==) avalanche breakdown is controlled by layer, i.e. =, and <and increases linearly with . Dependence 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 then grows and grows also reaching at maximal value (Fig. 19b). As a result, when then grows with increase and grows also. Note that when doping of and layers are equal then concentration , at which tunnel current density has minimal value, is determined by formula (106) with accuracy up to small corrections of order , as in the case of independent doping of and layers. Formula for may be obtained from expression (108), if we replace by in it.
5. Basic performance of avalanche heterophotodiode
In punch-through conditions of absorber , current responsivity of heterostructure under study can be described by relation (4). In calculating quantum efficiency 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 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 and from interfaces of heterostructure (/(; – relative dielectric constant of environment; and quantum efficiency with respect to light ray which has penetrated into narrow-gap layer is written
quantum efficiency with respect to light ray which has reached to second interface of heterostructure,
, – reflection coefficient of light from not illuminated (backside) surface. From expressions (118)-(120) follow, that
Dependence on for heterostructure 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 is very thin.
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 is maximal, and effective noise factor 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 is also minimal. If , then (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (McIntyre 1966) , and therefore
In ratio in interval of fields of interest 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 multiplication layer and optimal doping is slightly less than value given by formula (125). When , (where satisfies equation (113) (see Fig. 21), in which is defined by formula (94) for ) then avalanche multiplication of charge carriers in narrow-gap layer does not occur. Under these conditions, field value at metallurgical boundary of junction (, Fig. 1) equals to (see (93) and (94)). For many semiconductors (see Sections 3.1-3.2) including , values and are defined by relations (95). In the case of heterostructure , in first approximation in parameters of smallness
we find that value of concentration at which function reaches its minimum
is given by
Formulas (127) and (128) are valid when , i.e., as follows from Section 4.2.1, when
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 and will be again determined by (127) and (128), in which is replaced by . It is shown from (127) and (128) that and are decreasing, moreover sharply, with increase (see Fig. 21, 22), and, also, although weakly, with increase . Decrease of values and with increase is caused by situation when the thicker the less and the greater fall of field on layer thickness. Slight decrease and with growth is due to increasing of length of interband tunneling in narrow-gap layer with increase and the fact that at minimum . For small values either or , field is so weakly dependent on in layer, that value in it is almost constant. Therefore, when then values and do no longer depend on and slightly decrease with increase 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 on heterostructure is less than voltage of avalanche breakdown . 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:
Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when quantum efficiency reaches almost its maximal value when μm. Therefore, for development of concentration – thickness nomogram in Fig. 21, namely this value was selected. Note that decrease in dispersion in results in increase in dispersion and , while increase gives the opposite result. Value of noise current density A/Hz1/2 corresponds to А/сm2, and value А/ Hz1/2 corresponds to А/сm2.
5.3. Operational speed
Minimal possible time-of-response of this class of devices
is determined by time-of-flight of charge carriers through multiplication layer and absorber , and also by value of function , which is close to 1 when , and is equal to when (Tsang, 1985), (Filachev et al, 2011), (Emmons, 1967), (Kurochkin & Kholodnov, 1996). It was noted above that in . Therefore, in SAM-APD
As is evident from Fig. 20b, in heterostructure quantum efficiency value lies in interval when and μm. It means that, because of not so much loss in quantum efficiency compared to maximal possible (only 27 % less), time-of-response value ps can be achieved by forming absorber with thickness μm and fully reflecting backside surface. Minimal value is determined by maximum allowable minimal value . When A/cm2, then as follows from Fig. 22, μm, and therefore ns.
6. Analytical model of avalanche photodiodes operation in Geiger mode
We consider possibility to describe transient phenomena in APDs by elementary functions, first of all, when initially applied voltage is greater than avalanche breakdown voltage . 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 . 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 structure. Obtained iterative relations allow to determine, although fairly easy, but only by numerical method, options for realization of Geiger mode when ratio differs very much from unity, where and – impact ionization coefficients of electrons and holes and – electric field in – layer (base , Fig. 23). "Continuous" model (Kholodnov, 2009) developed in this section admits value . 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 transverse to axis (Fig. 23). Then, in the most important single-photon process, area , according to uncertainty principle, shall not exceed in the order of magnitude, square of wavelength of light . Under these conditions, it is allowably to consider problem as one-dimensional (axis , 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 . 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: diode – load resistance – power supply source providing bias . Let and regions are heavily doped, so that prevailing share of bias falls across base . Then after charging process voltage on it can be considered equals to . When electron-hole pairs appear in the base then occurs their multiplication that results in decrease due to screening of field in base by major charge carriers inflowing into and regions (Fig. 23) in quantity and and voltage drop across load resistor and, hence, current in external circuit arise
In present structure charge is mainly concentrated in thin near border and layers (let's call them plates, Fig. 23). Therefore, as in (Vanyushin et al, 2007), field will be assumed uniform. Numerical value when 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 and holes are determined by their drift in electric field with velocity of saturation , i.e.,
where and – linear density (per unit length) of electrons and holes, – full conductive current, – absolute value of electron charge, - time.
Substituting volume charge density from Poisson equation in continuity equation for and integrating over depletion layer (DL) we obtain that, in approximation of zero-bias current, in quasi-neutral parts of structure
where – voltage on DL, and – DL capacity and thickness, –
dielectric constant of vacuum, – dielectric permittivity, let’s call avalanche current .
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 structure) – layer can be considered as DL, i.e., in (135) and below should be replaced by . By integrating continuity equation for and with respect to from to and marking linear density of photogeneration rate as we obtain equations
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 and regions
Strictly speaking, equations (139) are valid when , from which . Therefore, let’s assume uniform PhG along . Then, at , symmetry requires . Equations (139) are correct in concern of the order of magnitude both when 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 succeeded by averaging with respect to 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 gives 0.6 << 1, and < 0.4 (Fig. 24a), which corresponds to approximation of uniform field. Note that smallness does not mean smallness (curve 3 in Fig. 24a).
Relations (133)-(139) allow obtaining equations
with initial conditions
At delta-shaped time-evolving illumination relations (140) and (141) are converted into
where – number of absorbed photons.
If we take and then we find that breakdown is determined by condition , which at gives another value for breakdown field than obtained directly from solving of stationary problem in Section 3. However, discrepancy between and is no more than 20 %, if 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 , when
and problem is solved in quadratures. To solve in elementary functions let’s approximate exact dependence by piecewise-linear function passing through principal point (Fig. 25 and 26), where =, and reaches its peak during .
Suppose, for simplicity , where . Then is not more than value of break point of piecewise-linear approximation (Fig. 25). Under these conditions
where . Geiger mode occurs when during time of inverse recharge of avalanche diode, avalanche is able to develop and cancel itself in full. As seen from (145) it is happened when . Maximal voltage drop on load equals to . Since <<, 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 , where and – current and initial radii of charge "drop" of parabolic type. Implying under capacity in (Pospelov et al, 1974) value and putting μm, μm, in the case of single-photon process we get . This justifies our assumption that charge spreading over sample cross-section during avalanche Geiger process is not intensive.
The above analysis shows that to create high performance SAM-APD (in particular, based on widely used heterostructures) it is necessary to maintain close tolerances on dopants concentration in wide-gap multiplication layer I – and in narrow-gap absorption layer II – , and also on thickness of wide-gap multiplication layer (Fig.1). This is due to strong dependence of interband tunnel current in such heterostructures on ,and . Allowable variation intervals of values ,and , and, optimal thickness of absorber also, can be determined using results obtained in Sections 4 and 5. Value of minimal possible time-of-response depends not only on photocurrent’s gain but on allowable noise density at preset value of photocurrent’s gain also. The lower noise density, the larger is value . For example, for heterostructure minimal time-of-response equals to ns, when noise current equals to 3.3×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 exceeds breakdown voltage of no more than a few volts, then, for 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 APD. Moreover if cross-section area , then we can expect that in single-photon case under in (145) should imply value of order . This is due to finite size of single-photon spot and not intensive spreading of charge during time of avalanche Geiger process when photogeneration of charge carriers occurs in – region of structure depleted by charge carriers. Proposed approach allows describing Geiger mode by elementary functions at voltages higher as well. Note that equation (140) and physical grownds allow to expect three possible process modes at pulse illumination under . When then generated photocurrent will tend to reach some constant and flow indefinitely (unless, of course, ignore energy losses). When then generated photocurrent will be of infinitely long oscillatory character. When then Geiger mode is realized.