\r\n\tHydrogen gas is the key energy source for hydrogen-based society. Ozone dissolved water is expected as the sterilization and cleaning agent that can comply with the new law enacted by the US Food and Drug Administration (FDA). The law “FDA Food Safety Modernization Act” requires sterilization and washing of foods to prevent food poisoning and has a strict provision that vegetables, meat, and fish must be washed with non-chlorine cleaning agents to make E. coli adhering to food down to “zero”. If ozone dissolved water could be successively applied in this field, electrochemistry would make a significant contribution to society.\r\n
\r\n\tOxygen-enriched water is said to promote the growth of farmed fish. Hydrogen dissolved water is said to be able to efficiently remove minute dust on the silicon wafer when used in combination with ultrasonic irradiation.
\r\n\tAt present researches on direct water electrolysis have shown significant progress. For example, boron-doped diamonds and complex metal oxides are widely used as an electrode, and the interposing polymer electrolyte membrane (PEM) between electrodes has become one of the major processes of water electrolysis.
\r\n\tThe purpose of this book is to show the latest water electrolysis technology and the future of society applying it.
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 \n\t\t\t\t=104 сm-1; specific effective masses of light carriers = 0.06 and = 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).
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:
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 \n\t\t\t\t\tFig. 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.
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.
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.
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 \n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t 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