9.1.1 Internal Currents

Our first goal is to obtain equations for the four internal currents , and in terms of the device parameters and the terminal voltages and . We can then write expressions for the terminal currents , and , and from these we will obtain equations for important performance parameters such as current gain, specific on-resistance, and so on. We initially assume low-level injection in all regions, since this allows us to use the minority carrier diffusion equations (MCDEs) to obtain the desired internal currents. We will later remove this restriction in the base and collector.

The solution to the MCDE in the emitter and collector proceeds in the same way as in the neutral regions of a pn diode discussed in Section 7.3. We first write the general solution of the MCDE in the emitter and collector, subject to two boundary conditions in each region. One boundary condition, the carrier density at the edge of the depletion region, is provided by the law of the junction, Equation 7.25. The second boundary condition is based on the assumption that the minority carrier densities far from the junctions remain at their equilibrium values. However, in SiC BJTs it may not be accurate to assume that the emitter ohmic contact is infinitely far from the junction. In this case, we must set the minority carrier density at the position of the ohmic contact to its equilibrium value. The solution for in the emitter involves hyperbolic functions (see Appendix B), and can be written

9.1

Here is the intrinsic carrier concentration in the emitter, is the ionized dopant concentration in the emitter, and is the hole diffusion length in the emitter, where is the hole diffusion coefficient and is the hole lifetime. We specify in the emitter separately from in the other portions of the device, since the emitter is usually so heavily doped that bandgap narrowing occurs, resulting in a higher intrinsic carrier concentration. The hole current crossing the EB junction is obtained from the derivative of Equation 9.1 at the depletion edge x = 0,

9.2

where is the area of the junction. In the case of a “long” emitter where , Equation 9.2 reduces to

9.3

The solution in the collector is similar to that in the emitter. By analogy with Equation 9.2 we can write the hole current crossing the CB junction as

9.4

which in the case of a “long” collector reduces to

9.5

Equations 9.3 and 9.5 are similar in form to the Shockley diode Equation 7.26.

The solution for the minority carrier density in the base is obtained in the same manner – we find the general solution of the MCDE, subject to the boundary conditions at the EB and CB junctions provided by the law of the junction. In a typical SiC BJT we cannot always assume , and the full solution for will involve hyperbolic functions, namely

9.6

The electron current crossing the EB junction is obtained from the derivative of Equation 9.6 evaluated at , and the electron current crossing the CB junction is obtained from the derivative evaluated at . After some algebra, we can write

9.7

and

9.8

If the base width is much shorter than the minority carrier diffusion length in the base, we can set , and the above equations reduce to

9.9

Equations 9.7, and 9.8 are the desired equations for the four internal currents in the BJT of Figure 9.1.

9.1.2 Gain Parameters

We can now write equations for some of the important performance parameters of the BJT. The common-base current gain is defined as the ratio of collector current to emitter current. We assume the BJT is operated in the forward-active mode with the EB junction forward biased and the CB junction reverse biased, so that and . Under forward-active biasing the terms in the above equations are small compared to the terms, and the terms are small compared to the terms. The current component represents the reverse-bias leakage current of the CB junction, and from Equation 9.4 we see it is very small and can be neglected. Therefore, the common-base current gain can be written

9.10

where is defined as the base transport factor and is the emitter injection efficiency. The base transport factor is the fraction of electrons injected from the emitter that diffuse across the base and are swept across the collector junction. The emitter injection efficiency is the fraction of current crossing the EB junction that is due to electrons injected from the emitter into the base. The base transport factor is obtained by dividing Equation 9.8 by Equation 9.7. Under forward-active biasing, can be simply written

9.11

Likewise, using Equations 9.7 and 9.2, the emitter injection efficiency is

9.12

Inserting Equations 9.11 and 9.12 into Equation 9.10, we can write the common-base current gain as

9.13

The common-emitter current gain is the ratio of collector current to base current,

9.14

Inserting Equation 9.13 into Equation 9.14, we can write

9.15

We can simplify the above equations under certain conditions. If the base is narrow compared to a diffusion length, the hyperbolic functions involving can be replaced by the first term of their Taylor series expansions, namely,

9.16

Moreover, if the emitter is long compared to a diffusion length, the hyperbolic functions involving can be simplified using

9.17

Under these conditions, and we can write

9.18

From this, it follows that

9.19

If the emitter is short compared to a diffusion length, , we simply replace by in the above equations. Although these simplifications are appealing, a word of caution is in order. The “narrow base” and “long emitter” assumptions are often invoked for silicon BJTs, but they may not be accurate for all SiC power BJTs. When in doubt, it is advisable to use the full expressions that involve hyperbolic functions.

9.1.3 Terminal Currents

We next wish to obtain expressions for the terminal currents , and . This is easily done by recognizing that , and . Adding Equations 9.7 and 9.2 we obtain

9.20

Similarly, adding Equations 9.8 and 9.4 yields

9.21

The equation for follows by subtracting Equation 9.21 from Equation 9.20.

The above equations initially seem formidable, but a closer analysis reveals symmetries that lead to a simpler formulation. We note that each equation is the algebraic sum of two Shockley diode equations of the form given in Equation 7.26. We can exploit this symmetry by writing Equations 9.20 and 9.21 in the alternate forms

9.22

and

9.23

Equations 9.22 and 9.23 are known as the Ebers–Moll equations for the BJT. The four constants in the Ebers–Moll equations can be determined by comparing Equations 9.22 and 9.23 to Equations 9.20 and 9.21. For convenience, we summarize these parameters below.

9.24

As before, the approximations in Equations 9.16 and 9.17 can be applied to the above equations, if justified in a particular situation.

Equations 9.24 allow us to calculate all four constants in the Ebers–Moll equations for the BJT terminal currents. The Ebers–Moll equations are associated with a simple and easily-remembered equivalent circuit for the BJT known as the Ebers–Moll model, shown in Figure 9.2. The Ebers–Moll equations and model are valid in all biasing regimes of the BJT: forward-active, saturation, inverse-active, and cutoff, and they form the basis for the BJT model used in popular circuit analysis programs such as .

Consider the Ebers–Moll model for a BJT in the forward-active mode, with the EB junction forward biased and the CB junction reverse biased. With the CB junction reverse biased, the diode current is just the reverse leakage current −, which is small and can be neglected. Likewise, the dependent current generator can be neglected. Diode then represents the hole and electron currents crossing the EB junction under forward bias, and the dependent current generator represents the fraction of electron current injected from the emitter that reaches the collector (keep in mind that the flow of electrons is opposite to the positive direction of electron current). If the BJT were operated in the inverse-active mode with the EB junction reverse biased and the CB junction forward biased, the roles of the and diodes would be reversed: the collector would inject electrons into the base, and the emitter would sweep them out. In saturation, both junctions are forward biased and all four elements of the Ebers–Moll model are active.

9.1.4 Current–Voltage Relationship

The Ebers–Moll model can be used to obtain a single equation for the characteristics with base current as a parameter. We first write in terms of the Ebers–Moll circuit model as

9.25

where, as illustrated in Figure 9.2,

9.26

and

9.27

We can eliminate V_BC in Equation 9.27 by writing , resulting in

9.28

Inserting Equations 9.26 and 9.28 into Equation 9.25 then provides an equation for in terms of and ,

9.29

Equation 9.29 can be solved for , yielding

9.30

From the Ebers–Moll model we can write the collector current as

9.31

We now substitute Equation 9.30 into Equations 9.26 and 9.28 to eliminate , then insert Equations 9.26 and 9.28 into Equation 9.31 to obtain the desired equation for as a function of with as a parameter. After some algebra we can write

9.32

Equation 9.32 is valid in all biasing regimes: forward-active, saturation, inverse-active, and cut-off. Despite its apparent complexity, Equation 9.32 is a simple algebraic equation involving four constants (the four Ebers–Moll parameters) and the variables and . Equation 9.32 can be used to generate a plot of as a function of for various values of base current , and such a plot is shown in Figure 9.3. The parameters used to generate this plot are representative of a typical 4H-SiC BJT.

The equations developed to this point involve a number of assumptions that may not be valid in all regions of operation. This is especially true of power BJTs, since they operate at high current densities where our assumptions of low-level injection fail, first in the collector and then in the base. Other effects occur at high voltages, and still others at high temperatures. In addition, there are important lateral effects that are not captured in our one-dimensional analysis. In the following sections we will discuss the most important of these effects, and how they influence practical SiC power BJTs.

9.1.5 High-Current Effects in the Collector: Saturation and Quasi-Saturation

To support a high blocking voltage in the off state, the power BJT incorporates a thick, lightly-doped collector drift region between the CB junction and the substrate, as shown in Figure 9.4. As with the power junction field-effect transistor (JFET) and power MOSFET, this drift region is designed to support the desired blocking voltage, with doping and thickness given by Equations 7.10 and 7.11. In the absence of conductivity modulation, the drift region would add a series resistance given by Equation 7.12 to the basic BJT described above, but in the saturation regime the drift region is partially or totally conductivity modulated. The actual resistance and associated voltage drop added by the collector drift region will be considered next.

Figure 9.5 illustrates the relationships of an power BJT with a thick lightly-doped collector drift region, and Figure 9.6 shows the corresponding minority carrier densities in the device at operating points (a), (c), and (e). The behavior in saturation consists of two distinct regimes: saturation and quasi-saturation. This can be understood as follows. In the forward-active region (a), the CB junction is reverse biased and minority carriers are not injected into the collector drift region. The drift region is not conductivity modulated, and the carrier densities are illustrated in Figure 9.6a. As is reduced, the reverse bias on the CB junction is reduced and eventually reaches zero at point (b) in Figure 9.5. This is the boundary between the forward-active and saturation regimes. As is reduced further, the CB junction becomes forward biased, injecting holes into the collector drift region. Since the collector is lightly doped, even a small injection of holes places the region near the junction into high-level injection, and the minority carrier density at bias point (c) is shown in Figure 9.6c. Here the drift region is conductivity modulated out to a distance , but not beyond. The carrier densities vary linearly with distance from to , as will be explained below. As we continue to reduce , the CB junction becomes even more forward biased, increasing the injection of holes into the drift region, and at bias point (d) the modulated portion extends across the entire drift region, that is, . The resistance of the drift region has now reached a very low value, and the characteristic in Figure 9.5 follows a steep slope toward the origin. Thus, the term “saturation” in the power BJT refers to the condition where the drift region is fully conductivity-modulated, and the term “quasi-saturation” refers to the condition where the drift region is partially conductivity-modulated.

We will now develop equations describing the carrier density in the drift region and the voltage drop across the drift region in saturation and quasi-saturation. Since the region must remain charge neutral, we can write

9.33

The hole concentration at is given by the law of the junction, Equation 7.25,

9.34

Since the drift region is lightly doped, can exceed at even moderate forward biases , so high-level injection prevails, and in this region we can write

9.35

The electron and hole currents in the drift region are

9.36

9.37

We now assert that the injected hole current flowing into the drift region from the base, , is small compared to the electron current flowing across the base from the emitter, . The situation in the BJT is obviously different from the pin diode, where electrons are injected from the region and holes from the region. In the BJT drift region, the primary electron current comes from electrons injected from the emitter that diffuse across the base into the collector. The hole current injected from the base into the collector is much smaller, due to the gain of the transistor. Recall that . Assuming , the base current is much smaller than the collector current. The base current in saturation consists of holes injected into the emitter plus holes injected into the collector plus holes recombining in the base, so we can be sure that . Thus the collector current is almost totally due to electrons, and we can set in Equation 9.37. Solving for the electric field and employing the Einstein relationship, we find that

9.38

The collector current, which is due primarily to electrons, can be obtained by combining Equation 9.36 with Equations 9.35, and 9.38,

9.39

Again using the Einstein relationship, Equation 9.39 can be written

9.40

This equation can be solved by cross-multiplying both sides by and integrating with respect to . Performing the integration and solving for yields

9.41

Since in saturation, the third term can be dropped, making the hole density a linearly decreasing function of position,

9.42

The reader should recognize that the transport parameters and in Equation 9.39 and the following equations are those of majority carriers (electrons) in the n-type collector drift region. This is different from our earlier development in Equations 9.32, where the transport parameters in each region were those of minority carriers.

As an aside, we note that Equation 9.42 could have been deduced directly from the ambipolar diffusion equation (ADE) Equation 7.39. If recombination in the drift region is negligible, we can set . In steady-state , so the ADE reduces to simply . This means is a linear function of position, and we can write

9.43

Since , Equation 9.40 can be rewritten in the form

9.44

Inserting Equation 9.44 into Equation 9.43 leads directly to Equation 9.42.

Having obtained an expression for the electron and hole densities in the collector drift region as a function of current, Equation 9.42, we now wish to calculate the voltage drop across the drift region. This voltage drop can then be added to the in Equation 9.32 to obtain an relation for the power BJT. To begin, we note from Equation 9.42 that the value at which the hole density equals the background doping density is

9.45

Inserting Equation 9.34 for yields

9.46

Here, refers to the voltage drop across the internal CB junction, which is less than the voltage between the collector and base terminals because of the voltage developed across the collector drift region and the substrate. In Equation 9.46, is obtained from the internal by setting , with given by Equation 9.30.

We assume the drift region is conductivity modulated over the region where the hole density exceeds the background doping density, but not in the region . The voltage drop across the unmodulated portion of the drift region is then given by

9.47

The voltage drop across the conductivity-modulated portion is obtained by integrating the electric field given by Equation 9.38 over the region . Thus we can write

9.48

where is given by Equation 9.34. In evaluating Equation 9.34, we set , with given by Equation 9.30. For SiC power BJTs, is typically less than and can be neglected compared to other voltage drops in the device.

Having obtained expressions for the voltage drop across the collector drift region, we are now in a position to modify the relation given in Equation 9.32 to include these effects. The procedure is as follows. We chose a value for base current and step the internal to calculate using Equation 9.32. To obtain the terminal , we add Equations 9.47 and 9.48 to the internal using Equation 9.46 for . Figure 9.7 shows curves for the transistor of Figure 9.3 with the drift region voltage included (solid lines). For reference, the dotted lines show the curves of Figure 9.3 that do not include the drift region voltage. Points (a) and (b) on the curve denote boundaries between different modulation modes. From the origin to point (a), the drift region is fully conductivity modulated and its differential resistance is very small. Between (a) and (b), the drift region is partially conductivity modulated and the differential resistance is higher. To the right of point (b), the injection level is insufficient to produce any conductivity modulation , and the differential resistance is that of the full unmodulated drift region.

Characteristics like those in Figure 9.7 are often not seen in experimental devices, for several reasons. First, at the somewhat higher dopings needed to maximize , the drift region often never reaches full conductivity modulation as , and point (a) lies very close to the origin. Figure 9.8 shows the device of Figure 9.7 with a drift region doping of instead of . For the drift region used here, this doping provides the optimum and a theoretical blocking voltage of 1580 V. At this higher doping, the device remains in quasi-saturation almost to the origin. A second reason is that lateral effects such as current spreading in the drift region and the voltage drop due to spreading resistance in the base often obscure the breakpoints in the idealized situation of Figure 9.7. Finally, we should caution that the development leading to Equation 9.32 assumes low-level injection in all regions, whereas in quasi-saturation and saturation the collector drift region is in high-level injection. This will alter the parameters and in Equation 9.24 and subsequent equations, and this effect has not been included in Equation 9.32 or Figures 9.7 and 9.8.

9.1.6 High-Current Effects in the Base: the Rittner Effect

We have discussed high-level injection in the collector and examined how conductivity modulation can reduce the drift region voltage drop in saturation. We now turn to high-level injection in the base. When the BJT is operated at high currents, the injection of electrons from the emitter into the base can become large enough that the electron density in the base exceeds the ionized dopant density. When this occurs, charge neutrality demands that the hole concentration in the base also increase so that . The higher hole concentration increases the back injection of holes from the base into the emitter. This is intuitively reasonable, since if we have more holes per unit volume in the base, we expect a greater flow of holes from base to emitter under forward bias conditions. The increased hole flow must be supplied by an increased base current, and this means a lower current gain, since . This phenomenon is known as the Rittner Effect. We will now develop equations to describe this effect, culminating in an expression for as a function of collector current.

To solve for , we need to obtain expressions for the collector and base currents under conditions of high-level injection in the base. If we neglect recombination in the base, the base current in the forward-active region consists exclusively of holes injected from the base into the emitter, the component identified as in Figure 9.1. Since the emitter remains in low-level injection, this current can be found by solving the MCDE in the emitter, subject to the boundary condition imposed by the hole density at the depletion edge, . Our first task is to find this boundary condition.

Invoking the law of the junction on the emitter side of the EB depletion region, we can write

9.49

Likewise, on the base side of the EB depletion region, we can write

9.50

In these equations, denotes the edge of the depletion region in the emitter and 0 denotes the edge of the depletion region in the base. The above equations can be combined to give the hole density in the emitter at the depletion edge,

9.51

Since the emitter remains in low-level injection, the electron density remains at its equilibrium value . However, if the base is in high-level injection, the hole density will be greater than the doping density in the base. Charge neutrality in the base requires that

9.52

Solving for , we can write

9.53

Evaluating Equation 9.53 at and inserting into Equation 9.51 yields

9.54

We see that the boundary condition for holes in the emitter now depends on the injection level in the base . From the law of the junction, we can write

9.55

Inserting Equation 9.55 into Equation 9.54 yields

9.56

Having obtained the boundary condition for the hole density in the emitter, we are now in a position to write expressions for the base and collector currents, and from their ratio we can find . Assuming the emitter width is long compared to the minority diffusion length , the solution of the MCDE for holes in the emitter is a decaying exponential,

9.57

The hole current flowing into the neutral emitter at is given by

9.58

As stated earlier, with negligible recombination in the base, . The collector current is the electron current flowing across the base from the emitter, namely, the components in Figure 9.1 (recall that electron particle flow is in the opposite direction to electron current). With negligible recombination in the base, the electron density decreases linearly from the emitter edge to the collector edge (see Figure 9.6a), and the collector current can be written

9.59

This gives us another expression for , namely

9.60

We now divide Equation 9.59 by Equation 9.58 to obtain

9.61

Inserting Equation 9.55 for and Equation 9.56 for , and substituting Equation 9.60 into Equation 9.56 yields

9.62

The last factor in Equation 9.62 describes the decrease in due to high-level injection in the base. As the collector current density is reduced, Equation 9.62 approaches the low-injection given in Equation 9.19, and we can rewrite Equation 9.62 in the form

9.63

where is the Rittner current density given by

9.64

The Rittner current can be regarded as the collector current density where has fallen to half of its low-injection value. It is obviously desirable to have a high value of the Rittner current, which suggests increasing the base doping and/or reducing the base width. However, given in Equation 9.19 is inversely proportional to , so increasing the doping has the undesirable effect of reducing . Reducing base width increases both and , but to avoid punch-through in the base, the doping-thickness product needs to obey

9.65

Taking the equality in Equation 9.65, the optimum and can be written in terms of the critical field as

9.66

and

9.67

Thus, for highest gain and highest Rittner current, we should reduce while increasing to satisfy the equality in Equation 9.65, thereby preventing punch-through. Equation 9.67 indicates another advantage of SiC for power BJTs, namely a higher Rittner current due to the higher critical field, but the Rittner current is reduced somewhat at room temperature by the incomplete ionization of acceptor dopants in the base.

9.1.7 High-Current Effects in the Collector: Second Breakdown and the Kirk Effect

In the forward-active region of operation, the emitter injects electrons into the base, where they diffuse to the reverse-biased CB junction and are swept into the collector drift region. At high current densities, typical of power BJTs, a significant density of electrons exists in the base, throughout the CB depletion region, and in the collector drift region. As discussed in the previous section, when the density of electrons in the base exceeds the base doping, the base enters high-level injection, and this leads to a reduction in current gain through the Rittner Effect. We now wish to consider conditions in the CB depletion region and the collector drift region.

Focusing first on the CB depletion region, the electric field at any point within the depletion region must obey Poisson's equation. On the collector side of the metallurgical junction we may write

9.68

where is the density of electrons drifting across the depletion region, and we have redefined our coordinate system to place the origin at the CB metallurgical junction. here represents the total donor density, since donors atoms in the depletion region are fully ionized. The CB junction is strongly reverse biased and the electric field is high, so we may assume that electrons are moving at their saturated drift velocity . Under this assumption, the electron density in the depletion region is uniform with respect to position, and we can write

9.69

We can calculate the electric field by inserting Equation 9.69 into Equation 9.68 and integrating with respect to , resulting in

9.70

In equilibrium the collector current is zero, and Equation 9.69 shows there are no electrons in the depletion region. The electric field in this case is illustrated by curve (a) in Figure 9.9. The depletion region spreads much further into the collector due to the asymmetry in doping between the base and collector. As the collector current is increased, the electron density given by Equation 9.69 becomes significant compared to the (very low) doping density , and the field taper given by Equation 9.68 is reduced, resulting in profile (b) in Figure 9.9. Here the depletion region extends through the entire collector drift region, and the field falls rapidly inside the substrate. If the current is increased even more, the electron density given by Equation 9.69 will eventually equal the doping density , at which point the slope of the field profile becomes zero, curve (c). At even higher currents, the electron density will exceed the doping density and the net charge density changes sign. The electric field now increases with distance, resulting in profile (d). Here the peak field has shifted from the CB junction to the junction. If the current continues to increase, the field at the CB junction eventually goes to zero, profile (e). The current density corresponding to profile (e) is called the Kirk current, and can be calculated from Equation 9.70 by setting , and , resulting in

9.71

If exceeds the Kirk current , the electric field shifts toward profile (f). When the peak field at the junction reaches the critical field for avalanche breakdown, as illustrated by profile (f), the device enters second breakdown. Holes generated by impact ionization in the high-field region at the junction flow toward the base and are injected into the emitter, acting as additional base current. This results in a much larger electron current from the emitter, further increasing the collector current and providing positive feedback to the breakdown process. It is important to note that this can occur at collector voltages below that required to cause avalanche breakdown at zero current. This is because at high current densities the field taper in the collector drift region given by Equation 9.70 and shown as profile (f) can exceed the normal field taper when the current is zero, profile (a). Since the area under the electric field remains equal to , which is held constant, the peak field in (f) can be higher than the peak field in (a), even though the voltage has not increased.

Now let us see how these concepts can be related to the critical field of the semiconductor. When the device is in the blocking state with , the electric field is shown in profile (a). As discussed in Section 7.1.1, in a non-punch-through (or NPT) design, the drift layer is totally depleted when the peak field just equals the critical field. Gauss' law and Poisson's equation lead to the relations

9.72

and

9.73

Inserting these expressions into Equation 9.71 yields

9.74

This indicates that for a given blocking voltage , the Kirk current increases as the square of the critical field. The higher critical field of SiC leads to a much higher value for the Kirk current, effectively eliminating second breakdown as an issue in SiC BJTs.

We note that when the electric field has the form of profile (f), the field is zero over the portion of the collector drift region nearest the CB metallurgical junction. Poisson's equation requires that when the field is uniform (and in this case it is uniform at zero), the region must be charge neutral. This neutral region is referred to as the current-induced base, and has width as indicated in Figure 9.9. Since the field is zero in this region, there can be no electron drift, and the collector current is supported entirely by electron diffusion. Since diffusion requires a concentration gradient, and since the region must remain charge neutral throughout, additional holes must be present in a concentration to exactly balance the additional electrons at each point. Electron diffusion in a charge-neutral region is the same process that takes place in the neutral base and, therefore, the current-induced base acts as an extension of the metallurgical base. This effect is referred to as base push-out, and results in a reduction in the current gain in the forward-active region at high current densities.

9.1.8 Common Emitter Current Gain: Temperature Dependence

The common emitter current gain of a narrow-base npn BJT under low-level injection was given in Equation 9.19, where the density of ionized acceptors in the base appears in the denominator. As discussed in Appendix A, aluminum acceptors in 4H-SiC have an ionization energy around 200 meV and not all the acceptors are ionized at room temperature. Figure A.1 shows that at a doping density of only about 30% of the acceptors in the base are ionized and contribute holes at 23 °C. The low density of holes in the base increases both the emitter injection efficiency (Equation 9.18) and (Equation 9.19). However, as temperature goes up, a larger fraction of the acceptors become ionized, and this causes to decrease. For a BJT with base doping of , the common emitter current gain will fall to one-third of its room temperature value at 300 °C. For this reason, it is important to evaluate the performance of the BJT, indeed all SiC power devices, at the highest junction temperature envisioned for the particular application.

The decrease in with temperature is actually beneficial in one respect, since it helps prevent thermal runaway. This makes it possible to parallel multiple BJTs in high-current modules.

9.1.9 Common Emitter Current Gain: the Effect of Recombination

The current gain in SiC BJTs is profoundly affected by recombination taking place in the base and emitter regions, at the surfaces of the base and emitter, at the EB and CB junctions, and in the implanted regions typically used to facilitate base contacts. We will discuss each of these effects below.

Recombination (and generation) is treated in most standard device textbooks. We are concerned here with recombination through deep levels in the bandgap associated with crystal defects, a process known as Shockley–Read–Hall (SRH) recombination. The net recombination rate per unit volume through single-level SRH centers in a bulk crystal can be written

9.75

where and are hole and electron capture cross sections, is the thermal velocity ( at room temperature), is the density of SRH centers per unit volume, and are the hole and electron minority carrier lifetimes, and and are given by

9.76

where is the energy of the SRH center in the bandgap. For midgap centers, and are each equal to .

With appropriate modifications, Equation 9.75 can also be applied to two-dimensional crystal surfaces or interfaces where we have a distribution of states with respect to energy across the bandgap. In these cases, the net recombination rate per unit area can be written

9.77

where is the density of recombination centers per unit area at the surface or interface, and and are the surface recombination velocities for holes and electrons, respectively. In both Equations 9.75 and 9.77 the net recombination rate goes to zero in equilibrium , and becomes negative (net generation) when . Both surface and bulk recombination rates are proportional to the density of SRH centers or , so minimizing the density of such centers is essential to reducing recombination.

In SiC BJTs, recombination events remove electrons injected from the emitter into the base, preventing them from contributing to collector current. Recombination also removes holes injected from the base into the emitter. This increases the gradient of hole density in the emitter, which increases the base current. Both of these effects reduce .

The dominant recombination sites in the BJT are illustrated schematically in Figure 9.10. Recombination can occur through defects in the neutral base and neutral emitter, indicated by (a) in the figure. We will discuss this recombination momentarily. Surface recombination can occur at the top surfaces of the base and emitter, and at the sidewalls of the emitter, labeled (b) in the figure. To minimize surface recombination, it is important to employ the best possible surface passivation, typically a thermal or deposited oxide followed by post-oxidation anneal in nitric oxide [1]. In addition, orienting the emitter fingers so their sidewalls lie on the plane can reduce emitter sidewall recombination, since this plane has a lower surface recombination velocity [1].

Recombination can also occur in the implanted regions under the base contacts, region (c) in the figure, due to crystal damage caused by the implantation [2]. An effective solution is to locate the implants several diffusion lengths (or several base widths, whichever is shorter) away from the emitter edge. Recombination also occurs through defects at the pn junctions, shown as (d) in the figure, especially when the epigrowth is interrupted at these interfaces. This recombination can be reduced by growing the entire npn structure using continuous epigrowth [1]. Finally, we must consider recombination at the emitter ohmic contact, region (e) in the figure. This increases the hole current injected from the base and reduces , but this degradation can be minimized by making the emitter thicker than the hole diffusion length .

Since surface recombination occurs along or close to the emitter edges, can often be increased by using wide emitter fingers, thereby decreasing the perimeter-to-area ratio of the emitter [1]. However, this has to be balanced against the increase in specific on-resistance . At first glance, increasing the emitter area relative to cell area would seem to make the cell more efficient, but we need to consider the effect of base spreading resistance. The lateral sheet resistivity of the base is given by

9.78

This resistance can be significant in SiC BJTs due to the low hole mobility and low acceptor ionization percentage . Base current flowing from the base contact through the lateral resistance of the base produces a lateral voltage drop that reduces under the center of the emitter, reducing both electron and hole currents and . This does not affect , since and are reduced by the same factor, but it makes the interior of the emitter inactive and increases the on-resistance. In choosing the width of the emitter, the designer must evaluate the trade-off between enhancing and degrading .

Process (a) in Figure 9.10 represents SRH recombination within the neutral base and neutral emitter, and this can obviously limit . The dominant SRH centers in bulk 4H-SiC are centers associated with carbon–silicon divacancies, and centers associated with carbon vacancies. These deep levels can be minimized by a 5-h thermal oxidation at 1150 °C prior to the implant activation anneal, followed by another 5-h oxidation after the implant activation anneal [1]. SiC power BJTs receiving such anneals, along with the other measures discussed above, have demonstrated above 250 at room temperature [1].

9.1.10 Blocking Voltage

In the previous sections we considered the on-state performance of the BJT. We now turn to the blocking voltage. In the blocking state, the CB junction is reverse biased and supports the entire collector voltage. The maximum voltage that can be applied to the collector is limited either by punch-through of the neutral base or by avalanche breakdown of the CB junction.

Punch-through occurs when the depletion region of the CB junction extends across the neutral base and merges with the depletion region of the EB junction. When this occurs, the potential barrier confining electrons to the emitter is reduced and a large electron current flows from emitter to collector. The current is no longer controlled by the base, since the neutral portion of the base has vanished. As discussed in Section 10.1, punch-through can be prevented by insuring that the doping-thickness product of the base is large enough that the base cannot be completely depleted before the onset of avalanche breakdown at the CB junction. This condition is given by Equation 10.1. Since Equation 10.1 is satisfied in a well-designed BJT, the blocking voltage will normally be limited by avalanche breakdown.

Avalanche breakdown and edge termination techniques are discussed in Chapter 10, but there is one aspect of the BJT that requires special attention, namely the role of the internal current gain of the BJT. In the blocking state the base current is held at zero, which is equivalent to an open-circuit condition at the base terminal. As will be discussed in Section 10.1.1, avalanche breakdown is the result of impact ionization of carriers in the high-field region of the reverse-biased CB junction. Impact ionization generates electron–hole pairs in the depletion region, and the electric field separates the carriers, drawing the electrons into the collector and holes into the base. With set to zero, these holes cannot flow out the base terminal and instead must flow into the emitter. This is equivalent to an internal base current that is amplified by the gain of the BJT. In other words, for every hole flowing into the emitter, approximately electrons are injected from the emitter into the base. These electrons diffuse across the base and are swept into the CB depletion region where they initiate additional impact ionization. The holes generated by these new ionization events are themselves swept toward the emitter where they cause the injection of even more electrons, and so on. Once initiated, this process increases without bound, resulting in an uncontrolled increase in collector current at a reverse voltage that would be insufficient to cause avalanche breakdown in an isolated CB junction. As discussed in many textbooks, the blocking voltage of the BJT with base open-circuited is related to the breakdown voltage of an isolated CB junction by the equation

9.79

where is a constant, typically between 3 and 6. Since the denominator in Equation 9.79 is greater than unity, the open-base blocking voltage of the BJT is less than the breakdown voltage of the CB junction in isolation.