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<title>Dr. Michael J. Chudobiak - Thesis, Chapter 8</title>
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<h1 align="center"><a name="_Toc366312055" id="_Toc366312055">Chapter 8 - The Theory of Drift Step Recovery Diodes
(DSRD)</a><br></h1>
<h2><a name="s81" id="s81">8.1 - Introduction</a><br></h2>
<p><font size="2">In traditional step recovery diodes charge is stored in the diode by means of a nearly
steady-state forward current flow. That is, the forward bias exists continuously for times comparable to or longer
than the hole and electron lifetimes in the active region. However, the more recent high-power drift step-recovery
diode (DSRD) uses a short forward bias pulse to introduce stored charge to the device [Grek85], [Grek89], [Belk94],
[Foci96]. Since the pulse width is considerably less than the carrier lifetimes, the charge is concentrated near
the junctions, which is desirable for a sharp reverse step recovery. Other more complicated high-power devices have
been designed with similar pulsed biasing in mind [Grek83],[Gorb88]. For instance, by using this "reversible
injection control" [Grek89] two-terminal functional equivalents of the thyristor have been made, which do not
suffer from current localization effects characteristic of three terminal devices. Because of this, these new
structures have been shown to be capable of operating at much higher power levels than conventional structures, and
as such, it has become more important to examine the nature of the forward transient in the basic
p<sup>+</sup>sn<sup>+</sup> diode structure.<br></font></p>
<h2><a name="s82" id="s82">8.2 - The Forward Transient In pin Structures</a><br></h2>
<p><font size="2">It is possible to obtain an analytical solution for the forward transient for two cases, those
being the p<sup>+</sup>in<sup>+</sup> structure, where high injection is implicitly assumed, and the
p<sup>+</sup>sn<sup>+</sup> structure if it is assumed that the s layer (either p-type or n-type) is under low
injection. Since the devices mentioned above are power devices, the low injection case is not of interest here. The
forward current is assumed to be constant for the duration of the transient.<br></font></p>
<p><font size="2">The solution for a rectangular pin structure can be obtained by solving the differential equation
[Vars69]<br></font></p>
<p><img alt="thesis image" src="Img00001.gif" width="198" height="46"><font size="2">(8.1)<br></font></p>
<p><font size="2">where p(x,t) is the excess hole density, X is the distance normalized to the ambipolar diffusion
length L<sub>d</sub>, and T is the time t normalized to the lifetime</font> τ<font size="2">.<br></font></p>
<p><font size="2">Equation (8.1) is subject to the boundary conditions [Vars69],[Bend67]<br></font></p>
<p><img alt="thesis image" src="Img00002.gif" width="122" height="50"><font size="2">(8.2)<br></font></p>
<p><img alt="thesis image" src="Img00003.gif" width="120" height="50"><font size="2">(8.3)<br></font></p>
<p><font size="2">and the initial condition [Vars69]:<br></font></p>
<p><img alt="thesis image" src="Img00004.gif" width="70" height="24"><font size="2">(8.4)<br></font></p>
<p><font size="2">where W<sub>L</sub> is the width of the I region, normalized by L<sub>d</sub>, and J<sub>F</sub>
is the forward current density. In obtaining (8.3), it has been assumed that neutrality exists in the I region,
such that p(X,T) = n(X,T).<br></font></p>
<p><font size="2">Solving (8.1) using Laplace transform methods, subject to (8.2) and (8.3) yields the intermediate
solution<br></font></p>
<p><img alt="thesis image" src="Img00005.gif" width="420" height="62"><font size="2">(8.5)<br></font></p>
<p><font size="2">which can be rewritten as<br></font></p>
<p><img alt="thesis image" src="Img00006.gif" width="425" height="62"><font size="2">(8.6)<br></font></p>
<p><font size="2">Taking the inverse Laplace transform of both sides, under consideration of equation (8.4),
yields<br></font></p>
<p><img alt="thesis image" src="Img00007.gif" width="516" height="65"><font size="2">(8.7)</font></p>
<p><font size="2">or, equivalently,<br></font></p>
<p><img alt="thesis image" src="Img00008.gif" width="534" height="64"><font size="2">(8.8)</font></p>
<p><font size="2">By using the Laplace transform pair [Spie65],<br></font></p>
<p><img alt="thesis image" src="Img00009.gif" width="358" height="62"><font size="2">(8.9)<br></font></p>
<p><font size="2">and integrating both sides of (8.8), the final solution can be obtained:<br></font></p>
<p><img alt="thesis image" src="Img00010.gif" width="570" height="134"></p>
<p><font size="2">(8.10)</font></p>
<p><font size="2">This solution is considerably simpler and more compact than an equivalent solution presented in
[Vars69]. (It should be noted that both equation (8.10) and the solution given in [Vars69] only satisfy the initial
condition given in (8.4) in the limit of t 0, not at t = 0. This is because the boundary conditions (8.2) and (8.3)
are in fact inconsistent with the initial condition, a fact which is not always appreciated.)<br></font></p>
<p><font size="2">The evolution of p(X,T) is shown for</font> τ <font size="2">= 10</font> μ<font size=
"2">s, W = 250</font> μ<font size="2">m, J<sub>F</sub> = 10 A/cm<sup>2</sup> in Figure 8.1. (The steady state
solution is denoted as p<sub>SS</sub>(X)). It is immediately evident that substantial charge injection occurs at
both x = 0 (the p<sup>+</sup>i junction) and at x = W (the in<sup>+</sup> junction) throughout the entire
transient.<br></font></p>
<h2><a name="s83" id="s83">8.3 - The Forward Transient In psn Structures</a><br></h2>
<p><font size="2">If the forward current in a p<sup>+</sup>sn<sup>+</sup> diode is sufficiently large, the injected
carriers will overwhelm the background doping, allowing the analytical results for a pin diode to be used to
determine the transient response. This section quantifies the critical current density above which these results
can be used. It will be shown below that even for very light doping, the expression derived above rapidly becomes
inaccurate.<br></font></p>
<p align="center"><img alt="thesis image" src="Img00011.gif" width="431" height="377"></p>
<p><a name="_Toc366312717" id="_Toc366312717"><font size="2"><i>Figure 8.1. Calculated charge injection in a pin
diode during the forward transient, for several different values of t/</i></font>τ<font size="2"><i>. The
horizontal axis is linear, and the vertical axis is logarithmic. Note that substantial charge injection occurs at
both junctions throughout the entire transient. p</i><sub><i>SS</i></sub><i>(X) is the steady-state
distribution.</i></font><br></a></p>
<p><font size="2">Consider a device with a lightly doped n-type middle layer, and assume that quasi-neutrality
exists in this layer. Then,<br></font></p>
<p><img alt="thesis image" src="Img00012.gif" width="74" height="22"> <font size="2">(8.11)<br></font></p>
<p><font size="2">Also assume that the doping in the p<sup>+</sup> and n<sup>+</sup> regions is much higher than in
the n<sup>-</sup> middle region. Then the current at the p<sup>+</sup>n<sup>-</sup> junction (x=0) will be almost
entirely a hole current, and the current at the n<sup>-</sup>n<sup>+</sup> junction (x=w) will be an electron
current. Mathematically,<br></font></p>
<p><img alt="thesis image" src="Img00013.gif" width="77" height="26"> <font size="2">(8.12)</font></p>
<p><img alt="thesis image" src="Img00014.gif" width="73" height="25"> <font size="2">(8.13)</font></p>
<p><img alt="thesis image" src="Img00015.gif" width="84" height="27"> <font size="2">(8.14)</font></p>
<p><img alt="thesis image" src="Img00016.gif" width="90" height="24"> <font size="2">(8.15)<br></font></p>
<p><font size="2">The transport equations can be written as<br></font></p>
<p><img alt="thesis image" src="Img00017.gif" width="274" height="51"> <font size="2">(8.16)</font></p>
<p><font size="2">and</font></p>
<p><img alt="thesis image" src="Img00018.gif" width="266" height="51"> <font size="2">(8.17)<br></font></p>
<p><font size="2">where V<sub>T</sub> is the thermal voltage kT/q. If (8.14) is substituted into (8.16), the
electric field at x = W can be found:<br></font></p>
<p><img alt="thesis image" src="Img00019.gif" width="172" height="47"> <font size="2">(8.18)<br></font></p>
<p><font size="2">and similarly, from (8.13) and (8.17),<br></font></p>
<p><img alt="thesis image" src="Img00020.gif" width="153" height="50"> <font size="2">(8.19)<br></font></p>
<p><font size="2">This allows the current at either junction to be calculated in terms of the doping and carrier
distribution. If (8.19), (8.12) and (8.11) are substituted into (8.16), one obtains<br></font></p>
<p><img alt="thesis image" src="Img00021.gif" width="220" height="54"> <font size="2">(8.20)<br></font></p>
<p><font size="2">Similarly, using (8.18), (8.15) and (8.11) in (8.17) gives<br></font></p>
<p><img alt="thesis image" src="Img00022.gif" width="265" height="51"> <font size="2">(8.21)<br></font></p>
<p><font size="2">If the first terms in equations (8.16) and (8.17) are identified as diffusion terms, and written
as J</font><font size="4"><sub>diff</sub></font><font size="2">, and if we define<br></font></p>
<p><img alt="thesis image" src="Img00023.gif" width="140" height="47"> <font size="2">(8.22)<br></font></p>
<p><font size="2">then (8.20) and (8.21) can be rewritten in the form:<br></font></p>
<p><img alt="thesis image" src="Img00024.gif" width="166" height="71"> <font size="2">(8.23)</font></p>
<p><font size="2">and</font></p>
<p><img alt="thesis image" src="Img00025.gif" width="161" height="47"> <font size="2">(8.24)<br></font></p>
<p><font size="2">Equations (8.23) and (8.24) show that the balance of the drift and diffusion currents at the
junctions is affected by the presence of doping in the middle layer. These two functions are plotted as a function
of f in Figure 8.2. As N<sub>D</sub></font> → <font size="2">0, and hence f</font> → <font size="2">0,
both functions approach the value 0.5, which of course leads to equations (8.2) and (8.3). In other words, the
drift and diffusion currents at both junctions in a pin diode are equal. This remains largely true for f < 0.1.
As N<sub>D</sub> increases, and f increases correspondingly, the current at the n<sup>-</sup>n<sup>+</sup> junction
is dominated by a drift current, and the current at the p<sup>+</sup>n<sup>-</sup> junction is dominated by the
diffusion component. Since the diffusion current at the high-low junction becomes small, dn/dx is also small and
very little charge is built up at the high-low junction.</font></p>
<p align="center"><img alt="thesis image" src="Img00026.gif" width="326" height="300"></p>
<p><a name="_Toc366312718" id="_Toc366312718"><font size="2"><i>Figure 8.2. Diffusion current as a fraction of the
total current, at both junctions. In the limiting case of no middle-layer doping, f = 0 and the diffusion and drift
components are equal. For very heavy doping, f</i></font> → ∞ <font size="2"><i>and the high-low
junction current is almost entirely drift current, and the p</i><sup><i>+</i></sup><i>n junction current is almost
entirely diffusion current.<br></i></font></a></p>
<p><font size="2">It is straightforward to show that f increases rapidly with N<sub>D</sub>, if one considers the
time imediately after the beginning of the forward transient current pulse. At this early time, no holes will have
yet traveled to the n<sup>-</sup>n<sup>+</sup> junction, so we can write:<br></font></p>
<p><img alt="thesis image" src="Img00027.gif" width="162" height="42"> <font size="2">(8.25)<br></font></p>
<p><font size="2">The current in the bulk of the middle layer must be ohmic, since little charge has been injected
and n</font>>> <font size="2">N<sub>D</sub>. Thus, in this region,<br></font></p>
<p><img alt="thesis image" src="Img00028.gif" width="102" height="46"> <font size="2">(8.26)<br></font></p>
<p><font size="2">If this bulk electric field is assumed to build up from zero over a short distance</font>
Δ<font size="2">x near the n<sup>-</sup>n<sup>+</sup> junction, then the derivative dE/dx in (8.25) can be
approximated as E<sub>bulk</sub>/</font>Δ<font size="2">x. Then, combining (8.25), (8.26) and (8.22) so as to
eliminate n(w,0<sup>+</sup>) yields<br></font></p>
<p><img alt="thesis image" src="Img00029.gif" width="266" height="52"> <font size="2">(8.27)<br></font></p>
<p><font size="2">An estimate for</font> Δ<font size="2">x can be obtained by writing<br></font></p>
<p><img alt="thesis image" src="Img00030.gif" width="220" height="134"> <font size="2">(8.28)<br></font></p>
<p><font size="2">The value of dn/dx in (8.28) can be estimated from the electron diffusion current. Of course, the
diffusion current to total current ratio varies with f, as discussed above. The most interesting case is for f =1,
where the diffusion current is 1/3 of J<sub>F</sub>, since the diode behaviors for f >> 1 and f << 1
are quite different. The case of f =1 is a "critical" boundary case. Thus,<br></font></p>
<p><img alt="thesis image" src="Img00031.gif" width="102" height="46"> <font size="2">(8.29)<br></font></p>
<p><font size="2">Then combining (8.26) to (8.29) to eliminate</font> Δ<font size="2">x
yields:<br></font></p>
<p><img alt="thesis image" src="Img00032.gif" width="96" height="46"> <font size="2">(8.30)</font></p>
<p><font size="2">where</font></p>
<p><img alt="thesis image" src="Img00033.gif" width="166" height="54"> <font size="2">(8.31)<br></font></p>
<p><font size="2">Hence, for current densities substantially larger than J<sub>0</sub> (say J<sub>F</sub> > 10
J<sub>0</sub>), the diode acts as though it were intrinsic, leading to balanced drift and diffusion currents, and
charge injection from both junctions. For current densities substantially less than J<sub>0</sub> (say
J<sub>F</sub> < 0.1 J<sub>0</sub>), the charge injection will be dominated by the p<sup>+</sup>n<sup>-</sup>
junction, and relatively little charge will be stored at the high-low junction. Since J<sub>0</sub></font> ∝
<font size="2">N<sub>D</sub><sup>3/2</sup>, the critical current density J<sub>F</sub> = J<sub>0</sub>
(corresponding to f = 1) increases moderately quickly with doping, and the usefulness of the intrinsic
approximation becomes restricted for even relatively light doping levels.<br></font></p>
<p><font size="2">Physically, equation (8.22) shows that the condition f(x,t) = 1 corresponds to an injected
carrier density of n(x,t) = 2 N<sub>D</sub>. Thus, if the forward current is large enough such that
n(w,0<sup>+</sup>) >> 2 N<sub>D</sub> at the high-low junction immediately after the beginning of the
transient (i.e. J<sub>F</sub> >> J<sub>0</sub>), the diode will act as a pin diode.<br></font></p>
<p><font size="2">The validity of this estimate of J<sub>0</sub> is shown by comparison with MEDICI simulations in
Figure 8.3. The hole distribution at t = 60 ns is shown for several values of N<sub>D</sub>, and hence
J<sub>0</sub> (calculated from (8.31)), with = 10</font> μ<font size="2">s, W = 250</font> μ<font size="2">m,
and J<sub>F</sub> = 10 A/cm<sup>2</sup>. (Since the time is the same in each case, the total charge in each of the
diode structures is approximately equal.) Clearly, the hole distributions for J<sub>F</sub>/J<sub>0</sub> equal to
(i.e., a pin diode), 100, and 10 are very similar and show significant charge injection from both junctions, as
expected. In contrast, the hole distributions for J<sub>F</sub>/J<sub>0</sub> equal to 0.1 and 0.01 show injection
at the p<sup>+</sup>n junction only, as expected. The curve for J<sub>F</sub>/J<sub>0</sub> =1 is an intermediate
case, showing some charge injection from the high-low junction, but much less than for the cases with larger
J<sub>F</sub>/J<sub>0</sub> ratios. These simulations confirm the theoretical results derived above. (It should be
noted that the doping corresponding to J<sub>0</sub> is quite small - only 5.6x10<sup>13</sup>
cm<sup>-3</sup>.)<br></font></p>
<p align="center"><img alt="thesis image" src="Img00034.gif" width="386" height="354"></p>
<p><a name="_Toc366312719" id="_Toc366312719"><font size="2"><i>Figure 8.3. Simulations calculated using the MEDICI
simulator to confirm the validity of the derived expression for J</i><sub><i>0</i></sub><i>.
J</i><sub><i>0</i></sub> <i>is calculated from (8.31). For
J</i><sub><i>F</i></sub><i>/J</i><sub><i>0</i></sub><i>> 10, the diode behaves like a pin diode, with
substantial charge injection at both junctions. For J</i><sub><i>F</i></sub><i>/J</i><sub><i>0</i></sub><i><
0.1, charge injection occurs exclusively at the p</i><sup><i>+</i></sup><i>n</i><sup><i>-</i></sup><i>junction.
J</i><sub><i>F</i></sub> <i>= J</i><sub><i>0</i></sub><i>is an intermediate case. In each case
J</i><sub><i>F</i></sub><i>= 10 A/cm</i><sup><i>2</i></sup><i>, and N</i><sub><i>D</i></sub><i>is varied to change
J</i><sub><i>0</i></sub><i>. In order of decreasing J</i><sub><i>F</i></sub><i>/J</i><sub><i>0</i></sub><i>the
corresponding values of N</i><sub><i>D</i></sub> <i>are 2.6x10</i><sup><i>12</i></sup><i>,
1.2x10</i><sup><i>13</i></sup><i>, 5.6x10</i><sup><i>13</i></sup><i>, 2.6x10</i><sup><i>14</i></sup><i>, and
1.2x10</i><sup><i>15</i></sup> <i>cm</i><sup><i>-3</i></sup><i>.<br>
<br></i></font></a></p>
<p><font size="2">Equation (8.30) predicts how the charge is injected at short times after the beginning of the
transient. Obviously, as time progresses, the value of f(W,t) will change, since substantial charge is stored near
the high-low junction in the steady state. The key turning point occurs when injected holes from the p+n junction
reach the high-low junction. This of course will increase n(W,t), and increase f(W,t). In other words, double
injection occurs [Dean69], and injected charge will rapidly build up at the high-low junction after this time. This
time can be estimated by dividing the middle region width W, by the drift velocity, such that<br></font></p>
<p><img alt="thesis image" src="Img00035.gif" width="92" height="49"> <font size="2">(8.32)<br></font></p>
<p><font size="2">This can be rewritten using (8.26):<br></font></p>
<p><img alt="thesis image" src="Img00036.gif" width="106" height="46"> <font size="2">(8.33)</font></p>
<p><font size="2">where</font></p>
<p><img alt="thesis image" src="Img00037.gif" width="50" height="47"> <font size="2">(8.34)<br></font></p>
<p><font size="2">Figure 8.4 illustrates the validity of this calculation for W = 250 m and J<sub>F</sub> = 10
A/cm<sup>2</sup>, for a range of N<sub>D</sub> from 10<sup>14</sup> to 5x10<sup>14</sup> cm<sup>-3</sup>. The hole
concentration is plotted at t = t<sub>di</sub> for each particular doping. In each case, the peak injected charge
at the high-low junction is just beginning to become significant, reaching a density approximately equal to
N<sub>D</sub>. (Figure 8.1 shows that the steady state distribution is between 10<sup>16</sup> and 10<sup>17</sup>
cm<sup>-3</sup>, about two orders of magnitude higher than N<sub>D</sub>.)<br></font></p>
<p><font size="2">Of course, even after t = t<sub>di</sub>, diodes with J<sub>F</sub> < J<sub>0</sub> will still
have less charge injected at the high-low junction than diodes with J<sub>F</sub> > J<sub>0</sub>, but the
difference will be less noticeable than for t < t<sub>di</sub>.<br></font></p>
<p align="center"><img alt="thesis image" src="Img00038.gif" width="365" height="336"></p>
<p><a name="_Toc366312720" id="_Toc366312720"><font size="2"><i>Figure 8.4. Simulations calculated using the MEDICI
simulator. The injected hole density at the high-low junction at t = t</i><sub><i>di</i></sub><i>is shown for
several different dopings. In each case, the peak density</i></font> >><font size=
"2"><i>N</i><sub><i>D</i></sub><i>. The charge injected at the high-low junction grows rapidly after t =
t</i><sub><i>di</i></sub><i>, due to the onset of double injection.</i></font><br>
<br></a></p>
<h2><a name="s84" id="s84">8.4 - Implications For Pulse Sharpening Diodes Design Theory</a><br></h2>
<p><font size="2">The use of drift step-recovery diodes in pulse sharpening applications has been described in
[Grek85], [Grek89], [Belk94], [Foci96]. An important characteristic describing the reverse transient of any step
recovery diode is the ramp voltage, which is the voltage built up across the diode just before the fast sharpening
transient begins. The ramp voltage should be as small as possible to obtain an ideal step waveform. In the DSRD,
the fast transient begins when the charge sweeping-out boundary [Bend67] emanating from the p<sup>+</sup>n junction
meets the sweeping-out boundary emanating from the high-low junction. If the distance between the p<sup>+</sup>n
junction and the meeting point of the sweeping-out boundaries is termed W<sub>Q</sub>, the ramp voltage
V<sub>RAMP</sub> can be found by applying Poisson's equation to the fixed ionized charge and the mobile charge,
assumed to be moving at the saturation velocity v<sub>S</sub>. Thus,<br></font></p>
<p><img alt="thesis image" src="Img00039.gif" width="185" height="51"> <font size="2">(8.35)<br></font></p>
<p><font size="2">The voltage developed across the quasineutral region between x = W<sub>Q</sub> and the high-low
junction will be much smaller than the voltage developed across the p-n junction space charge region [Bend67], and
is ignored.<br></font></p>
<p><font size="2">To minimize V<sub>RAMP</sub>, it is necessary to minimize W<sub>Q</sub>. This implies maximizing
|dn/dx| at the p<sup>+</sup>n<sup>-</sup> junction such that the charge injected by the forward transient is kept
very close to the junction. From the preceding section, we can see that this requires that J<sub>F</sub> <
J<sub>0</sub>. Significant improvement in ramp voltage will occur as J<sub>F</sub> is brought down from
10J<sub>0</sub> to J<sub>0</sub>/10, and relatively little improvement will occur below this, as suggested by
Figure 8.2. For instance, a DSRD designed to operate at 1700 V, with N<sub>D</sub> = 10<sup>14</sup>
cm<sup>-3</sup>, will have J<sub>0</sub> = 24.8 A/cm<sup>2</sup>. For a cross sectional area of 0.3 cm<sup>2</sup>,
this corresponds to I<sub>0</sub> = 7.4 A. A diode with these parameters was manufactured and presented in
[Grek85], where a value of I<sub>F</sub> = 3 A was used. Clearly these values of J<sub>F</sub> and I<sub>F</sub>
fall within the predicted desirable range. Decreasing J<sub>F</sub> further would have had the undesirable effect
of either increasing the cross-sectional area and the junction capacitance, or increasing the forward pulse width
t<sub>F</sub>, which has its own limits as discussed below.<br></font></p>
<p><font size="2">The expression derived for J<sub>0</sub> can also be used to estimate the maximum charge
consistent with good step recovery action that can be stored in a DSRD. The charge stored in a DSRD during the
forward bias pulse is given by<br></font></p>
<p><img alt="thesis image" src="Img00040.gif" width="73" height="22"> <font size="2">(8.36)<br></font></p>
<p><font size="2">where t<sub>F</sub> is the duration of the forward pulse. Grekhov noted in [Grek85] that for the
injected charge to remain near the junction, t<sub>F</sub> should be much smaller than the diode transit time
t<sub>T</sub>, where<br></font></p>
<p><img alt="thesis image" src="Img00041.gif" width="69" height="49"> <font size="2">(8.37)<br></font></p>
<p><font size="2">If we choose</font></p>
<p><img alt="thesis image" src="Img00042.gif" width="54" height="42"> <font size="2">(8.38)</font></p>
<p><font size="2">and</font></p>
<p><img alt="thesis image" src="Img00043.gif" width="62" height="42"> <font size="2">(8.39)<br></font></p>
<p><font size="2">(such that the effective characteristic length, L<sub>eff</sub>, of the injected carrier
distribution is W/10) as reasonable maximum values, the maximum Q<sub>+</sub> can be determined as a function of
N<sub>D</sub> and W. A more useful exercise is to calculate the reverse transient storage time t<sub>S</sub> as a
function of V<sub>BR</sub> and W, where<br></font></p>
<p><img alt="thesis image" src="Img00044.gif" width="102" height="46"> <font size="2">(8.40)<br></font></p>
<p><font size="2">where Q<sub>-</sub> is the maximum charge that can be removed from the diode during the reverse
transient. It is important to note that Q<sub>+</sub> and Q<sub>-</sub> are not necessarily the same thing. For a
wide-base diode under low injection, the maximum net charge that can be extracted from a diode is Q<sub>+</sub>/2
[Lind65]. Similarly, for a narrow-base diode, Q<sub>-</sub> = 2/3 Q<sub>+</sub>. However, for the high-injection
case examined here,<br></font></p>
<p><img alt="thesis image" src="Img00045.gif" width="69" height="22"> <font size="2">(8.41)<br></font></p>
<p><font size="2">as Belkin and Shulzchenko [Belk94] and Focia et al. [Foci96] have reported experimentally.
Grekhov's [Grek85] reported experimental values of Q<sub>+</sub> = (400 ns)(3 A) and Q- = ½ (50 ns)(34 A),
resulting in Q<sub>-</sub>/Q<sub>+</sub> = 0.71; however computer simulations reported below cast some doubt on the
accuracy of this.<br></font></p>
<p><font size="2">The breakdown voltage V<sub>BR</sub> can be calculated from [Bali87]<br></font></p>
<p><img alt="thesis image" src="Img00046.gif" width="145" height="35"> <font size="2">(8.42)<br></font></p>
<p><font size="2">where N<sub>D</sub> is in cm<sup>-3</sup>. In equation (8.40), it is assumed that the diode is
operated just below breakdown, and that the pulse to be sharpened is a linear ramp (hence the average voltage of
V<sub>BR</sub>/2). To determine the cross-sectional area of the device, the DSRD design equation from [Grek85] is
used:<br></font></p>
<p><img alt="thesis image" src="Img00047.gif" width="102" height="42"> <font size="2">(8.43)<br></font></p>
<p><font size="2">Consideration of equations (8.31) and (8.36)-(8.43), and assuming R = 50</font>
Ω<font size="2">, allows N<sub>D</sub> and A to be chosen for a particular V<sub>BR</sub>. However, this
leaves one unspecified physical parameter, W. Making W large will increase the maximum stored charge Q<sub>+</sub>,
but it will also increase V<sub>RAMP</sub>. Ultimately, computer simulations are required to confirm the proper
choice of W. However, for the convenience of calculations, the width W with be normalized as a parameter, called
the "width factor", or WF, where:<br></font></p>
<p><img alt="thesis image" src="Img00048.gif" width="182" height="27"> <font size="2">(8.44)<br></font></p>
<p><font size="2">For WF = 1, W is equal to the width of the depletion region at breakdown [Bali87]. (In (8.44), W
is in cm and N<sub>D</sub> is in cm<sup>-3</sup>.) In other words, for WF=1, the depletion region consumes the
entire middle layer at V<sub>BR</sub>. For WF > 1, portions of the middle layers are never covered by the
depletion region, and for WF < 1 the diode is a punchthrough device. (If a punchthrough structure is used,
equation (8.42) no longer applies.)<br></font></p>
<p><font size="2">Consideration of equations (8.34)-(8.44) produces the plot shown in Figure 8.5. The device
presented in [Grek85] corresponds to WF = 1.65, and V<sub>BR</sub> = 1700 V, for which the ideal t<sub>S</sub> is
predicted to be about 57 ns. This agrees rather well with the 50 ns value that was used in the experiment. It was
reported that the step recovery action degraded noticeably above 50 ns, as one would expect.<br></font></p>
<p align="center"><img alt="thesis image" src="Img00049.gif" width="427" height="303"></p>
<p><a name="_Toc366312721" id="_Toc366312721"><font size="2"><i>Figure 8.5. The curves on this design chart show
the maximum practical storage time t</i><sub><i>S</i></sub><i>, in nanoseconds, for a psn diode with a middle-layer
width factor WF of 1, 1.65, 2, or 3, and breakdown voltage V</i><sub><i>BR</i></sub> <i>(in
Volts).<br></i></font></a></p>
<p><font size="2">It is apparent from Figure 8.5 that the DSRD structure is of little use below 500 V, as the
maximum useful storage times become very short.<br></font></p>
<p><font size="2">Previous design approaches for DSRDs [Grek85] did not specify a simple method of choosing
J<sub>F</sub> and w. The equations presented above, in the form of (8.31) and Figure 8.5, partially rectify this
situation. The equations given above do not guarantee that a given diode structure can be used as a DSRD. Choosing
J<sub>F</sub> << J<sub>0</sub> ensures that the ramp voltage is minimized as much as possible for a
particular structure, but it does not ensure that the ramp voltage is insignificant relative to V<sub>BR</sub>. To
calculate V<sub>RAMP</sub> exactly computer simulations are required. The next section reports the results of such
simulations.<br></font></p>
<h2><a name="s85" id="s85">8.5 - DSRD Ramp Voltage</a><br></h2>
<p><font size="2">To develop a design approach for the DSRD ramp voltage, simulations were performed using the
MEDICI device simulator, for fifteen devices. Values of N<sub>D</sub>, A, and I<sub>F</sub> were determined using
the theory presented in the last section for devices with V<sub>BR</sub> values of 500 V, 1000 V, 1500 V, 2000 V,
and 2500 V. For each of these voltages, three values of WF were considered: 1, 2, and 3. Knowledge of WF allowed
t<sub>F</sub> and t<sub>S</sub> to be calculated for each device.<br></font></p>
<p><font size="2">The results of the transient simulations are very simple. For WF = 1,
V<sub>RAMP</sub>/V<sub>BR</sub> = 0.1, regardless of V<sub>BR</sub>. Similarly, for WF = 2,
V<sub>RAMP</sub>/V<sub>BR</sub> = 0.25, regardless of V<sub>BR</sub>, and for WF = 3,
V<sub>RAMP</sub>/V<sub>BR</sub> = 0.4, regardless of V<sub>BR</sub>. It is not surprising that for a given WF the
V<sub>RAMP</sub>/V<sub>BR</sub> ratio is independent of V<sub>BR</sub>, because for each diode with the same WF the
shape of the injected charge is identical, if it is normalized to the width of the middle layer. Thus, the
normalized position where the sweeping-out boundaries from the left and right meet (initiating the step recovery
action) will be identical for diodes with the same WF. Similarly, the normalized width of the depletion region at
V<sub>BR</sub> is also identical for devices with a given WF, by definition. Thus the ratio of the two positions,
and hence the ratio of the corresponding voltages (V<sub>RAMP</sub> and V<sub>BR</sub>) will be
identical.<br></font></p>
<p><font size="2">This considerably simplifies the design of DSRDs. Typically, one wishes to have
V<sub>RAMP</sub>/V<sub>BR</sub></font> ≤ <font size="2">0.1, and as large a storage time as possible, so WF = 1
is the ideal choice. With WF fixed, all parameters are now uniquely specified for a given V<sub>BR</sub>. With this
in mind, Figure 8.5 can be simplified and enlarged, as in Figure 8.6. It is now evident that the DSRD is restricted
to high voltages, of at least 1 kV, if V<sub>RAMP</sub>/V<sub>BR</sub></font> ≤ <font size="2">0.1 is to be
achieved. Below 1 kV, the storage time becomes too short to work with, as does the forward bias pulse width,
t<sub>F</sub>.<br></font></p>
<p align="center"><img alt="thesis image" src="Img00050.gif" width="411" height="305"></p>
<p><a name="_Toc366312722" id="_Toc366312722"><font size="2"><i>Figure 8.6. Maximum practical storage time
t</i><sub><i>S</i></sub><i>, and the corresponding forward bias time t</i><sub><i>F</i></sub><i>, in nanoseconds,
for a psn diode with the ideal middle-layer width factor WF of 1 and breakdown voltage V</i><sub><i>BR</i></sub>
<i>(in Volts).<br></i></font></a></p>
<h2><a name="s86" id="s86">8.6 - DSRD Transition Times</a><br></h2>
<p><font size="2">Grekhov et al [Grek85] calculated that the maximum voltage rate of change for saturation-velocity
limited silicon devices is 2000 V/ns. Since the current density and carrier distributions change radically during
the diode reverse transient, it is not possible to sustain the extraction velocity of the carriers at the
saturation velocity for the entire step recovery transient, so in practice one can only expect to achieve a
fraction of this maximum transition speed. The switching time results (t<sub>R</sub>) from some of the simulations
described in the previous section (with the addition several high-voltage devices) are summarized in Table 8.1,
along with the physical and electric parameters used in the simulations. Table 8.1 shows that the maximum
realizable voltage-rate-of-change is about 500 V/ns, which is the approximate maximum value for conventional SRDs
and WFSRDs as well (see Table 1.1). (It is not clear why there appears to be a rate-of-change minimum around the
2500 V device.) Also, the low values of</font> τ<font size="2"><sub>EFF</sub> at and below 1 kV again show the
operating voltage restrictions of the DSRD.<br></font></p>
<p><font size="2">Figure 8.7 shows a typical simulated waveform for the 4000 V device specified in Table 8.1.<br>
<br></font> <a name="_Toc366312389" id="_Toc366312389"><font size="2">Table 8.1 - Switching times for various
DSRDs, with WF = 1.</font> τ<font size="2"><sub>EFF</sub> is calculated from equation (3.4), using the values
in the table.</font></a> <font size="2"><br></font></p>
<table border="1">
<tr>
<td width="48"> </td>
<td colspan="3" width="198">
Calculated
<p align="center">Physical Parameters</p>
</td>
<td colspan="3" width="144">Calc. Electrical Parameters</td>
<td colspan="4" width="232">Simulation Results</td>
</tr>
<tr>
<td width="48">V<sub>BR</sub>, V</td>
<td width="72">N<sub>D</sub>, cm<sup>-3</sup></td>
<td width="63">A, mm<sup>2</sup></td>
<td width="63">L, μm</td>
<td width="48">
I<sub>F</sub>,
<p align="center">A</p>
</td>
<td width="48">
t<sub>F</sub>,
<p align="center">ns</p>
</td>
<td width="48">
t<sub>S</sub>,
<p align="center">ns</p>
</td>
<td width="56">
t<sub>R</sub>,
<p align="center">ns</p>
</td>
<td width="60">V<sub>BR</sub>/t<sub>R</sub>, V/ns</td>
<td width="60">V<sub>RAMP</sub>,V</td>
<td width="56">τ<sub>EFF</sub>, ns</td>
</tr>
<tr>
<td width="48">500</td>
<td width="72">5.110<sup>14</sup></td>
<td width="63">1.75</td>
<td width="63">36.3</td>
<td width="48">2.47</td>
<td width="48">5.5</td>
<td width="48">2.7</td>
<td width="56">1.0</td>
<td width="60">500</td>
<td width="60">50</td>
<td width="56">12.2</td>
</tr>
<tr>
<td width="48">1000</td>
<td width="72">2.010<sup>14</sup></td>
<td width="63">8.83</td>
<td width="63">81.5</td>
<td width="48">3.11</td>
<td width="48">27.7</td>
<td width="48">8.6</td>
<td width="56">2.1</td>
<td width="60">476</td>
<td width="60">100</td>
<td width="56">59.5</td>
</tr>
<tr>
<td width="48">1500</td>
<td width="72">1.210<sup>14</sup></td>
<td width="63">22.7</td>
<td width="63">131</td>
<td width="48">3.56</td>
<td width="48">71.2</td>
<td width="48">16.9</td>
<td width="56">3.1</td>
<td width="60">484</td>
<td width="60">150</td>
<td width="56">151</td>
</tr>
<tr>
<td width="48">1700</td>
<td width="72">9.910<sup>13</sup></td>
<td width="63">30.5</td>
<td width="63">151</td>
<td width="48">3.71</td>
<td width="48">95.4</td>
<td width="48">20.8</td>
<td width="56">3.6</td>
<td width="60">472</td>
<td width="60">170</td>
<td width="56">208</td>
</tr>
<tr>
<td width="48">2000</td>
<td width="72">8.010<sup>13</sup></td>
<td width="63">44.5</td>
<td width="63">183</td>
<td width="48">3.92</td>
<td width="48">139</td>
<td width="48">27.3</td>
<td width="56">5.5</td>
<td width="60">364</td>
<td width="60">200</td>
<td width="56">292</td>
</tr>
<tr>
<td width="48">2500</td>
<td width="72">5.910<sup>13</sup></td>
<td width="63">74.9</td>
<td width="63">237</td>
<td width="48">4.22</td>
<td width="48">235</td>
<td width="48">39.7</td>
<td width="56">8.9</td>
<td width="60">281</td>
<td width="60">250</td>
<td width="56">590</td>
</tr>
<tr>
<td width="48">3000</td>
<td width="72">4.710<sup>13</sup></td>
<td width="63">114.6</td>
<td width="63">293</td>
<td width="48">4.49</td>
<td width="48">359</td>
<td width="48">53.7</td>
<td width="56">7.4</td>
<td width="60">405</td>
<td width="60">300</td>
<td width="56">744</td>
</tr>
<tr>
<td width="48">4000</td>
<td width="72">3.210<sup>13</sup></td>
<td width="63">224.3</td>
<td width="63">411</td>
<td width="48">4.94</td>
<td width="48">703</td>
<td width="48">86.8</td>
<td width="56">7.8</td>
<td width="60">513</td>
<td width="60">400</td>
<td width="56">1450</td>
</tr>
<tr>
<td width="48">5000</td>
<td width="72">2.410<sup>13</sup></td>
<td width="63">377.5</td>
<td width="63">533</td>
<td width="48">5.32</td>
<td width="48">1183</td>
<td width="48">126</td>
<td width="56">9.9</td>
<td width="60">505</td>
<td width="60">500</td>
<td width="56">2430</td>
</tr>
</table>
<p align="center"><img alt="thesis image" src="Img00051.gif" width="562" height="393"></p>
<p><a name="_Toc366312723" id="_Toc366312723"><i>Figure 8.7 - Optimum pulse sharpening action of the 4000 V device
described in Table 8.1. The sharpened output 10%-90% rise time is 7.8 ns.<br></i></a></p>
<p>These results do not agree entirely with Grekhov's [Grek85]. In [Grek85], a 1700 V diode was reported with a 1.5
ns rise time, giving a V<sub>BR</sub>/t<sub>R</sub> ratio of 1133 V/ns. This seems overly fast, compared to the
results listed in Tables 8.1 and 1.1. Figure 8.8 shows the results of a Medici simulation reproducing the
conditions described in [Grek85] (i.e., N<sub>D</sub> = 10<sup>14</sup> cm<sup>-3</sup>, A = 30 mm<sup>2</sup>, L =
250 μm, I<sub>F</sub> = 3 A, t<sub>F</sub> = 400 ns, t<sub>S</sub> = 40 ns). From the simulation, the 10%-90%
rise time is calculated to be 15.2 ns. If just the fast part of the transient is considered, from 22% - 90%, the
corresponding rise time is 3.7 ns, which is in line with the results of the simulations given in Table 8.1. It is
not clear what transition time definition was used in [Grek85], particularly since the output waveform appears to
be hand-drawn, rather than photographed. Similarly, Grekhov has reported a 2000 V, 2 ns device in [Grek89],
yielding 1000 V/ns, but again the actual output waveform photo is not shown. For the same voltage, the optimized
device of Table 8.1 indicates a 10%-90% rise time of 5.5 ns.<br></p>
<p align="center"><img alt="thesis image" src="Img00052.gif" width="572" height="381"></p>
<p><a name="_Toc366312724" id="_Toc366312724"><i>Figure 8.8 - Simulation results for the diode and circuit
conditions in [Grek85].</i><br></a></p>
<p>Also, Figure 8.8 suggests that the value of Q<sub>-</sub>/Q<sub>+</sub> of 0.71, calculated from the values
given in [Grek85] is too low, as additional charge is removed from the diode after the input voltage has reached
1700 V. In other words, the voltage ramp time could have been increased to Q<sub>-</sub>/Q<sub>+</sub> = 1. This
simulation result tends to support the use of equation (8.41), and agrees with [Belk94].<br></p>
<p>The simulated results are in better agreement with the recent experimental results reported in [Foci96]. A rise
time of approximately 5 ns is reported for a voltage swing of 1700 V, for an average switching rate of 340 V/ns.
Also, the reported values of storage time and I<sub>F</sub>/I<sub>R</sub> yield an effective lifetime of 250 ns.
Both of these values agree reasonably well with the results for the 1700V device given in Table 8.1.<br></p>
<h2><a name="s87" id="s87">8.7 - Other DSRD Issues</a><br></h2>
<p><font size="2">The maximum operating voltage for a single DSRD is somewhat limited by the fact that the diode
area increases very rapidly (and undesirably) with voltage. Combining equations (8.42) and (8.43) shows
that<br></font></p>
<p><img alt="thesis image" src="Img00053.gif" width="65" height="31"> <font size="2">(8.45)<br></font></p>
<p><font size="2">whereas the desirable increase in storage time t<sub>S</sub> is much slower:<br></font></p>
<p><img alt="thesis image" src="Img00054.gif" width="68" height="31"> <font size="2">(8.46)<br></font></p>
<p><font size="2">For this reason, at higher voltages it may be advantageous to use multiple series-connected DSRDs
rather than a single device. Belkin and Shulzchenko [Belk94] used this approach to obtain 6 kV pulses, with four
lower-voltage DSRDs connected in series. This approach also has the advantage that higher middle layer dopings can
be used, which makes device fabrication easier.<br></font></p>
<h2><a name="s88" id="s88">8.8 - Conclusion</a><br></h2>
<p><font size="2">In this section, the evolution of the carrier distributions in p<sup>+</sup>sn<sup>+</sup> diodes
during the forward transient has been considered. A critical current density, J<sub>0</sub>, has been derived. For
J >> J<sub>0</sub>, a p<sup>+</sup>sn<sup>+</sup> diode will behave as a pin diode, with significant charge
injection at both junctions. For J << J<sub>0</sub>, significant charge injection will occur only at the pn
junction for times t < t<sub>di</sub>. For t > t<sub>di</sub>, carriers will be injected by both junctions.
Interestingly, doping levels in the middle layer (typically 10<sup>14</sup> cm<sup>-3</sup>) can be orders of
magnitude less than the forward steady state carrier concentrations (typically > 10<sup>16</sup>
cm<sup>-3</sup>), and yet can dramatically affect the evolution of the carrier distributions.<br></font></p>
<p><font size="2">The critical current J<sub>0</sub> has been shown to be an important parameter in the design of
drift step recovery diodes. To minimize the ramp voltage, J<sub>F</sub> should be less than J<sub>0</sub>. Also,
knowledge of J<sub>0</sub> allows an estimate of the maximum usable stored charge in a DSRD. This results in a much
more comprehensive design theory for DSRDs than that which was previously available. This parameter should prove
useful in the design of several other high-power devices that rely upon transient forward biasing, or reversible
injection control.<br></font></p>
</div>
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