Breakdown Voltage for Superjunction Power Devices With
Charge Imbalance: An Analytical Model Valid for Both
Punch Through and Non Punch Through Devices
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Citation
Han Wang, E. Napoli, and F. Udrea. “Breakdown Voltage for
Superjunction Power Devices With Charge Imbalance: An
Analytical Model Valid for Both Punch Through and Non Punch
Through Devices.” Electron Devices, IEEE Transactions on
56.12 (2009): 3175-3183.© 2009 Institute of Electrical and
Electronics Engineers.
As Published
http://dx.doi.org/10.1109/ted.2009.2032595
Publisher
Institute of Electrical and Electronics Engineers
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Final published version
Accessed
Wed May 25 21:41:02 EDT 2016
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http://hdl.handle.net/1721.1/54726
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Detailed Terms
IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 12, DECEMBER 2009
3175
Breakdown Voltage for Superjunction Power Devices
With Charge Imbalance: An Analytical Model
Valid for Both Punch Through and
Non Punch Through Devices
Han Wang, Ettore Napoli, and Florin Udrea
Abstract—An analytical model for the electric field and the
breakdown voltage (BV) of an unbalanced superjunction (SJ)
device is presented in this paper. The analytical technique uses
a superposition approach treating the asymmetric charge in the
pillars as an excess charge component superimposed on a balanced
charge component. The proposed double-exponential model is able
to accurately predict the electric field and the BV for unbalanced
SJ devices in both punch through and non punch through conditions. The model is also reasonably accurate at extremely high
levels of charge imbalance when the devices behave similarly to a
PiN diode or to a high-conductance layer. The analytical model is
compared against numerical simulations of charge unbalanced SJ
devices and against experimental results.
Index Terms—Analytical model, charge imbalance (C.I.), power
semiconductor devices, semiconductor device modeling, SJ modeling, superjunction (SJ).
I. I NTRODUCTION
S
UPERJUNCTION (SJ) is a power device concept that
allows a favorable tradeoff between breakdown voltage
(BV) and ON-state loss for power MOSFETs [1]–[10]. In SJ
MOSFETs, the drift region is replaced by alternatively stacked
heavily doped N and P regions (pillars). Unlike in conventional
power MOSFETs, where the specific ON-state resistance is proportional to BV2.5 , the SJ MOSFET specific ON-state resistance
is virtually linear with the BV, allowing a cut in the specific
ON -state resistance of 5X to 20X limited by the technology
of forming thin and deep N and P pillars. The superior static
and dynamic performance is demonstrated by Infineon in their
CoolMOS series and by STMicroelectronics in their MDmesh
power MOSFETs [11], [12]. Several analytical models for the
BV of SJ have been proposed [13]–[17]. These models are of
utmost importance since they provide a fast and reliable way to
devise the optimal performance of an SJ structure. However,
Manuscript received March 17, 2009; revised July 29, 2009. First published
October 30, 2009; current version published November 20, 2009. The review
of this paper was arranged by Editor M. A. Shibib.
H. Wang is with the Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(e-mail: hanw@mtl.mit.edu).
E. Napoli is with the Electronic and Telecommunication Engineering
Department, University of Napoli Federico II, 80125 Napoli, Italy.
F. Udrea is with the Engineering Department, University of Cambridge, CB3
0FA Cambridge, U.K.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TED.2009.2032595
most of the previous work assumes an ideal charge balance
between P and N pillars, while it is well known that a tiny
imbalance in the charge significantly affects the BV [14], [18],
[19]. Since charge imbalance (C.I.) is unavoidable in a real
device, a physical insight into the degradation mechanisms and
an analytical solution for it are highly desirable.
In [13], a 1-D model that only works for balanced SJ devices
with very narrow pillars is presented. Reference [14] presents
a simple model of the BV that is suitable for balanced pillars.
Reference [15] provides a rigorous treatment of the interaction
between the depletion regions in the two directions, but only
the balanced case is considered, and the model is complex. A
simple and accurate model for the 2-D potential distribution of
an SJ is proposed in [16] and [17], but the general solution is
solely used to derive two BV models for the balanced case. A
novel approach to model an unbalanced SJ is introduced in [18],
which gives an analytic BV model using particularly simple and
intuitive equations. A limitation of this model, however, is that
it can only predict the BV up to a relatively small level of C.I.
since the electric field and BV equations do not account for
the non punch through (NPT) case. On the other hand, a model
that is also valid at high levels of C.I. is very useful in offering
insights into the physics of power device behavior when the
C.I. also causes the device to move from a punch through (PT)
state to an NPT one. In this paper, an extended formulation
of the models proposed in [16]–[18] is proposed. The model
uses a superposition approach treating the asymmetric charge
in the pillars as an excess charge component superimposed
on a balanced charge component. The electric field of the
unbalanced SJ is modeled using a pair of exponential equations.
Furthermore, this paper derives an analytical equation for the
BV based on this double-exponential formulation. This new
model improves the prediction of the BV provided in [18]
for the PT case and makes it also valid for the NPT case.
Extensive 2-D numerical simulations confirm the validity of the
proposed model. The analytical results are also compared with
the experimental data from [19] to verify its accuracy.
This paper is organized as follows. Section II recalls the
analytical model for the balanced SJ proposed in [16] and
[17]. Section III presents the superposition principle and recalls the linear model for the unbalanced SJ proposed in [18].
Section IV presents the double-exponential model, which leads
to an analytical solution for both the electric field and the BV
0018-9383/$26.00 © 2009 IEEE
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 12, DECEMBER 2009
Fig. 1 Schematic of the SJ sustaining layer that is composed of alternating N
and P doped pillars. (a) Elementary cell of the sustaining layer. (b) Boundary
conditions for the solution of the Poisson equation.
of symmetric unbalanced SJ devices. The proposed model is
the only proposed one to date that is valid for both PT and
NPT SJ devices. Section V presents the BV results of the
double-exponential model compared against both numerical
simulations and experimental results. Furthermore, the model
gives physical insights into the behavior of the balanced and
unbalanced SJ, as well as the extreme cases when the SJ
effectively becomes a 1-D PiN diode.
II. E LECTRIC F IELD M ODEL FOR BALANCED
C HARGE SJ D EVICES
The basis of the proposed model for the unbalanced SJ
is described in [16] and [17]. In [16] and [17], the Poisson
equation is solved for a one unit cell of the SJ layer [Fig. 1(a)]
with boundary conditions specified in Fig. 1(b). This leads
to an analytical expression for the 2-D electrostatic potential
and the electric field. In the balanced symmetrical case in
which both pillar widths and pillar doping are equal (YN = YP
and ND = NA = N ), the peak electric field occurs both at
(x = 0, y = YN ) and at (x = W, y = −YP ). This is shown in
Fig. 2(a) in which the electric field distribution for a balanced
device with W = 30 μm, YN = YP = 2.5 μm, and ND =
NA = 4 × 1015 cm−3 is presented. The applied reverse voltage
is 500 V. Note the symmetric behavior of the electric field and
the presence of two equivalent points of the maximum electric
field.
The BV is obtained by analyzing the section y = YN in
which the maximum electric field is located at x = 0 and the y
component of the electric field is zero. The electric field along
this path, for a balanced symmetric device, as demonstrated in
[16] and [17], is
∞
VD x VR +VD Kn γn cos(Kn x)
+
Ebal (x)|y=YN = −2 2 +
W
W
2 cosh(Kn YN )
n=1
(1)
Δ
where VD = (qN/2εs )W 2 , W and YN are the length and width
of the pillars, VR is the applied reverse voltage, N is the doping
in each pillar, Kn = nπ/W , and γn = 8VD /(nπ)3 [(−1)n − 1].
Fig. 2 Two-dimensional electric field distribution for the SJ devices with
W = 30 μm and YN = YP = 2.5 μm. (a) Balanced SJ device with ND =
NA = 4 × 1015 cm−3 and an applied reverse voltage of 500 V. (b) Unbalanced SJ device with ND = 4 × 1015 cm−3 and NA = 3.0 × 1015 cm−3
and an applied reverse voltage of 220 V. Note that the C.I. increases the peak
electric field for a given applied voltage, thus reducing the voltage sustaining
capability of the device.
Equation (1) is exploited in [16] and [17] and provides an analytic model for the balanced SJ case that allows the calculation
of the BV as a function of pillar doping and dimension and also
provides guidelines for the minimum ON-state resistance design
of an SJ device.
III. S UPERPOSITION P RINCIPLE AND A NALYTICAL M ODEL
FOR C HARGE I MBALANCED SJ D EVICES
A. Superposition Concept
In this paper, it is assumed that the P and N pillars have
symmetrical geometry (YN = YP ) and uniform doping. At this
stage, it is also assumed that the pillars are fully depleted at the
onset of breakdown, and, hence, the SJ device is in a PT mode.
The PT assumption is justified as, in balanced SJ devices, full
depletion normally occurs at a much lower voltage than the BV.
The PT hypothesis will however be removed in order to account
for the NPT case in Section IV of this paper. With respect to
[16] and [17], the hypothesis of equal doping in the N and in
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WANG et al.: BREAKDOWN VOLTAGE FOR SUPERJUNCTION POWER DEVICES WITH CHARGE IMBALANCE
3177
Fig. 3 Charge superposition principle for the C.I. The unbalanced SJ sustaining layer is considered as the superposition of a balanced SJ with doping equal
to (ND + NA )/2 and a PiN diode with N doping equal to (ND − NA )/2. The proposed subdivision provides an exact analytical solution of the considered
Poisson problem if the SJ sustaining layer is completely depleted (PT condition).
the P pillars is removed, and, hence, in Fig. 1, it is assumed
that ND = NA . The resulting SJ sustaining layer is, therefore,
unbalanced. It is well known that an unbalanced charge reduces
the voltage sustaining capability of SJ devices [14], [18], [19].
In fact, the unbalanced charge breaks the symmetry of the electric field in the device and exalts one peak on the electric field
with respect to the other. This is shown in Fig. 2(b) in which the
electric field distribution for an unbalanced device with W =
30 μm, YN = YP = 2.5 μm, ND = 4 × 1015 cm−3 , and NA =
3.0 × 1015 cm−3 is presented. The applied reverse voltage is
220 V. Note the asymmetric behavior of the electric field and
the dominant peak electric field located at (x = 0, y = YN ).
The unbalanced charge is considered as a balanced charge
component superimposed to a differential charge component.
We suppose that ND > NA without loss of generality. As
shown in Fig. 3, the balanced component has a p-type doping
of (ND + NA )/2 in the P pillar and an equal n-type doping in
the N pillar. The differential component consists of a negative
amount of p-type doping whose concentration is (ND − NA )/2
in the P pillar and a positive amount of n-type doping whose
concentration is (ND − NA )/2 in the N pillar. From the potential point of view, the negative amount of p-type doping is
equivalent to an n-type doping of the same magnitude. Combining the contribution of the P pillar and of the N pillar, the
differential component in the entire drift region has a uniform
n-type doping that is equal to N ∗ = (ND − NA )/2. Note that
the structure formed by the differential component is equivalent
to a PiN diode. In conclusion, the unbalanced SJ sustaining
layer considered in this paper is studied as the superposition
of a balanced SJ sustaining layer with N = (ND + NA )/2 and
a PiN diode with N ∗ = (ND − NA )/2 n-type doping.
B. Electric Field Distribution
The potential distribution due to the balanced and differential
charge components is separately calculated with the boundary
conditions in Fig. 1. The electric field due to the differential
component is calculated assuming that it is at the edge of the
PT condition (that is, with the electric field equal to zero for
x = W ). The resulting equation of the electric field has the
well-known triangular shape, typical of the PiN diode, in which
Δ
Vu = qN ∗ W 2 /2εs is the voltage supported by the differential
charge component
x
2Vu 1−
.
(2)
Ediff (x, y) =
W
W
The electric field due to the balanced SJ charge component
Ebal , reported in [16] and [17], is given by (1). The electric
field equation along the section y = YN , where Vu and VB are
the voltage supported by the differential charge and balanced
charge components, is the sum of Ebal and Ediff
ESJ (x)|y=YN = Ebal (x)|y=YN + Ediff (x)|y=YN
x VB
VD x
2Vu 1−
+
+
1−2
=
W
W
W
W
W
∞
Kn γn cos(Kn x)
.
(3)
+
2 cosh(Kn Yn )
n=1
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 12, DECEMBER 2009
It is worth noting that the two solutions Ebal and Ediff
are both an exact analytical solution of the particular Poisson
problem. Furthermore, if the PT condition holds, the sum of the
two solutions is also an exact analytical solution of the complete
unbalanced SJ electrostatic problem.
C. Previously Proposed BV Model
E ∗ (x) that satisfies the following boundary conditions:
Y
VR VD
E ∗ (0) =Ebal (0) = Emax =
+
L
(8)
W W
W
dEbal (x) 2VD
dE ∗ (x) =
=− 2
(9)
dx x=0
dx
W
x=0
lim E ∗ (x) =
Since we assumed, without loss of generality, that ND >
NA , the unbalanced SJ sustaining layer will have the peak
electric field and, hence, the breakdown at the corner on the
P+ side (y = YN , x = 0). The applied reverse voltage is the
integral of the electric field in the x-direction at y = YN given
by (3) and is VR = VB + Vu . The calculation of the BV is
carried out using the critical electric field EC approximation,
that is, assuming that the device breaks when the peak electric
field is equal to EC . The BV is therefore obtained solving the
following equation:
x→∞
VR
.
W
(10)
Equations (8) and (9) guarantee that E ∗ and Ebal have
equal value and derivative at (x = 0, y = YN ). Equation (10)
states that E ∗ behaves similarly to the electric field of an ideal
intrinsic sustaining layer as x → ∞.
The resulting approximation of the electric field along the
y = YN section is [17]
YN
VD
VR
2x
E ∗ (x)|y=YN =
+
L
exp −
.
W
W
W
L(YN /W )W
(11)
∞
2Vu VB VD Kn γn cos(Kn · 0)
This approximation is remarkably good near x = 0 but be+
+
+
= EC .
W
W W n=1 2 cosh(Kn Yn )
comes increasingly worse as x is close to W . We can sig(4) nificantly improve the estimation of (1) by adding a further
exponential term to (11). The additional exponential term is
The previous equation can be simplified by substituting in
negligible near x = 0 and is meaningful when x is close to
(4), as shown in [16] and [17], the L(·) function defined as
W . The constants that characterize the additional exponen∞
tial term are defined imposing proper boundary conditions.
n
1 − (−1)
Δ
L(t) = 1 − 4
.
(5)
The new approximation is the double-exponential model that,
(nπ)2 cosh(nπt)
n=1
using the simplified notation L(·) = L(YN /W ) and VSJ =
VD L(YN /W ), provides the following estimation for the elecThe result is
tric field:
VB
VD
2Vu
VSJ
2x
VB
+
+
L(YN /W ) = EC .
(6) E ∗ (x)|
+
exp
−
=
bal
y=YN
W
W
W
W
W
L(·)W
The BV is the applied voltage VR = VB + Vu when (6) is
2(x − W )
VSJ
exp
−
. (12)
verified
W
L(·)W
ESJ (0)|y=YN =
VBV = Vu + VB = W · EC − Vu − VD L(YN /W ).
(7)
Equation (7) has been reported in [18] and is a particularly
meaningful equation since it states that the actual BV for an
unbalanced SJ sustaining layer is equal to the ideal BV, given by
W · Ec diminished by the effect of the balanced SJ component
(VD L(·)) and by the effect of the unbalanced SJ component
(Vu ). Equation (7) is, however, only valid for SJ sustaining
layers in which the breakdown arises in PT conditions.
∗
, in the limit
The conditions that have been imposed on Ebal
(YN /W ) 1, in order to properly fit the Ebal distribution are
∗
Ebal
(0)|y=YN = Ebal (0)|y=YN
= Emax
A. Double-Exponential Electric Field Model
In [17], in order to allow the analytical solution of the ionization integral, (1) is approximated by an exponential function
YN
W
(13)
∗
(W )|y=YN = Ebal (W )|y=YN
Ebal
YN
VD
VB
−
L
(14)
W
W
W
∂Ebal (x, y) 2VD
=
=− 2
∂x
W
x=0,y=YN
= Emin =
IV. D OUBLE -E XPONENTIAL BV M ODEL
In this section, a new method for calculating the BV of an
unbalanced SJ device is proposed. The new method, compared
to the model proposed in [18], is also accurate at high levels of
C.I. and for both PT and NPT cases.
VD
VB
+
L
=
W
W
∗
∂Ebal
(x, y) ∂x
x=0,y=YN
(15)
∗
(x, y) ∂Ebal
∂x
x=W,y=YN
∂Ebal (x, y) =
∂x
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=−
x=W,y=YN
2VD
.
W2
(16)
WANG et al.: BREAKDOWN VOLTAGE FOR SUPERJUNCTION POWER DEVICES WITH CHARGE IMBALANCE
3179
device in which the electric field is minimum, particularly in
the C.I. = −10% and C.I. = 0% cases. The double-exponential
model in (17), shown with solid lines in Fig. 4, is instead able
to accurately fit the electric field in every portion of the device
and for different amounts of C.I.
B. Double-Exponential BV Model for the PT Case
Equation (17) can conveniently be used for the calculation of
the BV. After imposing that the maximum electric field E(x =
∗
(0) is equal to EC , the BV is accurately
0, y = YN ) = ESJ
obtained by integrating (17) with respect to x along the y = YN
section
Fig. 4 Electric field along line y = YN on the verge of breakdown. The
structure has W = 30 μm and YN = YP = 2.5 μm. In the negative imbalance
cases, ND is fixed at 4 × 1015 cm−3 , and P pillar doping NA is 3.6 ×
1015 cm−3 and 2.8 × 1015 cm−3 , corresponding to −10% and −30% C.I.,
respectively. Ec is 3.2 × 105 V/cm for all cases. The (solid line) doubleexponential model proposed in this paper accurately fits the (circles) numerical
simulations throughout the whole device. The (thick dashed line) singleexponential model proposed in [17] is not accurate in predicting the electric
field value in proximity of the x = W point.
∗
ESJ
(0)|y=YN =
VB
VSJ
2Vu
+
+
W
W
W
−2
VSJ
exp
−
= EC
W
L(·)
W
BVPT =
The double-exponential model provides a much better approximation to (1) with respect to the single-exponential function in (11) that has been used in [17].
In order to account for the unbalanced charge, a differential
component of the electric field, which is defined by (2), is
superimposed to the balanced component in (12). The resulting
double-exponential model for the electric field at the y = YN
section is
x VB VSJ
2x
2Vu ∗
+
exp −
1−
+
ESJ (x)|y=YN =
W
W
W
W
L(·)W
2(x − W )
VSJ
exp
. (17)
−
W
L(·)W
The discrepancy between (3) and (17) is very small also
when the imbalance is relevant and the SJ sustaining layer
is at the edge of the NPT condition. The accuracy of (17)
is demonstrated in Fig. 4, where the electric field along the
y = YN section is shown for three different devices on the
verge of breakdown. The critical electric field Ec used for this
figure is 3.2 × 105 V/cm. The devices are characterized by
W = 30 μm and YN = YP = 2.5 μm. The balanced device,
that is, with C.I. equal to 0% (C.I. = 0%), is characterized
by ND = NA = 4 × 1015 cm−3 . The imbalance is indicated in
percentage and is defined as
A
100 · NDN−N
, ND ≥ 4 · 1015 ; NA = 4 · 1015
D
C.I. =
D
100 · NAN−N
, ND = 4 · 1015 ; NA ≤ 4 · 1015 .
D
(18)
In this way, while keeping the ND > NA hypothesis, it is
possible to explore the whole SJ design space. The −10%
charge imbalanced device is characterized by ND = 4 ×
1015 cm−3 and NA = 3.6 × 1015 cm−3 . The −30% charge imbalanced device is characterized by ND = 4 × 1015 cm−3 and
NA = 2.8 × 1015 cm−3 . In every case, the single-exponential
model accurately fits the maximum electric field and the central
portion of the electric field. It fails to fit in the region of the
(19)
∗
ESJ
(x)dx
0
−2
= EC W − Vu + VSJ exp
−1
L(·)
−2
(20)
≈ VBV .
= VBV + VSJ exp
L(·)
Equation (20) shows that the proposed double-exponential
model provides, in the PT case, a BV prediction that is, as
should be, very similar to the prediction of the model proposed
in [18].
C. Double-Exponential BV Model for the NPT Case
So far, the model assumes full depletion of the device at the
onset of breakdown, i.e., PT devices. However, an optimized
device, as shown in [17], is more often designed at the edge of
the PT–NPT condition or even slightly in the NPT region. The
NPT condition is also enhanced by the amount of unbalanced
charge in the device. An SJ device working in the NPT region
or an SJ device with a certain amount of unbalanced charge
is characterized by the presence of epilayer undepleted regions
when the device is at the edge of the breakdown condition. The
undepleted regions appear near (x = W, y = YN ) and (x =
0, y = −YP ).
As shown in Fig. 4 for the C.I. = −30% case, the doubleexponential model in (17) accurately predicts the electric field
in the SJ device also for the NPT case. The relevant difference
of the NPT case is the presence of an unphysical negative
electric field region that is close to x = W . In Fig. 4, the electric
field in the undepleted region is therefore set to zero.
Our target is using the double-exponential model for the
electric field (17) to calculate the BV also for the NPT devices.
Using (17) for the calculation of the BV requires two steps. The
first one is determining the NPT condition. The second one is
the integration of (17) from x = 0 to the edge of the undepleted
region, identified with x∗ in Fig. 4.
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Identifying the NPT Condition: The boundary between the
NPT and the PT regions occurs when the electric field reaches
zero at x = W with the device on the verge of breakdown. An example is the device with C.I. = −10% shown in
∗
(0) = EC with
Fig. 4. The NPT–PT boundary is, therefore, ESJ
∗
ESJ (W ) = 0. From (17), we have
⎧
⎨ 2Vu + VB + VSJ − VSJ exp −2 = EC W
L(·)
(21)
⎩ VB + VSJ exp −2 − VSJ = 0.
L(·)
From (21), subtracting the second equation from the first, it
is possible to define the NPT–PT boundary as
−2
≈ Vu + VSJ = EC W/2. (22)
Vu + VSJ − VSJ exp
L(·)
In this particular condition, summing the first and the second
equations in (21), we obtain an important result that is the
analytical equation for the BV of the unbalanced SJ device as
a function of the unbalanced charge when the device is at the
NPT–PT boundary. The result is
BVNPT−PT = EC W/2 − Vu .
(23)
Equation (23) is the extension to the unbalanced case of the
BV equation for the balanced SJ provided in [16].
If the SJ device is in the NPT condition, the following condition holds: Vu + VSJ > EC W/2. Let us define a dimensionless
measure of the amount of NPT condition. Such measure will be
indicated as N P
EC W/2
.
VSJ + Vu
NP = 1 −
(24)
Integrate the Electric Field: It is first necessary to deter∗
(x∗ ) = 0 while keeping the
mine the x∗ value for which ESJ
∗
condition that ESJ (0) = EC . The two conditions bring to the
solution of the following equation for x∗ :
−2
2Vu x∗
VSJ
VSJ
VSJ
exp
−
+
EC +
−
W
L(·)
W
W W
W
∗
∗
2(x − W )
2x
VSJ
exp
× exp −
−
= 0. (25)
L(·)W
W
L(·)W
Since (25) has no closed-form solution for x∗ , an approximate solution has been determined. The mathematical details of
the calculations are provided in the Appendix. The approximate
solution for x∗ is then
2
S /L(·))
x∗ = W − W L(·) (VuV+V
(β + β 2 )
S
(26)
VS (VS +Vu )
β = N P L(·)2 (V +V /L(·))2 .
u
S
The integration of the electric field provides
x∗
VNPT =
∗
ESJ
(x)dx =
2Vu x∗
W
x∗
VB ∗
x
1−
+
2W
W
0
−2x∗
−2
VSJ L(·)
+
1 − exp
1 − exp
.
2
L(·)W
L(·)
(27)
Fig. 5 BV for two different charge unbalanced SJ devices. (Bullets and stars)
Two-dimensional numerical simulations. (Solid and dashed lines) Proposed
analytical model in (29) with numerical calculation of x∗ . (Squares and
triangles) Full analytical model using (29) with x∗ calculated using (26). Both
SJ devices have W = 30 μm and YN = YP = 2.5 μm but with different
doping. The (stars, dashed line, and squares) top curves refer to a device whose
balanced condition is ND = NA = 2 × 1015 cm−3 . The (bullets, solid line,
and triangles) bottom curves refer to a device whose balanced condition is
ND = NA = 4 × 1015 cm−3 .
The BV is obtained by substituting the breakdown condition
given by (19) in (27). This means eliminating VB from (27)
using
−2
VB = EC W − 2Vu − VSJ 1 − exp
.
(28)
L(·)
The resulting BV for the NPT condition is
Vu x∗
x∗
BVNPT = BVPT − BVPT −
1−
W
W
−2x∗
−2
VSJ L(·)
+
1 − exp
1 − exp
2
L(·)W
L(·)
Vu x∗
x∗
≈ BVPT − BVPT −
1−
.
(29)
W
W
Note that BVNPT is lower than BVPT . Furthermore, for x∗ =
W , we have BVNPT ≈ BVPT confirming the continuity of the
BV prediction ranging from the PT to the NPT conditions.
V. P ERFORMANCE OF THE
D OUBLE -E XPONENTIAL M ODEL
With reference to the electric field, the double-exponential
model shown in (17), as previously reported commenting Fig. 4,
accurately predicts the electric field throughout the device and
provides a much better prediction of the electric field than the
model proposed in [17].
The first results regarding the double-exponential model
prediction for the BV are reported in Fig. 5 that shows the
BV for two SJ structures with a doping of 4 × 1015 cm−3 and
2 × 1015 cm−3 in the balanced case. The dimensions of both
SJ devices are W = 30 μm and YN = 2.5 μm. The bullets and
stars in Fig. 5 are the results from the 2-D numerical simulations. The solid and the dashed lines show the BV predicted
by the double-exponential model (29) when the value of x∗ is
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WANG et al.: BREAKDOWN VOLTAGE FOR SUPERJUNCTION POWER DEVICES WITH CHARGE IMBALANCE
Fig. 6 BV as a function of doping imbalance between the P and N pillars
for an unbalanced SJ structure with W = 40 μm and YN = YP = 4 μm.
(Solid lines) Proposed analytical model in (29) with numerical calculation of
x∗ . (Squares) Full analytical model using (29) with x∗ calculated using (26).
(Dashed lines) Linear model from [18]. (Bullets) Experimental data from [19].
The doping concentration of the SJ in the balanced case is 3 × 1015 cm−3 . The
negative and positive imbalances are defined in the same way as that in Fig. 5.
The circles indicate the onset of NPT as predicted by the proposed analytical
model. The double-exponential model is able to predict the reduction of the BV
and the different impact of the positive C.I. with respect to the negative C.I.
obtained by solving (25) numerically. The squares and triangles
report the BV predicted by the complete analytical model that
uses (29) and calculates the value of x∗ using (26). EC is 3.2 ×
105 and 2.7 × 105 V/cm for ND = NA = 4 × 1015 cm−3 and
ND = NA = 2 × 1015 cm−3 , respectively. The circles show
the PT–NPT boundary for both SJ devices and for both positive
and negative C.I.
Due to the definition of C.I. used in this paper, the degradation of the BV for positive C.I. shown in Fig. 5 is higher than the
degradation of the BV for negative C.I. The double-exponential
model performs very well for both PT and NPT devices. The
discrepancies are slightly larger for very high levels of C.I.
beyond −80% and +60%, where the model underestimates
the BV. The closed-form approximation of x∗ in (22) and,
hence, the closed-form approximation of the BV agree with
the numerical solutions perfectly for a low level of C.I. in both
PT and NPT devices. The approximation is very good for the
2 × 1015 cm−3 structure for the entire range of C.I. conditions.
The approximation for the 4 × 1015 cm−3 structure is good up
to C.I. = −80% and C.I. = +40%.
The proposed double-exponential model offers a valuable
insight into the performance of SJ devices at various levels
of C.I. The best BV performance of SJ is obtained when the
charge in the alternating P and N pillars is perfectly balanced.
In this optimal case, the peak electric fields at the two opposite
corners of the device reach EC simultaneously. As the level
of C.I. is increased, the differential component significantly
modifies the 2-D electric field profile to a more triangular one
with the peak electric field starting to be located only at the
more heavily doped side. This reduces the BV of the device
sharply.
At extremely high levels of negative C.I., the BV predicted by
the double-exponential model converges to that of a PiN diode,
in exact agreement with the simulation results. This is because,
at extremely high levels of negative C.I., the differential com-
3181
ponent dominates; the pillar with lower doping behaves like an
intrinsic semiconductor, while the BV arises in the pillar with
higher doping that behaves similarly to a PiN diode.
At extremely high levels of positive doping imbalance, the
differential component also dominates, but, now, both pillars
are doped. The pillar with higher doping is now very heavily
doped and asymptotically supports zero BV. This explains why
the BV drops faster when the C.I. is positive as compared to
negative C.I.
Fig. 6 shows the double-exponential model for the BV
against the experimental results regarding an SJ device whose
dimensions are W = 40 μm and YN = YP = 4 μm and the
doping in the balanced case is 3 × 1015 cm−3 . The solid line
in Fig. 6 refers to the double-exponential model (29) with x∗
solved numerically; the squares refer to the double-exponential
model (29) with x∗ given by (26); the dashed line refers to
the linear model (7), whereas the bullets are the experimental
results from [19]. The considered EC is 3.2 × 105 V/cm. The
experimental results are in good agreement with the doubleexponential analytical model. It is worth highlighting that
Saito et al. [19] state that the termination edge effect is responsible for the experimental BV results. Our model is instead
able to properly devise the trend of the BV as a function of
the C.I. without including any edge termination effect. This
hypothesis is confirmed by the subsequent results proposed by
Ono et al. [20]. Note also that both the experimental results in
[19] and [20] confirm that the impact of positive C.I. on the BV
is higher than the effect of the negative C.I.
VI. C ONCLUSION
An extended analytical model for the electric field and the
BV of SJ structures with C.I. has been presented. The model
uses a superposition approach treating the asymmetric charge
in the pillars as an excess charge component superimposed on a
balanced charge component. A pair of exponential equations is
used to model the electric field due to the balanced charge component, while the unbalanced charge component is modeled as a
PiN diode. This paper presents analytical relations for the BV of
balanced and unbalanced SJ devices. The presented analytical
relations are, for the first time, valid for SJ operating in both
PT and NPT modes. The model accuracy is demonstrated
against the 2-D numerical simulations and experimental results
from [19].
A PPENDIX
A PPROXIMATE A NALYTICAL S OLUTION FOR THE
x∗ VALUE IN THE NPT C ONDITION
This Appendix reports the approximate calculation of the x∗
value that satisfies (25), which is reported in the following for
clarity:
−2
2Vu x∗
VSJ
VSJ
VSJ
exp
−
+
EC +
−
W
L(·)
W
W W
W
2 (x∗ −W )
2x∗
VSJ
× exp −
exp
−
= 0. (30)
L(·)W
W
L(·)W
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3182
IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 12, DECEMBER 2009
Since L(·) 1, it is also true that exp(−(2/L(·))) 1
and exp(−(2x∗/L(·)W )) 1. Equation (30) is then approximated as
∗
2(x − W )
VSJ 2Vu (x∗ −W ) 2Vu VSJ
−
−
−
exp
EC −
= 0.
W
W
W
W
W
L(·)W
(31)
Since (2(x∗ − W )/L(·)W ) 1 when x∗ W , it is reasonable to expand the exponential term in Taylor series. The
expansion is up to the second order
2Vu (x∗ − W ) 2Vu
VSJ
VSJ
−
−
−
EC −
W
W
W
W
W
2 2 (x∗ − W )
2 (x∗ − W ) 1
+
= 0.
× 1+
L(·)
W
2 L(·)
W
(32)
The second-order equation is solved with respect to the
variable α = (x∗ − W )/W , obtaining
L(·)2
α=
− Vu − VSJ /L(·) + (Vu + VSJ /L(·))
2VSJ
4VSJ N P (VSJ + Vu )
. (33)
× 1−
L(·)2 (Vu + VSJ /L(·))2
Since the required solutions are close to the balanced condition for which N P = 0, the square root is also conveniently
Taylor expanded up to the second order. With the position
β = NP
VSJ (VSJ + Vu )
.
L(·)2 (Vu + VSJ /L(·))2
(34)
we have
α=
L(·)2 −Vu −VSJ /L(·)+(Vu +VSJ /L(·)) 1−4β
2VSJ
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L(·)2 −Vu −VSJ /L(·)+(Vu +VSJ /L(·)) (1−2β −2β 2 )
2VSJ
=−
L(·)2 (Vu +VSJ /L(·))
[β +β 2 ].
VSJ
(35)
Equation (35) results in (26) reported in the following for
clarity:
x∗ = W − W
L(·)2 (Vu + VS /L(·))
(β + β 2 ).
VS
(36)
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pp. 243–244, Mar. 1979.
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Han Wang received the B.A. and M.Eng. degrees
(with highest honors) in electrical and information science from the University of Cambridge,
Cambridge, U.K., in 2006 and 2007, respectively.
He is currently working toward the Ph.D. degree
in electrical engineering and computer science at
the Massachusetts Institute of Technology (MIT),
Cambridge, MA.
During the summer of 2005, he was a Student Intern with the Mobile Device Research Group, British
Telecom, Ipswich, U.K. In the summer of 2006, he
also worked as a UROP Student with the Communications and Signal Processing Group, Imperial College, London, U.K. From 2006 to 2007, he was with
the University of Cambridge, where he worked on the modeling and simulation
of power electronic devices. Since 2008 with MIT, he has been working on
GaN- and graphene-based devices. His current research interest focuses on the
development of GaN-based transistors for millimeter-wave applications and the
search of novel graphene-based ambipolar devices for applications in nonlinear
electronics.
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WANG et al.: BREAKDOWN VOLTAGE FOR SUPERJUNCTION POWER DEVICES WITH CHARGE IMBALANCE
Ettore Napoli was born in Italy in 1971. He received the M.Sc. degree in electronic engineering
(with honors) in 1995, the Ph.D. degree in electronic
engineering in 1999, and the B.Sc. degree in physics
(with honors) in 2009, all from the University of
Napoli Federico II, Napoli, Italy.
In 2004, he was a Research Associate with the
Engineering Department, University of Cambridge,
Cambridge, U.K. Since 2005, he has been an Associate Professor with the University of Napoli Federico II, Napoli, Italy. His scientific interests include
the modeling and design of power semiconductor devices and VLSI circuit
design. In the power devices field, his main interests are PiN diodes, vertical
IGBTs, superjunction devices, and lateral IGBTs. In the VLSI field, his interests
are high-speed arithmetic subsystems and advanced flip-flops. He is the author
or coauthor of more than 80 papers published in international journals and
conferences.
3183
Florin Udrea received the Ph.D. degree from the
Engineering Department, University of Cambridge,
Cambridge, U.K., in 1995.
He was an Advanced EPSRC Research Fellow
from August 1998 to July 2003 and, prior to this,
a College Research Fellow with Girton College,
University of Cambridge. Since October 1998, he
has been a Reader with the Engineering Department,
University of Cambridge. He has published over
250 papers in journals and international conferences
and is the holder of over 50 patents in power semiconductor devices and sensors. He is currently leading a research group in
power semiconductor devices and solid-state sensors. In August 2000, he
cofounded with Prof. Amaratunga Cambridge Semiconductor (also called
CamSemi), a start-up company in the field of power integrated circuits. In 2008,
he cofounded Cambridge CMOS Sensors with Prof. Gardner and Prof. Milne.
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