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
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Institute of Electrical and Electronics Engineers
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Final published version
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http://hdl.handle.net/1721.1/54726
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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 con-ditions. 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 model-ing, superjunction (SJ).
I. INTRODUCTION
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 specificON-state resistance is pro-portional to BV2.5, the SJ MOSFET specificON-state resistance is virtually linear with the BV, allowing a cut in the specificON-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 re-calls 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
3176 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. ELECTRICFIELDMODEL FORBALANCED
CHARGESJ DEVICES
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× 1015cm−3is 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 Ebal(x)|y=YN=−2 VDx W2 + VR+VD W + ∞ n=1 Knγn 2 cos(Knx) cosh(KnYN) (1) where VD Δ
= (qN/2εs)W2, W and YN are the length and width
of the pillars, VRis 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× 1015cm−3 and an applied reverse voltage of 500 V. (b)
Unbal-anced SJ device with ND= 4× 1015 cm−3 and NA= 3.0× 1015cm−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 ana-lytic 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 minimumON-state resistance design of an SJ device.
III. SUPERPOSITIONPRINCIPLE ANDANALYTICALMODEL FORCHARGEIMBALANCEDSJ DEVICES
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
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 elec-tric field in the device and exalts one peak on the elecelec-tric 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× 1015cm−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
po-tential point of view, the negative amount of p-type doping is equivalent to an n-type doping of the same magnitude. Com-bining 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
Vu
Δ
= qN∗W2/2ε
sis the voltage supported by the differential
charge component Ediff(x, y) = 2Vu W 1− x W . (2)
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 Vuand VB are
the voltage supported by the differential charge and balanced charge components, is the sum of Ebaland Ediff
ESJ(x)|y=YN = Ebal(x)|y=YN + Ediff(x)|y=YN
=2Vu W 1− x W +VB W + VD W 1− 2 x W + ∞ n=1 Knγn 2 cos(Knx) cosh(KnYn) . (3)
3178 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
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: ESJ(0)|y=YN= 2Vu W + VB W + VD W + ∞ n=1 Knγn 2 cos(Kn· 0) cosh(KnYn) = EC. (4) The previous equation can be simplified by substituting in (4), as shown in [16] and [17], the L(·) function defined as
L(t)= 1Δ − 4 ∞ n=1 1− (−1)n (nπ)2cosh(nπt). (5) The result is 2Vu W + VB W + VD W L(YN/W ) = EC. (6)
The BV is the applied voltage VR= VB+ Vu when (6) is
verified
VBV= Vu+ VB = W· EC− Vu− VDL(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· Ecdiminished by the effect of the balanced SJ component
(VDL(·)) 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.
IV. DOUBLE-EXPONENTIALBV MODEL
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.
A. Double-Exponential Electric Field Model
In [17], in order to allow the analytical solution of the ion-ization integral, (1) is approximated by an exponential function
E∗(x) that satisfies the following boundary conditions:
E∗(0) =Ebal(0) = Emax= VR W + VD W L Y W (8) dE∗(x) dx x=0 = dEbal(x) dx x=0 =−2VD W2 (9) lim x→∞E ∗(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] E∗(x)|y=YN =VR W + VD W L YN W exp − 2x L(YN/W )W . (11) This approximation is remarkably good near x = 0 but be-comes increasingly worse as x is close to W . We can sig-nificantly improve the estimation of (1) by adding a further exponential term to (11). The additional exponential term is negligible near x = 0 and is meaningful when x is close to
W . The constants that characterize the additional
exponen-tial term are defined imposing proper boundary conditions. The new approximation is the double-exponential model that, using the simplified notation L(·) = L(YN/W ) and VSJ= VDL(YN/W ), provides the following estimation for the
elec-tric field: Ebal∗ (x)|y=YN = VB W + VSJ W exp − 2x L(·)W −VSJ W exp 2(x− W ) L(·)W . (12)
The conditions that have been imposed on Ebal∗ , in the limit (YN/W ) 1, in order to properly fit the Ebaldistribution are
Ebal∗ (0)|y=YN = Ebal(0)|y=YN
= Emax= VB W + VD W L YN W (13)
Ebal∗ (W )|y=YN = Ebal(W )|y=YN
= Emin= VB W − VD W L YN W (14) ∂E∗bal(x, y) ∂x x=0,y=YN = ∂Ebal(x, y) ∂x x=0,y=YN =−2VD W2 (15) ∂E∗bal(x, y) ∂x x=W,y=YN = ∂Ebal(x, y) ∂x x=W,y=YN =−2VD W2. (16)
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×
1015cm−3and 2.8× 1015cm−3, corresponding to−10% and −30% C.I., respectively. Ec is 3.2× 105 V/cm for all cases. The (solid line)
double-exponential model proposed in this paper accurately fits the (circles) numerical simulations throughout the whole device. The (thick dashed line) single-exponential model proposed in [17] is not accurate in predicting the electric field value in proximity of the x = W point.
The double-exponential model provides a much better ap-proximation to (1) with respect to the single-exponential func-tion 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 ESJ∗ (x)|y=YN=2Vu W 1− x W +VB W + VSJ W exp − 2x L(·)W −VSJ W exp 2(x− W ) L(·)W . (17)
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 Ecused 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× 1015cm−3. The imbalance is indicated in
percentage and is defined as
C.I. = 100· ND−NA ND , ND≥ 4 · 10 15; N A= 4· 1015 100· NA−ND ND , ND= 4· 10 15; N A≤ 4 · 1015. (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×
1015cm−3and NA= 3.6× 1015cm−3. The−30% charge
im-balanced device is characterized by ND= 4× 1015cm−3and
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
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, y = YN) = ESJ∗ (0) is equal to EC, the BV is accurately
obtained by integrating (17) with respect to x along the y = YN
section ESJ∗ (0)|y=YN =2Vu W + VB W + VSJ W −VSJ W exp −2 L(·) = EC (19) BVPT= W 0 ESJ∗ (x)dx = ECW − Vu+ VSJ exp −2 L(·) − 1 = VBV+ VSJexp −2 L(·) ≈ VBV. (20)
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 double-exponential 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.
3180 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 56, NO. 12, DECEMBER 2009
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 break-down. An example is the device with C.I. =−10% shown in Fig. 4. The NPT–PT boundary is, therefore, ESJ∗ (0) = ECwith
ESJ∗ (W ) = 0. From (17), we have ⎧ ⎨ ⎩ 2Vu+ VB+ VSJ− VSJexp −2 L(·) = ECW VB+ VSJexp −2 L(·) − VSJ= 0. (21)
From (21), subtracting the second equation from the first, it is possible to define the NPT–PT boundary as
Vu+ VSJ− VSJexp −2 L(·) ≈ Vu+ VSJ= ECW/2. (22)
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= ECW/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 condi-tion holds: Vu+ VSJ> ECW/2. Let us define a dimensionless
measure of the amount of NPT condition. Such measure will be indicated as N P
N P = 1− ECW/2 VSJ+ Vu
. (24)
Integrate the Electric Field: It is first necessary to
deter-mine the x∗ value for which E∗SJ(x∗) = 0 while keeping the condition that ESJ∗ (0) = EC. The two conditions bring to the
solution of the following equation for x∗:
EC+ VSJ W exp −2 L(·) −VSJ W − 2Vu W x∗ W + VSJ W × exp − 2x∗ L(·)W −VSJ W exp 2(x∗− W ) L(·)W = 0. (25)
Since (25) has no closed-form solution for x∗, an approxi-mate solution has been determined. The mathematical details of the calculations are provided in the Appendix. The approximate solution for x∗is then
x∗= W− WL(·)2(Vu+VS/L(·)) VS (β + β 2) β = N P VS(VS+Vu) L(·)2(V u+VS/L(·))2. (26)
The integration of the electric field provides
VNPT = x∗ 0 ESJ∗ (x)dx = 2Vux ∗ W 1− x ∗ 2W +VB Wx ∗ +VSJL(·) 2 1− exp −2x∗ L(·)W 1− exp −2 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× 1015cm−3. The (bullets, solid line,
and triangles) bottom curves refer to a device whose balanced condition is
ND= NA= 4× 1015cm−3.
The BV is obtained by substituting the breakdown condition given by (19) in (27). This means eliminating VB from (27)
using VB= ECW− 2Vu− VSJ 1− exp −2 L(·) . (28)
The resulting BV for the NPT condition is
BVNPT= BVPT− BVPT− Vux∗ W 1− x ∗ W +VSJL(·) 2 1− exp −2x∗ L(·)W 1− exp −2 L(·) ≈ BVPT− BVPT− Vux∗ W 1− x ∗ W . (29)
Note that BVNPTis lower than BVPT. Furthermore, for x∗= W , we have BVNPT≈ BVPTconfirming the continuity of the
BV prediction ranging from the PT to the NPT conditions.
V. PERFORMANCE OF THE
DOUBLE-EXPONENTIALMODEL
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× 1015cm−3and 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 simula-tions. The solid and the dashed lines show the BV predicted by the double-exponential model (29) when the value of x∗ is
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× 1015cm−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). ECis 3.2×
105and 2.7× 105V/cm for ND= NA= 4× 1015cm−3and
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 degrada-tion 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× 1015cm−3structure for the entire range of C.I. conditions. The approximation for the 4× 1015cm−3structure 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-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× 105V/cm. The
experimental results are in good agreement with the double-exponential analytical model. It is worth highlighting that Saito et al. [19] state that the termination edge effect is re-sponsible 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. CONCLUSION
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 com-ponent, 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].
APPENDIX
APPROXIMATEANALYTICALSOLUTION FOR THE
x∗VALUE IN THENPT CONDITION
This Appendix reports the approximate calculation of the x∗ value that satisfies (25), which is reported in the following for clarity: EC+ VSJ W exp −2 L(·) −VSJ W − 2Vu W x∗ W + VSJ W × exp −L(·)W2x∗ −VSJ W exp 2 (x∗−W ) L(·)W = 0. (30)
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 approxi-mated as EC− VSJ W − 2Vu W (x∗−W ) W − 2Vu W − VSJ W exp 2(x∗− W ) L(·)W = 0. (31) Since (2(x∗− W )/L(·)W ) 1 when x∗ W , it is rea-sonable to expand the exponential term in Taylor series. The expansion is up to the second order
EC− VSJ W − 2Vu W (x∗− W ) W − 2Vu W − VSJ W × 1 + 2 L(·) (x∗− W ) W + 1 2 2 L(·) (x∗− W ) W 2 = 0. (32)
The second-order equation is solved with respect to the variable α = (x∗− W )/W , obtaining α = L(·) 2 2VSJ − Vu− VSJ/L(·) + (Vu+ VSJ/L(·)) × 1− 4VSJN P (VSJ+ Vu) L(·)2(V u+ VSJ/L(·))2 . (33)
Since the required solutions are close to the balanced con-dition for which N P = 0, the square root is also conveniently Taylor expanded up to the second order. With the position
β = N P VSJ(VSJ+ Vu) L(·)2(V u+ VSJ/L(·))2 . (34) we have α =L(·) 2 2VSJ −Vu−VSJ/L(·)+(Vu+VSJ/L(·)) 1−4β L(·)2 2VSJ −Vu−VSJ/L(·)+(Vu+VSJ/L(·)) (1−2β−2β2) = −L(·) 2(V u+VSJ/L(·)) VSJ [β +β2]. (35) Equation (35) results in (26) reported in the following for clarity: x∗= W− WL(·) 2(V u+ VS/L(·)) VS (β + β2). (36) REFERENCES
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Han Wang received the B.A. and M.Eng. degrees
(with highest honors) in electrical and informa-tion 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 In-tern 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 Process-ing 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.
Ettore Napoli was born in Italy in 1971. He
re-ceived 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 Asso-ciate Professor with the University of Napoli Fed-erico 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.
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 semi-conductor 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.
