A physics-informed uncertainty-aware surrogate for $p$-$n$ junction diodes

Suchita Kulkarni

cs.LGOct 29, 2026

A physics-informed uncertainty-aware surrogate for $p$ - $n$ junction diodes

Authors: Suchita Kulkarni

Organizations: Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria

Abstract

We develop an uncertainty-aware physics-informed surrogate for a one-dimensional $p$ - $n$ junction diode, conditioned on device length, doping and bias. Using the internal state of the device as predicted by the surrogate we develop a test-suite of quantities of interest including the terminal current, depletion width, peak field and stored charge. The formalism and the quantities are can be extended to asymmetric junction and two-dimensional diodes as shown by means of benchmark analysis. For the one-dimensional symmetric junction surrogate, we obtain 0.11% median $L_1$ error on the electrostatic potential, and 1.5% and 0.61% on the electron and hole densities. Finally, we develop an uncertainty analysis which determines the validity of surrogate answer within the user-specified tolerance.

Figures & tables

Figure 1: The 2D planar benchmark device (contact thickness exaggerated): a p + implant (

n_{A}=10^{18}

cm -3 , 1.26

\mu

m wide, 0.64

\mu

m deep) in an n - substrate (

n_{D}=10^{16}

cm -3 ) of width 3

\mu

m and height 2

\mu

m. The anode is a window on the top face, the cathode covers the bottom face, and the remaining boundary is insulating.

Set	Role	Devices (anchors)
Training	fit weights	27 (404)
Held-out	validation	3 (45)
Calibration	analysis, intervals	310 (4586)
Test	all reported claims	246 (3647)
Extrapolation	$V$ extrapolation	12 (—)

Table 1: Device sets and their roles. Anchors are counted after restricting to

V/V_{bi}<0.8

and, for the calibration and test sets, to the supported range

\kappa\geq-2.5

of Sec. 3.5 .

Figure 2: Network architecture representation

Quantity	1D symmetric	1D asymmetric	2D planar
$\psi$ ( $L_{1}$ )	0.42	0.49	0.11
$n$ ( $L_{1}$ )	1.1	0.47	0.50
$p$ ( $L_{1}$ )	13	7.5	5.0
Peak field $\mathcal{E}_{\rm peak}$	0.34	0.36	4.6
Depletion size $W=\|D\|$ (2D: area)	1.8	2.2	0.69
Stored charge $Q_{n}$	11	11	13

Table 2: Mean (over the four held-out voltages) relative error in percent for QoIs. For the current, the mean is over the held-out voltages at which the anode current density is at least

10^{-6}

A/cm 2 (0.28, 0.44 and 0.60 V; for the 2D device the current per unit depth divided by the 0.8

\mu

m anode window).

Figure 3: State-space reconstruction of the symmetric (left) and the asymmetric (right) 1D benchmark at the held-out voltage

V=0.44

V, on the trimmed domain. Rows: potential (DEVSIM, solid black; network, dashed blue); electron (solid) and hole (dash-dot) densities; and the relative error of

n

p

and

\psi

(dashed line: 20%).

Figure 4: State of the 2D planar benchmark at the held-out voltage $V=0.44$ V, on the trimmed model box (the anode is at the top, the cathode at the bottom). Rows: potential, $\log_{10}n$ and $\log_{10}p$ ; columns: DEVSIM, network, and network minus DEVSIM (the colour scale of the two density differences is clipped at the 99th percentile). Bottom: cuts along the vertical line through the middle of the anode window. The potential differs by at most 2.9 mV, at the corners of the junction. $\log_{10}p$ differs by up to 1.7 in the last row next to the cathode; over the whole box its 99th percentile is 0.04.

Scalar QoIs (relative error)
Quantity	Test (246 devices)	Training (27 devices)	Held-out (3 devices)
$\hat{J}_{\mathrm{RO}}$	0.70	0.21	1.4
Depletion width $W$	1.4	0.96	1.1
Peak field $\mathcal{E}_{\rm peak}$	0.39	0.20	0.25
Stored charge $Q_{n}$	0.99	0.77	0.44
Stored charge $Q_{p}$	2.9	1.5	4.3

Table 3: Median percent error of the ensemble against DEVSIM for the test, training and held-out devices.

Figure 5: State-space reconstruction on the best and worst mean $L_{1}$ error for anchors of the unseen devices, and for an anchor of the stiff high-doping, long-device regime.

Figure 6: Terminal current against voltage for six test devices spread over current level (top), and the ratio

\hat{J}_{\mathrm{RO}}/J

(bottom); each curve ends at

V/V_{bi}=0.8

or at 0.64 V. The shaded region and the open symbols mark points with

|J_{\rm DEVSIM}|<10^{-6}

A/cm 2 ; the ratio is dashed below this value. The devices are evenly spaced in mean log-current over the test devices; a device with fewer than five points above the floor or with a wrong-sign readout at any voltage is replaced by the eligible device nearest in current level. Above the floor the current is reproduced over up to seven decades and the largest error occurs at low voltage in the device with

L=0.12\,\mu

m, where the current is small. Below the floor the readout is up to about

14

times too large (device with

L=4.64\,\mu

m at

0.04

V).

Figure 7: Error factor of the terminal current over the $(L,n_{D})$ plane, averaged over voltage, for two different networks. Left: $\hat{J}_{\mathrm{RO}}$ of the six-network ensemble of the recipe. Right: $\hat{J}_{\mathrm{grad}}$ from a single modified network, see Appendix C for details of the modifications. The error factor is $\max(r,1/r)$ for the ratio $r$ of predicted to DEVSIM current.

Figure 8: Forward-bias extrapolation beyond the training range, for twelve devices. Top: relative error of the network’s terminal current, against

V

. The large error at low

V

is the small-current regime and is unrelated to extrapolation. Bottom: current against

V

for the best, median and worst device by median error beyond the end of the training range. The median over devices is drawn where at least six devices contribute.

Figure 9: Local 90% error bar for all five quantities, as a function of depleted fraction. The bar is narrow where the device is quasi-neutral and widens as the depletion region grows. For

W_{\rm{dep}}/L>0.6

the stored charges

Q_{n}

Q_{p}

and the depletion width are not applicable. The current has a bar up to

W_{\rm{dep}}/L\approx 7

; beyond it the interval has no upper limit.

Figure 10: Coverage of the 90% interval by depleted-fraction regime for the five quantities on the test devices (points: coverage; bars: 95% device-bootstrap intervals; dashed: target 0.90). A dash marks a regime that is not applicable (fully depleted: the width and the stored charges are undefined).

			$\tau=0.05$		$\tau=0.10$		$\tau=0.25$
QoI	Inform.	Coverage	Acc.	Hit	Acc.	Hit	Acc.	Hit
$J$	0.69	$0.91^{+0.03}_{-0.03}$	0.80	$0.96^{+0.02}_{-0.03}$	0.85	$0.98^{+0.01}_{-0.02}$	0.92	$1.00^{+0.00}_{-0.00}$
$W$	0.78	$0.90^{+0.03}_{-0.03}$	0.66	$0.99^{+0.01}_{-0.01}$	0.80	$1.00^{+0.00}_{-0.00}$	1.00	$1.00^{+0.00}_{-0.00}$
$E_{\rm peak}$	1.00	$0.91^{+0.02}_{-0.02}$	0.99	$0.99^{+0.00}_{-0.01}$	1.00	$1.00^{+0.00}_{-0.00}$	1.00	$1.00^{+0.00}_{-0.00}$
$Q_{n}$	0.79	$0.92^{+0.02}_{-0.02}$	0.70	$0.96^{+0.01}_{-0.02}$	0.84	$0.99^{+0.00}_{-0.01}$	0.92	$0.99^{+0.00}_{-0.00}$
$Q_{p}$	0.79	$0.90^{+0.04}_{-0.04}$	0.00	–	0.23	$0.90^{+0.08}_{-0.09}$	0.84	$0.94^{+0.03}_{-0.04}$

Table 4: Tolerance statements on the test devices. “Accepted”: fraction of informative queries whose bar is

\leq\tau

; “hit”: fraction of those whose true relative error is

\leq\tau

(target 0.90; – : no query accepted). “Inform.”: fraction of all queries on which the quantity is defined; for

J

it is the fraction with

|\hat{J}_{\mathrm{RO}}|\geq 10^{-6}

A/cm 2 , and the other columns refer to these queries.

$L$	$n_{D}$	$V$	$W_{\rm dep}/L$	$J_{\rm pred}$	$J_{\rm truth}$	$b\leq\tau$ ? ( $\tau=0.10$ )
( $\mu$ m)	(cm -3 )	(V)		(A/cm 2 )	(A/cm 2 )	$J$	$W$	$E_{\rm peak}$	$Q_{n}$	$Q_{p}$
6.9	$1.81\times 10^{18}$	0.64	0.0032	$6.96^{+0.27}_{-0.25}\times 10^{-2}$	$7.00\times 10^{-2}$	yes	no	yes	no	no
0.91	$2.65\times 10^{18}$	0.48	0.025	$7.56^{+0.17}_{-0.16}\times 10^{-4}$	$7.60\times 10^{-4}$	yes	yes	yes	yes	no
0.88	$3.45\times 10^{17}$	0.64	0.05	$2.97^{+0.05}_{-0.05}$	$2.97$	yes	yes	yes	yes	yes
0.26	$1.50\times 10^{18}$	0.60	0.097	$5.10^{+0.14}_{-0.13}\times 10^{-1}$	$5.12\times 10^{-1}$	yes	yes	yes	yes	no
0.15	$1.41\times 10^{18}$	0.44	0.21	$2.25^{+0.05}_{-0.05}\times 10^{-3}$	$2.25\times 10^{-3}$	yes	yes	yes	yes	no

Table 5: Example test queries and their answers: eight measurable test queries at evenly spaced ranks of

W_{\rm dep}/L

. The columns

b\leq\tau

state whether the error bar of the quantity is at most

\tau=10\%

, which needs no DEVSIM value;

J_{\rm truth}

is shown for comparison. The last query lies outside the calibration support (

W_{\rm dep}/L\gtrsim 7

), so no interval is issued.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Distribution of the ensemble error (Eq. ( 2.18 ); relative

L_{1}

error for the profiles in the top row, relative error for the scalar QoIs in the bottom row) over the (device, voltage) points of the training, held-out and test devices, on a logarithmic axis. Boxes: quartiles with median; whiskers: 5th and 95th percentiles. Dots: the median error of each of the six single networks on that set, so the spread of the dots around the box is the variation between networks. The held-out set has three devices and is indicative only. Table 3 gives the test medians.

Scalar QoIs (relative error)
	Ensemble error	Single network	Single network
Quantity		mean $\pm$ std	min–max
Readout current $\hat{J}_{\mathrm{RO}}$	$0.70^{+0.14}_{-0.10}$	0.88 $\pm$ 0.13	0.73–1.1
Depletion width $W$	$1.4^{+0.24}_{-0.19}$	1.6 $\pm$ 0.14	1.4–1.8
Peak field $\mathcal{E}_{\rm peak}$	$0.39^{+0.043}_{-0.037}$	0.43 $\pm$ 0.095	0.35–0.58
Stored charge $Q_{n}$	$0.99^{+0.15}_{-0.14}$	1.3 $\pm$ 0.12	1.2–1.5

Appendix

Table 6: Dependence on the random seed. The six networks differ only in their seed and have a common training dataset.

Figure 12: Error factor of the readout current $\hat{J}_{\mathrm{RO}}$ of the ensemble (top row) and of the differentiated current $\hat{J}_{\mathrm{grad}}$ of the network with the $\eta$ column and the $R_{\rm peak,ref}$ divisor (bottom row) over the $(L,n_{D})$ plane, for the same 120 unseen devices, including those with $|J_{\rm truth}|<10^{-6}$ A/cm 2 . The hatched region at $0.64$ V marks the devices with $V\geq 0.8\,V_{bi}$ , which are excluded.

Figure 13: Quasi-Fermi levels

v_{n}

and

v_{p}

(top row) and their slopes

\mathrm{d}v_{n}/\mathrm{d}x

(middle) and

\mathrm{d}v_{p}/\mathrm{d}x

(bottom) along three unseen devices at

V=0.52

V, of the network with the

\eta

column and the SRH-peak divisor, for

L/\lambda_{D}

of 10, 97 and 858. Black: DEVSIM; dashed: network. The slope axes are symmetric logarithmic (linear below 100 V/cm). Each column is labelled with

L/\lambda_{D}

and

J^{\rm grad}_{\rm pred}/J_{\rm truth}

Figure 14: Distribution of the error of each (device, voltage) point of the test devices on a logarithmic axis, for the physics-informed and the data-only six-network ensembles. Left: relative

L_{1}

error of

\psi

; right: relative error of the peak field. Solid lines: median, dashed lines:

90

th percentile of each set. The text above each panel gives the median and the ratio of the

90

th percentile to the median. Without the physics term the typical error of

\psi

is the same and the tail is heavier; the distribution of the error of the peak field is shifted to larger values.

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