L10 February 17 1

EE5342 – Semiconductor Device Modeling and CharacterizationLecture 10 - Spring 2005

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc/

L10 February 17 2

vD=Vext

ln iD

Data

ln(IKF)

ln(IS)

ln[(IS*IKF) 1/2]

Effect

of Rs

t

a

VNFV

exp~

t

a

VNRV

exp~

VKF

ln(ISR)

Effect of high level injection

low level injection

recomb. current

Vext-

Va=iD*Rs

t

a

VNV

2exp~

L10 February 17 3

Interpreting a plotof log(iD) vs. VdIn the region where

iD ~ ISeff(exp (Vd/(NeffVt)) - 1)

For N = 1 and Vt = 25.852 mV, the slope of the plot of log(iD) vs. Vd is evaluated as

{dlog(iD)/dVd} = log (e)/(NVt) = 16.799 decades/V = 1decade/59.526mV

L10 February 17 4

Static Model Eqns.Parameter ExtractionIn the region where

iD ~ ISeffexp (Vd/(NeffVt) )

{diD/dVd}/iD = d[ln(iD)]/dVd = 1/(NVt)

so N ~ {dVd/d[ln(iD)]}/Vt Neff,

and ln(IS) ~ ln(iD) - Vd/(NVt) ln(ISeff).

Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

L10 February 17 5

I-V data and ISeff estimation

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0.4 0.6 0.8 1.0

Vext (V)

Id (

A)

1.E-16

1.E-15

1.E-14

1.E-13

1.E-12

0.4 0.6 0.8 1.0

Vext (V)

ISef

f

L10 February 17 6

Hints for RS and NFparameter extractionIn the region where vD > VKF. Defining

vD = vDext - iD*RS and IHLI = [ISIKF]1/2.

iD = IHLIexp (vD/2NVt) + ISRexp (vD/NRVt)

diD/diD = 1 (iD/2NVt)(dvDext/diD - RS) + …

Thus, for vD > VKF (highest voltages only)

plot iD-1 vs. (dvDext/diD) to get a line with

slope = (2NVt)-1, intercept = - RS/(2NVt)

L10 February 17 7

RSeff and Neff estimation

y = 0.0275x + 2.311

R2 = 1

y = 0.0287x + 1.9049

R2 = 0.9998

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.0 500.0 1000.0

1/Ia (1/Amp)

RS

eff

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

0.4 0.6 0.8 1.0

Vext (V)

Nef

f

L10 February 17 8

Application of RS tolower current dataIn the region where vD < VKF. We still have

vD = vDext - iD*RS and since.

iD = ISexp (vD/NVt) + ISRexp (vD/NRVt) Try applying the derivatives for methods

described to the variables iD and vD (using RS and vDext).

You also might try comparing the N value from the regular N extraction procedure to the value from the previous slide.

L10 February 17 9

Estimating Junction Capacitance Parameters

• Following L29 – EE 5340 Fall 2003• If CJ = CJO {1 – Va/VJ}-M

• Define y {d[ln(CJ)]/dV}-1

• A plot ofy = yi vs. Va = vi has

slope = -1/M, andintercept = VJ/MF

L10 February 17 10

Derivatives Defined

The central derivative is defined as (following Lecture 14 and 11)

yi,Central = (vi+1 – vi-1)/(lnCi+1 – lnCi-1), with vi = (vi+1 + vi-1)/2 Equation A1.1

The Forward derivative (as applied to the theory in L11 and L14) is defined in this case as

yi,Forward = (vi+1 – vi)/(lnCi+1 – lnCi), with vi,eff = (vi+1 + vi-1)/2 Equation A1.2

L10 February 17 11

Data calculationsTable A1.1. Calculations of yi and vi for the Central and Forward derivatives for the data in Table 1. The yi and vi are defined in Equations A1.1 and A1.2.

Va (V) Cj (Fd) viCentral Forward

0.40 2.51E-12 derivative derivative0.35 0.585

0.30 2.11E-12 0.30 0.8160.25 1.347

0.20 1.96E-12 0.20 1.1520.15 1.007

0.10 1.78E-12 0.10 1.3250.05 1.938

0.00 1.69E-12 -0.20 2.231-0.25 2.300

-0.50 1.36E-12 -0.50 2.946-0.75 4.096

-1.00 1.20E-12 -1.00 4.132

yi = (dlnc/dv)^-1

L10 February 17 12

y = -2.551x + 1.6326

R2 = 0.9977, Central

y = -2.9965x + 1.7788

R2 = 0.9517, Forward

0

1

2

3

4

5

-1.0 -0.5 0.0 0.5 1.0

Vi (Volts)

yi (V

olts^-1

)

Central

Forward

Linear (Central)

Linear (Forward)

y vs. Va plotsFigure A1.3. The yi and vi values from the theory in L11 and L14 with associa-ted trend lines and slope, intercept and R^2 values.

L10 February 17 13

Comments on thedata interpretationIt is clear the Central derivative gives the more reliable data as the R^2 value is larger. M is the reciprocal of the magnitude of the slope obtained by a least squares fit (linear) plot of yi vs. ViVJ is the horizontal axis intercept (computed as the vertical axis intercept divided by the slope)Cj0 is the vertical axis intercept of a least squares fit of Cj-1/M vs. V (must use the value of V for which the Cj was computed). The computations will be shown later.The results of plotting Cj-1/M vs. V for the M value quoted below are shown in Figure A1.4

L10 February 17 14

Calculating theparametersM = 1/2.551 = 0.392

(the data were generated using M = 0.389, thus we have a 0.77% error).

VJ = yi(vi=0)/slope =1.6326/2.551 = 0.640

(the data were generated using fi = 0.648, thus we have a 1.24% error).

Cj0 = 1.539E30^-.392 = 1.467 pF (the data were generated using Cj0 =

1.68 pF, thus we have a 12.6% error)

L10 February 17 15

Linearized C-V plotFigure A1.4. A plot of the data for Cj^-1/M vs. Va using the M value determined for this data (M = 0.392).

y = -1.539E+30x + 1.058E+30

R2 = 9.976E-01

0.00E+00

1.00E+30

2.00E+30

3.00E+30

-1.0 -0.5 0.0 0.5 1.0Va (Volts)

Cj^

-1/M

L10 February 17 16

Doping ProfileThe data were equal-ly spaced (V=0.1V), the central differ-ence was used, for -7.4V ≤ V ≤ 0.4V, which for Cj = /x, corresponds to a range of 2.81E-5 cm ≤ x ≤ 8.99E-5 cm. These data are shown. The trend line is also shown for a linear fit. Since R^2 = 1.000, a linear N(x) relationship can be assumed.

y = 1.888E+20x + 1.861E+15

R2 = 1.000E+00

6.0E+15

8.0E+15

1.0E+16

1.2E+16

1.4E+16

1.6E+16

1.8E+16

2.0E+16

2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04

Depletion depth, x (cm)

Dop

ing

Con

cent

ratio

n (c

m^-

3)

dV

CdqA

xN n )(

2)(

22

L10 February 17 17

PARAMETER definition and units default value

TT transit time sec 0.0CJO zero-bias p-n capacitance farad 0.0M p-n grading coefficient 0.5FC forward-bias depletion capacitance coeff 0.5VJ p-n potential volt 1.0

SPICE Diode Capacitance Pars.1

L10 February 17 18

Cd = Ct + area·CjCt = transit time capacitance = TT·GdGd = DC conductance = area * d (Inrm Kinj + Irec Kgen)/dVdKinj = high-injection factor

Cj = junction capacitanceIF: Vd < FC·VJ Cj = CJO*(1-Vd/VJ)^(-M) IF: Vd > FC·VJ Cj = CJO*(1-FC)^(-1-M)·(1-FC·(1+M)+M·Vd/VJ)

SPICE Diode Capacitance Eqns.1

L10 February 17 19

Junction Capacitance

• A plot of [Cj]-1/M vs. Vd hasSlope = -[(CJO)1/M/VJ]-1

vertical axis intercept = [CJO]-2 horizontal axis intercept = VJ

Cj-1/M

VJVd

CJO-1/M

L10 February 17 20

Junction Width and Debye Length

• LD estimates the transition length of a step-junction DR (concentrations Na and Nd with Neff =

NaNd/(Na +Nd)). Thus,

bi

efft

dabia

dDaD

VFC12

NV

N1

N1

VFCVWNLNL

*

• For Va=0, & 1E13 < Na,Nd < 1E19 cm-3

13% < < 28% => DA is OK

pnqVL tD / , qNVVW effdbi /

L10 February 17 21

Junction CapacitanceAdapted from Figure 1-16 in Text2

Cj = CJO/(1-Vd/VJ)^M

Cj = CJO/(1-FC)^(1+M)*(1-FC·(1+M)+M·Vd/VJ)

VJFC*VJ

L10 February 17 22

CV data and N(x) calculation

1.E+15

1.E+16

1.E+17

1.E+18

1.E+19

2.0E-05 3.0E-05 4.0E-05 5.0E-05

0.00E+00

1.00E-13

2.00E-13

3.00E-13

4.00E-13

5.00E-13

-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00

C

Ax

dVdCqA

C)x(N

n

2

3

n