Instability of the behaviour of high resistivity silicon detectors due to the presence of oxide charges

Nuclear Instruments and Methods in Physics Research A288 (1990) 35-43 North-Holland

35

INSTABILITY OF Tile BE~tAVIOUR OF HIGH RESISTIVITY SILICON DETECTORS DUE TO THE PRESENCE OF OXIDE CHARGES A. L O N G O N I 1), M. S A M P I E T R O 1) and L. S T R f 0 D E R 2) lj Politecnico di Milano, Dipartimento di Eletlronica e Centro di Elettronica Quantistica e Strumentazione ..:~tronica CNR, Piazza Leonardo da Vinci 32. 20133 Milano, Italy 2) Max Planek lnstitut, Fohringer Ring 6, 8000 Miinchen, FRG

1. Introduction

High resistivity silicon detectors for applications in high energy physics (microstrips, drift chambers, pixel devices) are usually designed to operate in fully depleted conditions. Nevertheless the positive charges which are always present in the oxide induce a thin accumulation layer of free electrons at the Si-SiO 2 interface. Some of the consequences are: a lack of electrical insulation between n + electrodes, increased interelectrode capacitance between p + electrodes, a local reduction of the energy sensitivity of the detector, increased electric field near the edges of the p+ jonctions causing localized breakdowns and higher reverse currents. Moreover, none of these effects are stable. In fact, in the case of semiconductor detectors, the outer surface of the oxide is not usually covered by a shielding metal. Therefore negative charges coming from the environment or from the electrical contacts by surface migration can settle over the outer surface of the oxide and partially compensate for the effects of the positive charges inside the oxide. The amount of the negative charge on the outer surface of the oxide depends on the biasing condition, the environment humidity, the time elapsed from the application of the biasing voltages and the aging. The measurement of the mutual capacitance between two contiguous reverse biased p+ junctions, separated by a regiori covered by a non-metallized oxide, can provide information about the combined effect of the positive oxide charges and of the negative ones settled at the outer surface of the SiO 2. When the contiguous p+ junctions are reverse biased with respect to the n bulk, at a given reverse voltage the two depleted regions punch-through laterally at some depth from the surface. A sudden increase of the mutual capacitance between the two strips indicates that the undepleted conductive region under the oxide has been electrically disconnected from the undepleted bulk. Both simulation and 0168-9002/90/$03.50 © Elsevier Science Pubfishers B.V. (North-Holland)

experiments show that the punch-through is a bulk effect not very sensitive to the amount of compensating charge over the oxide. At much higher reverse voltages the undepleted region is reduced to the thin accumulation layer of electrons at the SiO2-Si interface. The inzer~lectrode capacitance depends, in these conditions, on the width of the depleted gap between the p+ electrodes and the accumulation layer of electrons, which behaves as a thin conductor. The width of the gap strongly depends, at a given reverse voltage, on the oxide charge density and on the amount of negative charge settled on the external SiO 2 surface. The two p + junctions of the test device we used for such measurements are concentric closed rings of large radius (approximately 0.5 cm) concentric with a central n + anode [1]. Each p+ ring has a width of 100 ~tm and is separated from the other bY a 50 ~tm gap covered with a 2000 ~, SiO 2 layer. The junctions are therefore very narrow compared to their length. This geometry is useful in order to enhance the effects connected with the edges of the junctions, in particular those due to the presence of the accumulation layer under the oxid~ "The wafer is n - type neutron doped, 10 k~2 cm (Nd = 5 × 1011cm-3), 280 txm thick, (1, 1, 1) oriented. Boron ion implantation was used for the p+ contacts and phosphorus for the anode. The technological process used for the production of the test device is described in [1]. It introduces, as usual, positive oxide charges.

2.

The

simuiations

The test device was simulated using the program PROUDS [2]. A cross section of the test device is shown in fig. 1, where the dashed box represents the region which was simulated with PROUDS. For the simulation purposes the n + contact has been placed at the bottom side of the dashed box, and all the oxide charges have been placed at the Si-SiO 2 interface. Preliminary measurements of the oxide charge density were performed with MOS condensers integra:ed near the described test device, by the measurement of the flat band voltage 1. DETECTOR TECHNOLOGY

A. Longoni et aL / Instability of the behaviour of siflcon detectors

36

shift of the C - V curve. The measured density was Nox = 6.5 x 1011 cm -2. It is worth noting that the whole positive charge per square centimeter associated with the ionized dopants in 280 vLm of depleted 10 kfi cm silicon is more than one order of magnitude smaller than Nox: A~op = NdXw = 1.5 X 10 ~° cm -~ where ~d is the net donor density and X w is the thickness of the wafer. The simula~,ions shown in figs. 2 to 4 have been perfermed for such oxide charge density. Figs. 2(a), (b) and (c) show, respectively, a 3D plot of the electric potential, the equipotential lines and a 3D plot of the electron density (log scale) for I V of reverse bias. Figs. 3(a), (b) and (c) show the same for 4 V of reverse bias. Fig. 4 shows the electron density (log scale) in section a - a (see fig. 1) of the device for Vrev = 1 V, 2 V and 4 V. The punch-through of the two depleted regions happens at approximatively 30 ~tm depth from the silicon surface. At only 2 V of reverse bias the electron density at the punch-through point has decreased by nearly two orders of magnitude with respect to the equilibrium concentration. The thin accumulation layer at the interface is still present at reverse voltages which nearly completely deplete the bulk region under the oxide (see fig. 3(c)). Numerical simulations, like PROUD, can be affected by some inaccuracy in calculating the electron concentration in the accumulation layer due to its very small thickness compared with practical grid-spacing. Nevertheless, the electron density in the accumulation layer can be easily calculated in closed form (see Appeh~ix A). For the experimentally measured oxide charge density, the electron density at the Si-SiO 2 interface proves to be of the order 10 is cm -3, and the thickness of the accumulation layer of 40/k.

.5

0.

-.5

-1.0

-1,5 150

u 0

b

15

3. The experiments

3.1. The punch-through of the depleted regions

10

The mutual capacitance between the two p+ junctions has been measured as a function of the reverse l

)? R=5225

0

A

150 I t

~i

n

÷

~

i

_~ I

t

I _. r--i-

, ~__.__L-----J0", SIMULATED

REGION

--"7",

;

F'-

i~

o

Fig. 2. Numerical simulation of the test device for 1 V reverse bias. (a) electric potential, (b) equipotential lines, (c) el~tron density.

NO=5xlOZZCm'3

P+

0

P+

P+

Fig. 1. Cross-section of the test device (dimensions in ~m)~

bias applied at the n + contact. The measurements were performed with a capacitance bridge HP4274A with 10 kHz test signal frequency. A typical result is shown in

37

A. Longr,ni et a L / lnstabifity of the behaviour of silicon detectors lOZO-

J

5xIO~L,

1010-

0

-1

101-

-2

i

ioo

5O

140

~n)

Fig. 4. Simulation of the electron density in section A - A of fig. 1.

-3

-4

150

3 %

2

,,/'"

1 0 ____L_ 3

2 v rev {v)

b

Fig. 5. Measurement of the mutual capacitance between the two p + junctions of the test device (dotted line). Theoretical result of the model proposed in Appendix B (continuous line).

15 10

/[/J/2Z/L/_/J/_Z/_~/./A

0 150 0

tt

Fig. 3, The same as fig. 2, for 4 V reverse bias.

fig. 5 (dotted line). The steep step in the C - V ~ , curve at 2.4 V indicates the punch-through of the depleted regions. In figs. 6(a) and 6(b) are schematically indicated the capacitances and the resistances which are significant m the measurement performed. Fig. 6(a) refers to a

_

Fig. 6. Capacitances and resistances which are significant in the measurement of the mutual capacitance between the strips p+. (a) Biasing voltage under the punch-through voltage, (b) above the punch-through.

biasing voltage under the punch-through voltage (defined as the reverse bias at which the measured capacitance reaches one half of the whole step), while fig. 6(lo) refers to a bias above the punch-through. The lumped capacitors C represent the mutual capacitance between each p+ contact and the highly conductivz accumulation layer under the oxide. The value o | the lumped 1. DETECffOR TECHNOLOGY

38

A. Longoni et aL / Instability of the behaviour of silicon detectors

resistor R is small for V~ev well below the punch-through voltage, shorting the accumulation layer to ground, and tends to infinity above the punch-through. The capacitor Cs represents the capacitance between the accumulation layer and the undepleted bulk. The capacitance-meter measures the capacitance Cm between its input connectors H and L defined as the ratio between the currenl signal at L (held at a virtual ground) and the prt.2uct of the voltage test signal applied at H with its angular frequency. The values of Cm is nearly zero for R --, 0 and approximates C / 2 for R ~ oo. In Appendix B a simple model is proposed which allows a rough estimation of the value of R as a function of V, and consequently an evaluation of Cm as a function of V,. The result of such a model is plotted as a continuous line in fig. 5 for comparison with experimental dz~a. Different sets of experiments performed in different humidity conditions show that the punch-through voltage is practically insensitive to the humidity conditions. A : ~ f t of the pofition of the step of the capacitance of less than 50 mV was observed when going from very low humidity conditions (less than 10% R.H.) to nearly saturated conditions (more than 80% R.H.). This fact can be explained by considering that the punch-through is a bulk effect and therefore slightly sensitive to what happens at the surface. As seen with computer simulations, the punch-through of the conductive region under the oxide happens at approximately 30 I~m in the bulk. 3.2. The interstrip capacitance above the punch-through At reverse voltages much higher than the punchthrough voltage, the semiconductor region under the oxide between the ~wo p+ junctions is completely depleted up to a depth x d from the surface, except for the accumulation layer at the Si-SiO2 interface. The depth x d is approximately proportional to the square root of the reverse bias applied. In these biasing conditions the value of the resistance R between the accumulation layer and the undepleted bulk can be considered infinite, while the value of the parasitic capacitance Cs between the accumulation layer and the undepleted bulk is inversely proportional to x d. The value of the measured capacitance Cm depends on the value of the mutual capacitance C between each p+ contact and the highly conductive accumulation layer under the oxide and on the value of the capacitance Cs (see Appendix B). The value of C is a function of the width of the depleted gap under the oxide between the accumulation layer and the p+ electrodes. In fig. 7 the plot of C versus the width G of the gap is shown. The curve has been calculated with the conformal transform method [3] for the geometly shown in the insert, which approximates our physical situation (a conductive pla~.e at a distance xa from the surface, representing the unde-

G

G

v ,j

4 2

o .._.L._..L 0

I

I

I

I

t

6

2

]

I iO

8

G (~m)

Fig. 7. Mutual capacitance between the central electrode shown in the insert and each lateral electrode, calculated with the conformal transform method.

pleted bulk region, should be considered for a better approxir:ation). The dielectric constants considered are those c~f the silicon on one side of the plane of the electrodes and of the vacuum on the other side. The length of the strip is equal to the perimeter of the oxide region between the p+ strip considered (3.2 cm). The width of the gap G depends on the density of oxide charges, on the biasing conditions and on the amount of negative charge on the external surface of the oxide. The distribution of the mobile charges over the oxide, and therefore their compensation effect, depends on the biasing conditions. After variation of the biasing conditions the new equilibrium distribution of the surface charge is reached with time constants which are a function of the surface mobility, which in turn depends strongly on the ambient humidity [4]. Fig. 8 shows the capacitance transient which can be observed when a reverse bias of 20 V is applied to the junctions. Two measurements are shown, the first one performed at about 12% relative humidity (dry' conditions), the second one performed at about 67% (medium humidity). Irarnediately after the voltage step, the value

VR= 20V

R,U.=12%

5

R,U.=67%

S

2

o

[ 0

__I . . . . 2

4

L__

!

6

8

10

t i m (sec)

Fig. 8. Measurement of the capacitance transient observed when a reverse bias of 20 V is applied to the junctions, at two different humidities.

A. Longoni et al. / Instability of the behaviour of silicon detectors 10-11

i0-12 A

'....

1

#, 10-13

lO-V~

l

!

i

200

~00

600

800

1000

t lille (sec) 10-az

i0-I:

b

10-14 .L.

10-xS

I 1 200 100 150 time ¢sec) Fig. 9. (a) Semilog~itlunic plot of the transient at 67% R.U. of fig. 8, showingthe slow exponential decay. (b) Fast component of the exponential decay of the same transient. 50

of the capacitance is nearly the same in the two cases (3.5 pF for the dry conditions and about 5% less for the 67% humidity case). Later, in dry conditions the valae of the capacitance remains practically constant (a slow decrease with a time constant of the order of 140 h can be observed), while at medium humidity a decrease of the capacitance toward a lower equilibrium value (2.1 pF) is observed. This transient is the superposition of two different exponential decays. The slower one, shown in the semilogarithmic plot of fig. 9(a), has a time constant of 377 s. The faster one, shown in fig. 9(b), is obtained by deconvoluting the experimental data from the previous slow exponential. It has a time constant of 30 s. Note that in the absence of the accumulation layer of electrons the capacitance would be approximately 0.9 pF. From the measurement of Cm, the v~ue of C can be determined by means of eqs. (10) and (11) of Appendix B, and the width of the depleted gap G can be obtained by means of fig. 7. The width of the gap immediately after the voltage step is of the order of 0.5 I~m. After about two hours the gap is practically the same as in dry conditions, while it is increased to about 3.5 lxm in medium humidity conditions. The migration over the external oxide surliace of negative charges coming from the electric~ contacts at the p+ junctions and reaching the nearest regions of the

39

oxide can explain the observed capacitance transients. The negative charges over the oxide surface partially compensate for the positive oxide charges, decreasing the density of mobile electrons in the accumulation layer and increasing, as a consequer, ce, for a fixed bias, the width of the depleted gap under the oxide. The experiment performed at 12% humidity shows that the surface mobility of the negative charge carders can be reduced practically to zero in dry conditions. In this case, the charge distribution corresponding to the zero bias condition is frozen by the negligible mobility and practically does not change after zhe voltage step. The two time constants which appear in the measurement performed in medium humidity conditions (67%) can be attributed to two different mohilities of two different negative charge carriers. An extremely simplified model composed of a resistor and of a capacitor can be used in order to roughly estimate the sheet resistance. The area of the capacitor can be taken to be equal to the product of the increase of the width of the depleted gap (3 ~m in the medium humidity case) with the length of the junction (3 cm), while the thickness is that of the oxide (0.2 Ixm). The area of the sheet resistor can be taken to be equal to that of the capacitor, giving about 104 squares in parallel. If the 30 s time constant is considered, a resistance of about 2 x 1016 ~ per square is obtained. This value is in agreement with measurements of other authors [4]. The square resistance in dry conditions is orders of magnitude higher. It is worth emphasizing that, in the measurement performed in dry ambient conditions, the charge distribution after the voltage step is z non-equilibrium one. In fact, it is the charge distribution corresponding to zero bias, frozen by the extremely low mobility. A sequence of measurements is presented in the following which shows that the eqtfilibrium distribution of surface charges depends only on the biasing conditions and not on the humidity of the surroundings, but the equilibrium distribution can be reached only if, for some period, the humidity is sufficiently high to guarantee the redistribution of the charges. Fig. 10 is a schematic diagram which summarizes the whole sequence of measurements, while fig. 1t presents the experimental data relative to some details of the sequence. The bias of the junction was aiternatively switched from zero bias to a reverse bias of 20 V (respectively under and well above the pinch-off bias). The humidity was alternately changed from very low to nearly saturated conditions (respectively less than 10% and more than 80%). The value of the meast~,red capacitance at 20 V gives information about ~he amount of negative charge over the oxide surface. Low capacitance (2.2 pF in the sequence shown in fig. 10) is related to the high density of negative surface charge, which compensates for the I. DETECTOR TECHNOLOGY

A. Longoni et aL / Instability of the behaoiour of sificon detectors

40

Io[-

BIAS OVl VOLTAGE~ >

HUMIDITY

20V

20V

20V

I

o I > 80%

80%

ROO~M ~I

> 80%

DRY (15%) 3.7pF

3,2 oF

~

2,2 pF

CAPACITANCEo~I:)FI

3

1

L

-2

2.2 pF

"'4 ' ' .y

5

HIGH CHARGE DENSITY

7

8

2.2 pF ii

12

9 Y

LOWCHARGE DENSITY

Fig. 10. Schematic diagram which summarizes the sequence of measurements described in the text. positive charge in the oxide, increasing the depleted gap between the p+ electrcdes and the electron layer under the oxide. High capacitance (3.7 pF in fig. 10) is related to the low density of negative surface charge and therefore to a narrow depleted gap under the oxide. The sequence starts with the surface charge in poorly defined conditions (the device was previously kept in quite dry conditions, about 40% humidity, and subjected to a few biasing cycles many hours before the beginning of this experiment). The application of the reverse bias gives a measured capacitance of 3.2 pF (point 2 of the seqrJence). The rise of the humidity (point 3) to more than 80% allows the rearrangement of the negative charges over the oxide surface required to reach the equilibrium distribution relative to the 20 V reverse biasing condition. The decrease of the capacitance to 2.2 pF indicates that negative charges moved from the contacts toward the surface of the oxide. The 5

J.3. Some examples of instability due to the oxide charges

---%~

4

B3o/.

v ta3

/ CAPACITANCE

d

,3

2000 I 20V STEP

following decrease in humidity leaves the distribution of surface charge unchanged, which is therefore not dependent on tl':e ambient conditions. The removing of the reverse bias (point 5) cannot change the charge distribution because of their low surface mobility, and therefore the subsequent application of the same reverse bias gives the same 2.2 pF (point 6). In order to reach the low density charge distribution which corresponds to the equilibrium distribution for zero bias the humidity must be increased (point 8). The increased mobility allows the excess negative charges to be removed from the oxide surface. The device can therefore switch to the equilibrium surface charge distribution corresponding to the applied bias if a sufficiently long "pulse" of humidity, which temporarily increases the surface mobility, is applied. Fig. 11 shows the experimental data relative to the points from 9 to 12 of fig. 11.

2000

~000

6000

8000

IO000 120(X)

TiME (see)

Fig. 11. Humidit~ and capacitance transients (experimental da~a), from the sequence of fig. 10.

The depleted region under the oxide in the proximity of the'; edge of the reverse biased p+ junction is a high field region where localized avalanche breakdown can occur. The presence of positive charges in the oxide increases the intensity of the electric field near the junction. We ca~ therefore expect an increase of the breakdown voltage if the negative charges from the contact can reach the surface of the oxide near the junction compensating in this way for the effect of the positive oxide charges. Fig. 12(a) presents some measurements of the reverse current at the upper p + contact of the device shown in the insert. The bulk is n-type 10 kft cm, the junctions are implanted, the symmetry is

41

A. Longoni et aL / Instabilio, of the behaoiour of silicon detec~rs

cylindrical and the distance from the n + ring to the p+ central region is 200 I~m. The humidity was about 40%. The different curves of the figure represent measurements perfon~aed at time delays from the application of the 100 V reverse bias from 5 to 14 h. Each measurement lasted a few minutes and, between two measurements the bias voltage was kept constant at 100 V. Fig. 12(b) sunmmrizes the results of these measurements, showing the drift of the breakdown voltage (here defined as the voltage at which the reverse current exceeds 1 ILA). The drift can be attributed to compensation effects of the oxide charges. A secxmd example of the instabilities due to the charges over the oxide is presented in fig. 13. When a small voltage difference is applied across two contiguous p ÷ junctions reverse biased with the depleted n-bulk, a potential barrier between them prevents the injection of holes from the one at bigher potential toward the other [5]. The height of the potential barrier increases with the increase of the electric field component perpendicular to the oxide surface and due to the presence of positive oxide charges. When the potential difference

!!:!i lO"~

~o-~

.

.,......-" .,

5

.. .:"

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t

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. ..

..............<::;;jii.:::::::;;iiil--"".......: ...

10"9 F;: .......

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t

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150

200

?_

Ytl I i0"9

<~ l o - ' ° I r

0

//

20

I

40

50

80

VOLTAGEBEI~EEN STRIPS (v)

Fig. 13. Modulus of the current exchanged between two contigucus p+ junctions when a voltage difference is applied acro'.~s them. Each curve has been measured at the indicated time delay (h) after the application of 40 V voltage difference.

between the junction increases, the height of the potential barrier decreases and eventually it becomes so small (of the order of hundreds of millivolts) that a large raumber of holes can be injected from one junction to the other. This current grows nearly exponentially with the voltage difference. The reduction of the effect of the positive oxide charge, due to the compensation produced by the negative surface charges, reduces the voltage difference required for the high injection of holes. Fig. 13 shows in semilogarithmic scale the modulus of the hole current between two p+ junctions for measurements performed in room humidity conditions at different time delays after the application of the bias. Between measurements the bias was kept at 30 V. The @.stance between the p+ strips is 100 I~m, the width of each strip is i00 ~tm, the resistivity of the bulk is 2 k £ cm. The drift is apparent.

4. Conclusions

250

*

°

200

I D

150 ~

L1oo L _ L _ _ L _ _ L _ I . _ _ I _ _ I _ _ L _ _ L o 2 4 6

10"7

250

V~ (v)

t

10-6 _

!_ 8

]O

TIME (~}

Fig. 12. ia) Reverse current of a p+ j~nction implanted on a t0 k~l cm n-type wafer. Measurements performed at time-delays from 5 to 14 h after the application of t00 V reverse bias. (b) Drift of the breakdown voltage.

The measurement of the mutual capacitance between two contiguous reverse biased p+ junctions, separated by a region covered by a non-metall~ed oxide, has been used as a technique to investigate the combined effect of the positive oxide charges and of the negative ones which reach the external surface of the oxide. "Ilie presence of an accumulation layer of electrons under the oxide, induced by the positive oxide charges, has been experimentally verified and the width of the depleted gap between the accumulation layer and the p+ junction has been measured. The width of this gap is small under the usual bias conditions (of the order of a few microns), tt. ~bilities due ~o the migration of charges over the external surface of the oxide have been verified. It has been verified that the equilibrium charge distribution, for a given bias condition, can be reached in a short time if the mobility of the surface charges is L DETECTOR T E C H N O L ~ Y

42

A. Longoni et at / Instability of the behaviour of silicon detectors

increased, at least temporarily, by increasing the humidity of the surroundings. In order to avoid instabilities due to the surface migration of charges, the oxide should be covered by metal everywhere it is possible.

I-

0+

-

Lox

-s....+...

-

r

Yd

Acknowlledgements We wish to thank P. Rehak for helpful and stimulating discussions. The work is supported by the Italian MPI, INFN and CNR.

"0

-

-

Yd

- - - r

No/2

b

-

I~, Lox

n Appendix A: The oxide charges and the accumulation layer

C

By solving the Poisson equation in the proximity of the SiO2-Si interface and in equilibrium conditions we can obtain the density of free electrons in the accumulation layer [6]: n(x) =

n(O) (l + x / ( v ~ i , a ) ) 2 .

(A1)

This expression is valid for x not greater than some La. The variable x is the depth from the SiO2-Si interface and L d is the Debye length: L d = ¢(sVt/(qn ( 0 ) ) .

(A2)

The electron density a~; the interface can be calculated by supposing that all the positive oxide charge is compensated for by an eqt¢al amount of negative charge of accumulated elex:trons.

,,,o,

=

(0) No . =

(AS)

The poten*.i~l difference Vb between the Si-SiO 2 interface and the undepleted bulk is obtained from: , (0) = N a exp(Vb/V, r).

(A4)

tn the absence of compensation from the outside we can therefore expect an electron density at the interface of the order of 10 ~s c m - 3, a Debye length (which gives the thickness of the accumulation layer, a few Debye lengths) of the order of 40 ~, and a potential barrier between the silicon surface and the bulk of the order of a few hundreds of millivolts. Negative ions deposited on the outer surface of the SiO 2 partially compensate for the positive oxide charges, reducing the number of electrons in the accumulation layer.

Appendix B

Yd

'

:

op

~

A simple model of the depletion region allows a rough estimation of the value of R as a function of the reverse voltage Vr and consequently an evaluation of Cm as a function of E- Let us suppose that the shape of the depletion region is a rectangle (see fig. 14(a)). The lateral expansion Ya is approximately proportional to the vertical one x d, that is Ya = A x a , where: xd = ¢(2%/qNa )( Vr + Vi).

(A6)

In order to attenuate the harshness of the full depletion approximation we can suppose (as shown in fig. 14(b)) that the concentration of free carriers along the se:,tion A - A of the figure drops linearly to zero from the equilibrium concentration N a in a distance equal to the Debye length L a (where L d = ¢[%Vt/(qNa) ] . Fig. 14(c) shows the electron concentration when the lateral expansion of the depletion region is greater than (Lox Ld)/2. We suppose that the slope Nd/L d remains constant and correspondingly the maximum concentration n top decreases as Ya increases until it is reduced to zero. The value of R, for Ya < (Lox - L2)/2 is: R =

Pxo 2W(Lo,v/2 - Y d ) "

(A7)

This is the same as we could obtain using the simple full depletion approximation. W is the perimeter of the oxide ring. For Yd > (Loa - Ld)/2, R is: -

(AS)

Lox

Fig. 14. Model of the depletion region under the o~de.

-

PXd

R = 2W((Lox + L,t ) / 2 - Ya)2/La

By referring to the model of fig. 6: IL C ~24R2C2 Cm = t0(V, - VL) = 2 1 + a~Z4g2c 2"

No12

.

(A8) •

The proportionality factor A between Yd and xa can be determined by observing in eq. (A5) that the value of R which gives a Cm which is ½ of the asymptotic value is

A. Longoni et aL / Instability of the beha~iourOfsilicon detectors equal to 1.15 Mf~ and by imposing in eq. (AT) that this va~ue of R is reached at the reverse voltage of 2.,~ V, where the measured value of Cm reaches one half of the asymptotic value. "/'he proportionality factor A results to be 0.28. Inserting R into eq. (A5) we obtain Cm, which is plotted in fig. 5 and compared with the experimental data. The measured slope at the cap-city step is 7 p F / V , while the slope computed as described is approximately 9 p F / V confirming the acceptabiliry of this simple model. Some information about the pinch-off voltage can also be obtained from P R O U D simulations. What P R O U D produces are 3D plots of the electron density in the semiconductor. Nevertheless the link between the electron density and the step in the C-V~,~ measurement is not direct. ,~. way to easily obtain some approximate information about this link is to suppose that the electron density in the bulk of the semiconductor under the oxide progressively decreases as the reverse bias of the junction increases, while the depletion region of the junction propagates only in the direction perpendicular to the surface and not under the oxide. The length of t ~ s undepleted charnel wlfich connects the surface accumulation layer to the undepleted bulk increases as the square root of the reverse voltage. The resistance R can therefore be written as: R=

i Xd qt~n Lo~W'

(A9)

where n is the electron density in the conductive channel, Xd is the depth of the depletion region of a plane junction, Lox is the width of the region covered by the oxide between the two junctions (50 ~tra) and W is the perimeter of the oxide ring. For Xd we can use the value corresponding to the experimental pinch-off voltage. Let R be the resistance for which Cm reaches one half of the asymptotic value for R --* hafinite (R = 1.15 Mfl). From eq. (A9) we can therefore evaluate the entre-

43

sponding electron density. We find n = 2.5 × 10 9 cm -3. Interpolating the data of P R O U D simulations in fig. 4 we get for the pinch-off voltage approximately the value of 2.30 V wl"dch is in good agreement with the experimental one. We can approximate Cs with the capacitance of a plane capacitor whose area is that of the electron sheet (approximately equal to the: area of the oxide between the two p + contacts) and whose distance between plates is equal to the depletion width of a plane junction : t the same reverse bias, that is: Cs =

' sA .• ((2,s/qNd)(V., + Vr)

(A10)

From the measured values Cm and from the previous evaluation of Cs we can get the value of the capacitance C between the accumulation layer under the oxide and one of the two contiguous p+ contacts:

c = CmO + f l + C,?Cm ).

(An)

References

[ll P. Rehak, E. Gatti, A. Longoni, J. Kemmer, P. Holt, R. Klanner, G. Lutz and A. Wylie, Nucl. Instr. and Meth. A235 (1985) 224. [2] M. Berger and E. Stein, Nucl. Instr. and Meth. A253 (1987) 382. [31 E. Durand, Electrostatique, Vol. 2 (Masson, Paris, 1966) p. 273. [4] A.S. Grove, Physics and Technology of Semiconductor Devines (John Wiley, London, 1967) ch. 12. [5] P. Rehak, J. Walton, E. Gatti, A. Longoni, M. Sampietro, J. Kemmer, H. Dietl, P. HoU, R. Klanner, G. Lutz, A. Wylie and H. Becker, Nucl. Instr. and Meth. A248 (1986) 367. [6] R.S. Muller and T.I. Kamins, Device Et~tronics for Integrated Circuits (John Wiley, London, 1977) ch. 3, p. 151.

1. DETECTOR TECHNOLOGY