3 STATES MENT OF VtERCE

3ATION

- i Bureau ol Standards

Library,

tJ1 h

AUG 1 8 1970

2~

NBS TECHNICAL NOTE 537

Nomographs for Use in the

Fabrication and Testing

of Ge(Li) Detectors
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9-,

* <*

NBS TECHNICAL NOTE 537

ISSUED AUGUST 1970

Nat. Bur. Stand. (U.S.), Tech. Note 537, 18 pages (Aug. 1970)

CODEN: NBTNA

Nomographs for Use in the Fabrication and

Testing of Ge (Li) Detectors

Alvin H. Sher

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402.

(Order by SD Catalog No. CI 3.46:537). Price 30 cents.

CONTENTS

PAGE

1. INTRODUCTION 2

2. FABRICATION OF Ge(Li) DETECTORS. 3

2.1 Nomograph Relating the Time, Temperature, Voltage, and

Drifted Depth for Lithium-Ion Drifting in p-Type

Germanium 3

2.2 Nomograph Relating the Lithium-Ion Drift Mobility,

Resistivity, and Oxygen Concentration of a Germanium

Crystal ~[

3. TESTING AND EVALUATION OF Ge(Li) DETECTORS 9

3.1 Nomograph Relating the Junction Area, Capacitance, and

Drifted Depth for Planar Ge(Li) Detectors 9

3.2 Nomograph Relating the Length, Ratio of Outer to Inner

Radii, and Capacitance for Coaxial Ge(Li) Detectors ... 11

3.3 Nomograph Relating Total System Spectral Resolution,

Total Noise Contribution, and Detector Contribution

Due to All Factors Other Than Electrical Noise 13

3.4 Nomograph Relating Gamma-Ray Energy, Detector

Contribution Due to all Factors Other Than Electrical

Noise, and Effective Fano Factor 15

4. REFERENCES lb"

111

Nomographs for Use in the Fabrication and

Testing of Ge(Li) Detectors

A. H. Sher

Six nomographs which can facilitate the fabrication

and testing of lithium-drifted germanium gamma-ray

detectors [Ge(Li) detectors] have been constructed

which relate the following parameters: (1) time,

temperature, applied bias, and drifted depth; (2)

lithium mobility, crystal resistivity, and oxygen

concentration; (3) area, capacitance, and drifted

depth for planar Ge(Li) detectors; (4) drifted

depth, length, and capacitance for coaxial Ge(Li)

detectors; (5) total spectral resolution, system

noise, and detector resolution; and, (6) detector

resolution, gamma-ray energy, and effective Fano

factor. The use of these nomographs is described

and illustrative examples are given.

Key Words: Capacitance, detector resolution,

effective Fano factor, Ge(Li) detector, lithiumion drift, nomograph, oxygen concentration.
1. INTRODUCTION

During the fabrication of Ge(Li) detectors and in their testing,

the need often arises to compare various sets of parameters or to

estimate certain quantities from experimental data. By using nomographs, the need for repetitive or relatively complicated calculations

can often be eliminated. To this end, nomographs relating the most

commonly used detector parameters were constructed.

In constructing these nomographs, the aim was to provide charts

which would be easy to use and interpret, and legible enough so that

parameters .could be manipulated with some degree of accuracy. Thus,

five of the nomographs are in the form of alignment charts with parallel

scales. The sixth nomograph, shown in figure 2, is in the form of

a concurrency nomograph since construction of an alignment chart would

have necessitated using scales for certain of the parameters which

would not be easily readable or accurate over the range of interest.

Nomography , [1] * by A. S. Levins was a useful aid in preparing these

charts; the interested reader is directed to this volume for assistance in constructing similar nomographs.

Figures in brackets indicate literature references given at the end

of this paper.
2. FABRICATION OF Ge(Li) DETECTORS

During the lithium-ion drifting process in p-type germanium, it is

often desirable after some elapsed period of time to obtain an estimate

of the compensated or drifted depth. One reason might be to obtain an estimate of the oxygen concentration in the germanium crystal for a measured value of the lithium drift mobility to be used as one of the

criteria for judging the economic or qualitative suitability of the

germanium crystal for fabricating Ge(Li) detectors. The following

nomographs may be used for the above purposes.

2.1 Nomograph Relating the Time, Temperature, Voltage, and Drifted

Depth for Lithium-Ion Drifting in p-Type Germanium.

For lithium drifting in p-type Ge under ideal conditions (perfect

material and control of variable parameters) , the drifted depth W (cm) can be given by the following:

W = (2y .Vt)

1/2

(1) Li

2

where u . is the lithium-ion drift mobility (cm /V-s) , and is related

to the temperature T (K) , diffusion constant D (cm2/s) , and Boltzmann's

constant k, by y Li = D/kT , V is the applied bias (V) , and t is the

elapsed drift time (s) . This nomograph is based on recent experimental

measurements of y Lj_ in germanium [2] over the temperature range used

for most drifting. Assuming the drift behavior of the diode follows

Equation 1, the nomograph (Fig. 1) can be used to estimate the drifted

depth obtained after a certain period of drift time for a given

applied bias and temperature. The time it will take to drift to a given depth as a function of the applied voltage and temperature of

the drift (or any of the other drift parameters when related quantities

are held constant) can also be determined from the nomograph.

As an example of the use of this nomograph, the drifted depth

obtained for an applied bias of 1000 volts at a temperature of 15°C

after 400 hours of drift can be found by connecting with a straight

line the value of 1000 volts on the left-hand scale with a value of

15°C on the temperature scale. The value of 400 hours on the time

scale is then connected to the point of intersection on the unmarked

(pivotal) scale and the value of drifted depth obtained is 0.8 cm. Conversely, by connecting the value of 0.8 cm on the drifted depth

scale with the point of intersection on the unmarked scale, obtained

from connecting the values of 1000 volts and 15°C, and extending this

line to the time scale, the drifted depth of 0.8 cm is obtained after

400 hours of drift. Comparisons can also be obtained between sets of

values of applied bias and temperature, in terms of drift rate, by

2 All further references to "connecting values" of various parameters

on the nomograph scales shall be taken to mean using a straight line.

3
connecting the various values of temperature and bias under study and

marking the intersection on the pivotal scale. The higher the intersection on this unmarked scale, the higher the drift rate.

Use of this nomograph is valid only if the drift behavior of the

diodes follows Equation 1. It has been shown that for many germanium

crystals, the lithium drift behavior does not follow this ideal case

[3] . Loss of mobile lithium ions during drift can cause an apparent

decrease in the drift rate; in these cases [3] the drifted depth can usually be obtained by using the following equation:

W = [VLiVT(l - e"

2tA

)]

1/2

(2)

where t is the lithium loss time constant (s) . When T>>t, Equation 2 reduces to Equation 1. Thus the nomograph (Fig. 1) should only be used

to estimate the maximum drifted depth to be obtained when loss of

lithium during drift does not significantly affect drift rate.
5000

4000

2000 -

1000"

800

600 —

\

g 400-

200-

100—I

80

60

40 u

\ \

\

\

h-N

LU

Q_

-61.2

60

50

-47. 7

-40

" 36.2

30

-23.8

20

10

--10

20

3.0

2.0

I— 1.0

0.8^

0.6

0.4

0.2

Q

0.1 3

1— .01

90

60

30

-

20

3000

2000

- 1000

800

600

400

-200

76

5 - > 4 — 3

- 100 £

80 1

60 £

i— 40

20

- 10

- 6

- 4

- 2

Fig. 1 Nomograph relating time (t) , temperature (T) , applied bias

(V) , and drifted depth (W)

.
le

E

o

M

6

o

£l018

5

<rZ

LUO

z

o

z

LU

CD

>-X

o

14

,13

^- 1 i i 1

I

1 r—i—

|

i - i -! —

r

1 "^^

TEMPERATURE;:23.8°C"

~~ ^x^v^ 5&cm — \\\ -^-*^^J0& cm \ ; ^s^O «^^20iicm

N. ^^^30X2 cm *^v^

7 \$^&Jm^-^

v \ \r

i i | _jl_ ill J ill

10 •13 10-12 10 10 3X10 10

fj. u , LITHIUM ION DRIFT MOBILITY (cm2/V-s)

Fig. 2 Concurrency nomograph relating lithium-ion drift mobility

(V .) t resistivity of single-crystal, p-type germanium (fi-cm)

,

and oxygen impurity concentration (N )

.

o
2.2 Nomograph Relating the L1th1um-Ion Drift Mobility, Resistivity,

and Oxygen Concentration of a Germanium Crystal.

3 This concurrency nomograph relates the lithium-ion drift mobility

yLi (cm^/v-s) , to the sample resistivity (ohm-cm) , and oxygen concentration NQ

(cm"2) , at 23.8°C, through the following equation [4]:

N = n(l - f) + C(f

_1

- 1) (3) o -3

where n is the acceptor concentration (cm ) [5] , C is the dissociation

constant for the lithium-oxygen complex LiO+ (3.5 x 10^2 cm

- 3 [6]), and

f is the ratio of the observed lithium mobility to the maximum lithium

mobility at 23.8°C (taken as 3.04 x 1010 cm2/V-s [2]).

This nomograph (Fig. 2) is used in the following way: for an experimentally measured value of lithium mobility (see Reference 2)

of 2 x 10-1 cm^/V-s, for example, and a specimen resistivity of 10

ohm-cm, the oxygen concentration is found to be approximately 1.2 x 10 1 atoms/cm . The important effect of the sample resistivity, i.e.,

acceptor concentration for p-type crystals, on the lithium mobility for

a given oxygen concentration is illustrated. For example, at an oxygen

concentration of 1.2 x 10^-4 atoms/cm , the lithium-ion mobility in a 40 ohm-cm, p-type germanium crystal would be more than an order of

magnitude lower than the mobility in a 10 ohm-cm crystal. As the oxygen

concentration of germanium crystals increases above approximately lO-*-^ atoms/cm^ , the variation of the acceptor concentration has less effect

on lithium mobility. Thus, the usual assumption of an inverse linear

relationship between oxygen concentration and lithium-ion drift mobility

in germanium is found to be invalid [3] .

3 This nomograph has been previously published in Reference 3,
20'

10-

6 -- \

4 + \

2 --

.2

\

\ —.6 \

\

\

\ 2- \ — \

UJQ

.or

\

\

\

\

\

\

\ 3

.1

10

20

30

40

50

Fig. 3 Nomograph relating diode junction area (A) , capacitance (C)

,

and drifted depth (W) for planar Ge(Li) detectors.

8
3. TESTING AND EVALUATION OF Ge(Li) DETECTORS

The relationships of capacitance and size of both planar and

coaxial Ge(Li) detectors and detector resolution and system noise

are described by the following nomographs.

3.1 Nomograph Relating the Junction Area, Capacitance, and Drifted

Depth for Planar Ge(Li) Detectors

2 This chart relates the junction area A (cm ) , capacitance C (pF)

,

and drifted depth W (cm), of a Ge(Li) diode under reverse bias through

the following equation:

W = KA/(36^C) = 1.41 (A/C). (4)

If the value of the detector area is connected with the value of the

capacitance, the drifted depth is given by the intersection of the

line on the center scale as shown in Fig. 3. For example, a diode

with an area of 7 cm^ and a capacitance of 10 pF would have a drifted

depth of approximately 1 cm. Conversely, the nomograph may also be

used to determine the expected capacitance of a detector with known

drifted depth and junction area.
1.5-1

O

o

2.0 —

3.0

400 —I

200-

100-

80

60

40 -

Ll_QO

3-O

20

10

6 -

4 -

Fig. 4

4.0

5.0

6.0

7.0

8.0

9.0

10.0

15.0 -

20.0

25.0

Nomograph relating the ratio of outer to inner radii (R2/R1)/ length (L) , and capacitance (C) for true coaxial Ge(Li) detectors.

Inset shows the relationship between the

parameters R\ , R£/ and L.

1-15.0

10.0

^-8.0

-6.0

-4.0

-2.0

—1.0

-0.8

0.6

^0.5

n l p

10
3.2 Nomograph Relating the Length, Ratio of Outer to Inner Radii,

and Capacitance for Coaxial Ge(Li) Detectors

True or double-open-ended coaxial, cylindrical, Ge(Li) detectors

with length L (cm), outer radius R2

, and inner radius R]_ , can be

related to the capacitance C (pF) , through the following equation:

C = [1.41(2TTL)]/[ln(R /R )] (5)

The inset in Fig. 4 shows the relationship of these parameters to the

coaxial detector geometry. The difference (R2 - R]_) is the drifted

depth. Consider a crystal where R2i- s 1*5 cm and Rj is 0.5 cm; therefore, R2 /R]_ = 3, and for L = 4 cm, the expected detector capacitance

is 32 pF. Conversely, for a known detector capacitance, crystal length,

and known radius R2

, the chart may be used to obtain a value R-[_ by

finding the ratio R2/R-|_; the drifted depth may then be estimated from

the difference (R2

- Rj)

.

11
5 —

I

4 —

aj

Cl. CO

I—o

CO UJ <3

to JT Qi 3 o u.

1—

'_) >

< 0) u_ _*: O1— UJ

U~l UJ i—

<

3 O

a ~ZL z: O

o i—

i

i—

i

0£ 1— 1—

Z3 O

_l LU O _l CO UJ

UJ

C£ Z=a: q: 31 O h- h-o o:

UJ UJ

/ — 2 /

/

/

/

/

/

/

Q O

:/

O UJ <

/

/

2 —

/

2/~-

/ 1_

/

/

/

/

/

/

COO

CO

>-

CO

\—o — 4 <3

L— 5

Fig. 5 Nomograph relating total spectral peak width (A ) , total system

T noise (A ) , and detector resolution due to all factors other

than electric noise (A ) . 12
3.3 Nomograph Relating Total System Spectral Resolution, Total Noise

Contribution, and Detector Contribution Due to all Factors Other Than

Electrical Noise.

T

The total system noise AE(keV FWHM) , the detector contribution

due to all factors other than electrical noise Ag (keV FWHM) , and the

total system spectral peak width a| (keV FWHM) , are related through

the following equation [7] : A°= [CaV- (aV] 1/2

. (6)

hi ill hi The total system spectral resolution, A§, is obtained by measuring the

full width of the pulse-height spectral peak at half the maximum height

(FWHM) ; the total noise contribution, Aj£, is obtained by measuring the

FWHM of a peak generated by a precision pulser with the detector

connected into the system as described in Reference 6. As an example

of the use of this nomograph (Fig. 5) consider the case when the

total system spectral resolution is 3 keV FWHM and the total noise

contribution is 2 keV; the detector contribution due to all factors

other than electrical noise from the detector and amplifier is

2.25 keV FWHM. If the width of a pulser peak is found from a measurement in which the detector is replaced by an equivalent capacitor,

then the right-hand scale is the amplifier noise, A**, and the value

obtained from the center scale is the deteotov contribution, A^ [7].

13
io

4 ^

•- 27Al(n, Y :

6 ~

4 -

r

24 Na - 208Tl(ThC")

a;

>-

a: 10'

6 -

4 -

66 Ga

r

60 Co

22r }- 50 Co

-207 Bi

56 Co

137 Cs 207 Bi

133 Ba

" 51 Cr - 2i2 Pb

10<

57 Co

01 o

o

o21<c

10.

1.00'

I-.2

-.4

- .6

1.0

•2.0

-4.0

6.0

o

a:o

o UJ <

8.0

-10.0

"-20.

Fig. 6 Nomograph relating gamma-ray energy (E )

, detector resolution

Y due to all factors other than electric noise (A ) , and effective

Et Fano factor (F 1 ).

14
3.4 Nomograph Relating Gamma-Ray Energy, Detector Contribution Due to

all Factors Other Than Electric Noise, and Effective Fano Factor.

The gamma-ray energy E , the detector contribution due to all

factors other than electrical noise A~; (keV FWHM) , and the effective

Fano factor F', are related by the following equation;

F' = [A°/2.355(E e)

1/2

] 2

, (7) E Y

where e = 2.97 eV/ion pair [8]. For this nomograph (Fig. 6), the

example given shows that for the 662-keV gamma-ray of •''Cs and for

Ag of about 1 keV FWHM, the effective Fano factor is 0.1. Assuming

Poisson statistics, the nomograph may also be employed to obtain the

statistical width, AE

, for a particular gamma-ray spectral peak by

connecting the value of gamma-ray energy and F' = 1, and extending

the straight line to the right-hand scale to obtain the statistical

width. Discussions on recent measurements of the effective Fano factor

may be found in References 9 through 11.

15
4. REFERENCES

1. A. S. Levens , Nomography , 2nd ed. , Wiley and Sons, New York, 1959.

2. A. H. Sher , "Lithium-Ion Drift Mobility in Germanium," J. Appl.

Phys. 40_, 2600-2607 (1969).

3. A. H. Sher and J. A. Coleman, "Lithium Driftability in DetectorGrade Germanium, "IEEE Trans. Nucl. Sci. NS-17 , No. 3, 125-129

(1970)

.

4. H. Reiss and W. Kaiser, "Solid State Reactions of Oxygen in

Silicon," Properties of Elemental and Compound Semiconductors , H. C. Gatos, ed., Vol. 5, Interscience, New York, 1969, pp. 103-119.

5. R. J. Fox, "Determination of Oxygen in Germanium by Lithium

Precipitation" Semiconductor Nuclear-Particle Detectors and Circuits , W. L. Brown, et. al. , eds., National Academy of Sciences, Washington,

1969, pp. 198-200.

6. A. H. Sher, W. K. Croll, and W. R. Thurber, "Evaluation of Methods

for the Determination of Oxygen Impurity Concentration in Germanium

Single-Crystals," Nat. Bur. Stand. (U.S.), Tech. Note, to be

published.

7. "USA Standard and IEEE Test Procedure for Semiconductor Radiation

Detectors (For Ionizing Radiation)," Standards Publication IEEE

No. 300 - USAS N42.1 (February 19, 1969); published in IEEE Trans.

Nucl. Sci. NS-16 , No. 6, 271-282 (1969).

8. R. H. Pehl, F. S. Goulding, D. A. Landis, and M. Lenzlinger,

"Accurate Determination of the Ionization Energy in Semiconductor

Detectors," Nucl. Instr. and Meth. 59, 45-55 (1968).

9. A. H. Sher and W. J. Keery, "Variation of the Effective Fano

Factor in a Ge(Li) Detector," IEEE Trans. Nucl. Sci. NS-17 , No. 1,

39-43 (1970).

10. R. H. Pehl and F. S. Goulding, "Recent Observations on the Fano

Factor in Germanium," Nucl. Instr. and Meth. 81, 329-330

(1970).

11. H. R. Zulliger and D. W. Aitken, "Fano Factor: Fact and Fallacy,"

IEEE Trans. Nucl. Sci. NS-17 , No. 3, 187-195 (1970).

* U. S. GOVERNMENT PRINTING OFFICE : 1970—1*31-989/12

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