Charged grain boundaries in germanium

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Charged grain boundaries in germanium

A. Broniatowski

To cite this version:

A. Broniatowski. Charged grain boundaries in germanium. Journal de Physique, 1981, 42 (5), pp.741-

749. �10.1051/jphys:01981004205074100�. �jpa-00209060�

Charged grain boundaries in germanium ^(*)

A. Broniatowski

C.N.R.S., Laboratoire P.M.T.M., ^{avenue J.-B.} Clément, ⁹³⁴³⁰ Villetaneuse, ^France (Rep le 28 novembre 1980, accepté ^le 30 janvier 1981)

Résumé.

Nous étudions les propriétés d’équilibre ^d’un joint ^de flexion de faible désorientation dans le _germa-

nium, ^dans l’hypothèse ^où les états liés au joint ^{forment une bande} partiellement remplie ^{avec un niveau de neutra-}

lité situé dans la moitié inférieure de la bande interdite. L’énergie électrostatique ^et l’entropie ^des charges ^d’écran jouent ^{un rôle} majeur pour déterminer l’occupation des niveaux du joint. ^Les effets d’écran dépendent fortement de la température ^{et du taux de} dopage : ^{dans le} germanium ^de type n, il apparait ^{une couche de} déplétion ^{à basse} température, ^{avec une évolution} graduelle ^{vers une} couche d’inversion à haute température. ^{Dans le} germanium p la couche de déplétion ^se transforme en une couche d’accumulation au passage de la température de neutralité.

L’effet de barrière électrostatique ^est beaucoup plus fort dans le germanium ⁿ que dans le germanium p, ^en accord

avec les résultats expérimentaux actuellement connus.

Abstract.

The equilibrium properties ^{of a low} angle ^tilt boundary ^{are discussed on the} assumption ^{that the} boundary ^{states form a} partially filled band with a neutral level located in the lower half of the energy gap. The electrostatic energy and the entropy stored in the screening layer ^are major factors in determining ^the occupancy of the boundary levels. The screening ^effects depend strongly ^on temperature ^{and the} doping conditions : in n type germanium ^a depleted layer ^{is found at low} temperatures, ^{with a} gradual evolution towards an inversion layer

in the high temperature range. In p type germanium ^the depleted layer transforms into an accumulation layer through ^{a neutral} temperature. ^The barrier effect is much stronger ^{in n} than in p type germanium, ⁱⁿ agreement with available experimental ^results.

Classification Physics ^Abstracts

73.40L

1. Introduction.

It is well known that there are

internal potential barriers in polycrystalline ^germa- nium, associated with grain boundaries [1]. ^The

latter result in a large increase in the rate of electron- hole recombination [2, 3]. ^A discussion of these effects involves some description ^{of the boundary in thermo-} dynamic equilibrium ^{and the} purpose of this _paper is to provide ^{such a} description ^{in terms of a} simple

model.

First consider the following experimental ^results.

1) Grain boundaries have an important ^{effect on}

the conductivity ^{of n} type germanium ^{but, in contrast,}

have little or no effect on the conductivity ^of p type germanium [1, 4].

2) ^{In n} type germanium the barrier potential may be as much as a few tenths of a volt [1, 5] ; the barrier width is somewhat less than one micron [1].

3) ^The boundary charge ^is apparently ^screened by

a depleted layer [1]. ^{An inversion} layer might ^also

form in some cases [5-9].

(*) ^Work supported ⁱⁿ part by the C.N.R.S. under contract : ATP 3290 (1977).

These results have suggested the existence of electron states localized at the boundary, ^which according ^to Shockley [10] might ^{form a} partially

filled band. The properties ^{of low} angle tilt boundaries have been discussed along these lines by ^Mueller [11].

Other authors [9, 12] ^have suggested ^{that the} boundary

states could form a discrete set of acceptor ^{or donor} levels. Impurity segregation could then be an impor-

tant factor.

Let us consider a bicrystal ^{with a low} angle (a ^few degrees) ^tilt boundary. ^The boundary is assumed to consist of a single ^{set of} parallel, equally spaced edge

dislocations. Moreover, ^we suppose the bicrystal ^is uniformly doped, ^{so that} effects of impurity segrega-

’ tion can be neglected. Following Shockley [10] ^we

suppose the dislocation states form a partially ^filled

band. A characteristic level Eo is defined so that when the Fermi level coincides with Eo, the band is half filled and the dislocations are electrically ^neutral.

More generally, the dislocations _may act as donors

or acceptors according ^{to the} position of the Fermi level with respect ^to Eo, ^thus becoming electrically charged. This model has been used by Schroter and Labusch [13] ^to interpret the electrical properties ^of

deformed germanium, with the level Eo, located in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004205074100

742

the energy gap, ^at 0.09 eV from the top of the valence band. (In ^this paper ^we shall also consider other values for the _purposes of comparison.) ^Since Eo ^is

located in the lower half of the energy gap, quite

different effects are expected ^{for n and} p type germa- nium. In n type germanium, ^the boundary ^is always negatively charged, ^{as the} Fermi level lies above Eo.

The boundary charge is screened by ^a depleted ^{or an}

inversion layer, depending ^{on the} temperature ^and doping conditions (section 3) ^{but in either case a}

nonohmic conductivity ^is expected, because of the

repulsive ^{nature of the} potential ^barrier.

In _p type germanium ^the boundary ^{can be made} electrically ^neutral by adjusting the Fermi level to

equal Eo. The condition EF

Eo defines the neutral temperature, To, ^{as a} function of the doping ^ratio

and Eo (Fig. 1); ^{for a} given doping ratio, ^the higher Eo is, ^the higher the neutral temperature. ^For tempera-

tures below To ^the boundary ^is positively charged, resulting ^{in a} depleted layer ^{and a} repulsive ^barrier

Fig. ^1.

The neutral temperature To ^{as a} function of the doping ratio, for various values of the level Eo.

while for temperatures ^above To, ^the boundary charge

is negative. ^In the latter case the majority charge carriers, holes, ^{are attracted to the} boundary ^and

an accumulation layer forms. The boundary ^should

then have little effect on the conductivity, ^{as the}

barrier is of an attractive, rather than a repulsive type.

As in the case of a dislocation, the electrostatic interactions play a major ^{role in} determining ^the occupation ^{of the} boundary ^{states. The} equilibrium occupation ^{is derived in section 2} using ^{a Thomas-}

Fermi approximation. ^The dependence of the boun-

dary charge ^{and the} screening ^{effects on} temperature is discussed in sections 3 and 4. A gradual ^evolution

takes place through ^{the extrinsic range, as an inversion} layer ^{forms in n} type germanium, ^{and an accumula-}

tion layer ⁱⁿ p type germanium. Lastly (section ⁵⁾

the occupation ^{of the} boundary ^{states is discussed in}

terms of the electrostatic _energy and the entropy

stored in the screening layer. ^{At low} temperature, the equilibrium equation involves the electrostatic energy alone and becomes identical with Read’s condition for the case of a single dislocation [14].

2. Equilibrium occupancy of the boundary ^states.

Equations ^are derived for an n type bicrystal ^and

the corresponding conditions for a p type specimen

are given ^at the end of the section.

-The _occupancy of the boundary ^{states is obtained}

with the following approximations :

1) the electrons are assumed to move indepen- dently, and therefore to form a Fermi distribution;

and 2) the electron states are described within the

rigid ^band approximation.

We start with an electrically ^neutral boundary,

and consider an energy level Ei within the bicrystal (1).

When the boundary ^is charged, ^{the level} Ei is shifted

by ^{the amount -} q V(r), where q ^{is the} elementary positive charge ^and V(r) ^the electrostatic potential produced by ^the boundary ^{and the} screening charges (Fig. 2). ^At equilibrium, ^the average number of elec- trons in the level E; - q V(r) ^is given by :

Fig. ^2.

^The bending of the energy bands in ^{an n} type bicrystal.

Equation (1) ^is complemented by ^Poisson’s equation :

where g is the dielectric constant and p(r) is the volume

charge density, which, ^{in turn} is related to the occupa- tion of the various _energy levels at the point ^r.

For the sake of simplicity ^the boundary plane ^is

assumed to be uniformly charged, ^{with a surface} charge density ^a. Then Poisson’s equation ^{becomes :}

where _p and V depend only ^on the distance x from the boundary.

The boundary conditions for the potential ^{are as}

follows :

(1) Energies ^are measured from the top of the valence band in the

unperturbed ^material; the electrostatic potential is also taken to be

zero at this point.

and 2) ^{there is a} discontinuity ⁱⁿ d V/dx ^of (JIB ^across

the boundary plane.

We now consider the relationship ^{between a, p and}

the fractional occupation ratios of the various electron levels.

a) ^The expression ^{for a} depends ^{on the} density ^of

states in the dislocation band. We consider the

case where the latter reduces to a single peak ^{at the}

level Eo. ^{Let 2} Nt be the number of boundary ^states

per unit _area, and /s their fractional occupation

ratio. It then follows that :

so that the boundary ^is electrically neutral for fB = V2.

According ^to equation (1) fB ^is given by :

where Vo is the electrostatic potential ^{in the} boundary plane, i.e. the barrier height.

Nt is related to the misorientation angle ^{0 and the}

number of states per unit length of dislocation which

we denote by 2/c. ^{Note that in} Read’s model [14],

c would be the distance between consecutive dangling

bonds along the dislocation. For small values of 0,

not exceeding ^{a few} degrees, Nt ^is given by :

where b is the length ^{of the} Burgers ^{vector and D is}

the distance between neighbouring dislocations. If

we take b = c = 4 A and (J = 20, ^{then D-} 115 A and - ^{2.2 x} 1013 CM- 2.

b) ^The screening charge density is made up of three contributions ;

1) the ionized impurities ^with density n¡(x);

2) the conduction electrons with density n(x) ;

and 3) the holes in the valence band with density p(x).

Let n’(x) ^{be the} density of unionized impurities ^and Nd ^the doping ratio. Then

In the extrinsic temperature range, n, p and nd ^are given by :

where N, ^{P and} Nd ^{refer to} the densities of electrons, holes and unionized impurities in the bulk of the

material ^{which can} be written as follows :

where N c ^and N v ^are the effective number of states in respectively the conduction and the valence band,

Eg ^{is the} energy gap and Ed is the donor level. The

screening charge density ^is given by :

Although the Fermi level depends, ⁱⁿ principle, ^on

the occupation ^{of the} boundary states, ^this depen-

dence is only important when the size of the bicrystal

is comparable with the thickness of the screening layer. ^{We are} only concerned with large specimens

for which EF ^{is fixed} by ^the neutrality ^condition p = 0

in the unperturbed material. As an approximation

we shall use the well-known expression :

As is shown in appendix ^A, the electrostatic _energy

density ^is given by :

In particular consider the value of _we in the boundary plane.

According ^{to Coulomb’s theorem, the} magnitude

of the electric field is given by ! cr 1/2 ^{8 which means}

that _we ⁼ a2 /8 ^8. Comparison ^with equation (16) gives :

where _{no, po} and n’o ^are the densities of electrons, holes and unionized impurities ^{in the} boundary plane.

The equilibrium properties ^{of the} boundary ^are

determined by ^{the set of} equations (4), (5), (8) through (13), (15) ^and (17). ^An approximate solution is given

in appendix ^{B for the} particular ^{case in which} /B ^its

very close to 1/2. This solution is applicable ^for typical doping concentrations (between ^{1013 and 1018} cm- 3)

and temperatures.

A similar set of equations ^{is found} for p type bicrys-

tals as follows :

where Na ^{is the} doping ratio, n° ^{is the} density ^of

unionized impurities ^and Ea ^{is the} acceptor ^level.

744

3. The properties ^{of a tilt} boundary ^{in n} type germa- nium.

3.1 OCCUPANCY OF THE BOUNDARY STATES.

-

Let us consider a moderately doped (- ¹⁰¹⁵ cm- 3)

n type bicrystal ^{with a 20 tilt} boundary. ^{For numerical} computation the neutral level is taken to be at 0.09 eV from the top of the valence band. Figure ^{3a shows}

the dependence of fB ^on temperature. The fractional

occupation ^{ratio is} slightly higher ^than 1/2 ^{with a}

minimum at T¡ 1’-1 ^{140 K, which means that the} boundary charge ^remains negative, lying ^{between 1011}

and 1012 electrons. cm- 2.

Fig. ^3.

(a) ^The occupation ^{ratio as a} function of temperature in an n type bicrystal (doping ^{ratio 1015 at . cm-} 3, ^tilt angle : 2°).

fB goes through ^{a minimum at} Ti - ^{140 K.} (b) The barrier height Vo ^{as a} function of temperature. (c) ^{The hole} density po in the

boundary plane, ^{as a} function of temperature. ^At Ti, po

Nd, ^the doping ^ratio.

3 . 2 BARRIER HEIGHT.

Since fB ^is very close to

1/2 ^the boundary ^{level is} practically coincident with the Fermi level. Consequently ^the energy bands are

bent as shown in figure ² giving ^{rise to an} electrostatic barrier of height :

The dependence ^{of Yo on} temperature ^{is shown in}

figure 3b. The numerical estimates _compare well

. with experimental ^{data [1, 5].} According ^to equa- tion (18) the barrier height ^{is a} decreasing ^function

of temperature. ^{Such a} decrease has been observed

experimentally [1].

3.3 PROPERTIES OF THE SCREENING LAYER.

The presence of an inversion layer, depends ^{on the den-} sity of holes in the boundary plane ^as compared

with the densities of conduction electrons and ionized

impurities. ^The doping ^ratio may be expressed ^{as :}

Similar expressions ^{are found for} po, no and ndo ^as

follows :

no and ndo ^{are easily shown to be very small as com-} pared ^with Nd ^and therefore, only Nd ^and po need be considered. Figure 3c shows that _po is an increasing

function of temperature, equal ^to Nd (the doping ratio) ^{for T}

T;. ^{Therefore a} gradual evolution of

an inversion layer ^takes place throughout the extrinsic temperature range : for T« Ti ^the screening layer

consists of a depleted layer only (po Nd). ^At higher

temperatures (T Z TJ ^the depleted layer splits ^and

an inversion layer ^forms along ^the boundary plane.

Figure ⁴ shows for two temperatures ^{at T«} Ti

and T >> Ti, ^{the variation of the} potential ^{and the} screening charge density with distance from the

boundary plane. The thickness of the screening layer

is about one micron. It is not clear, whether the inversion layer ^{can be} properly described within the

rigid ^band approximation. ^It might ^{not be the} case, ^as the holes are narrowly ^confined along ^the boundary plane (Fig. 4d) ^{and a more} rigorous quantum ^treat-

ment is probably required.

3.4 THE FORMATION OF THE INVERSION LAYER. -

According ^to equations (19) ^and (20) the condition Po - Nd gives;

Equation (23) defines the temperature Ti ^{for the}

formation of the inversion layer, ^{as a} function of the doping ratio and the level Eo. ^This equation parallels that for the neutrality condition in _p type

germanium, EF

Eo discussed in section 1. Inserting

the appropriate expressions for the Fermi level given

above (Eqs. (15) ^and (15’)) ^we find that for T; :

and for To : Eo=kToLog(Nv/Na). ^{When Na=Nd,T}

is approximately equal ^to To ^since Nc ^and N, ^{are of}

the same magnitude ⁱⁿ germanium, ^{which means}

that figure ^{1 also} represents ^the dependence ^of Ti

on the doping ^{ratio in n} type germanium.

Using ^the equilibrium equations given in section 2,

it can be easily shown that fB ^{has a} minimum when

Fig. ^4.

The electrostatic potential ^{and the} screening charge density ^as functions of distance from the boundary plane. (a) ^and (b) : ^{77 K.}

(c) ^and (d) : ^{300 K.}

We now consider the relationship ^{of this minimum}

to the formation of the inversion layer.

First _suppose that T T ;. ^Then by equation (23),

From figure 2, Eo is the minimum amount of energy

required ^to promote ^a valence electron into a boun-

dary ^state. Similarly, Eg - ^EF is the minimum energy needed to promote ^a boundary electron into the conduction band. If the temperature is increased

slightly, there will be changes ^{in the} filling ^{of the} boundary ^{states. Since} Eg - ^{EF Eo, more elec-}

trons are transferred from the boundary ^{into the}

conduction band than from the valence band into the boundary. ^Therefore fB ^{is a} decreasing ^function

of temperature. Conversely ^{when T} > Ti, Eg - EF > ^Eo and fB ^{becomes an} increasing function of T. Thus

fB ^{must go} through ^{a minimum at} T;, corresponding

to the formation of the inversion layer.

3.5 THE SCREENING OF A SINGLE DISLOCATION.

The previous discussion suggests ^a comparison ^with

the properties ^{of a} single dislocation. Schroter and Labusch [13] ^{assume a low} screening charge density

around the dislocations, ^and consequently ^{use the} Debye approximation. However, ^this assumption

holds only when the dislocations are practically ^neu- tral, ^i.e. in p type germanium ^{when T N} To. ^The argu- ments in (3.3) ^and (3.4) ^also apply ^with figure ² representing ^the bending of the energy bands ^near the dislocation. Depending ^{on the} temperature ^and the doping ^{ratio the} dislocation is surrounded with

a depleted, ^{an inversion or an} accumulation layer.

We understand that this fact has not been taken into account in discussing the results of Hall effect measu- rements in deformed germanium [13, 15].

4. The properties ^{of a tilt} boundary in p type ger- manium.

Figure ^{5 shows the} dependence ^of fB, Vo

and po ^on temperature ^{in a} p type bicrystal ^{with a 20}

tilt boundary. Again ^a doping ^{ratio of 1015 cm- 3 is}

assumed. The neutral temperature is about 140 K.

When T To ^the boundary ^becomes positively charged ( fB 1/2) ^{and a} depleted layer ^forms ( po Na) resulting ^{in a} repulsive ^{barrier as in the}

case of n type germanium. ^However the barrier height

is much lower in the present ^case (Fig. 5b). ^When

T > To ^the boundary charge ^becomes negative, resulting ^{in an} accumulation layer (po > Na).

5. The electrostatic energy ^and the entropy ^{stored in}

a bicrystal.

^The occupancy of the boundary ^states

is discussed in terms of the electrostatic _energy and the entropy stored in the bicrystal. ^The electrostatic energy is of major importance ^{at low} temperatures,

as Read has shown for the case of a single ^disloca-

tion [14]. ^A different behaviour however prevails ⁱⁿ

the high temperature range, due to changes ^{in the} screening conditions, ^{so that the} dependence ^{on the}

electrostatic _energy is much reduced. On the other hand, ^the entropy ^{stored in the} screening layer ^{is no} longer negligible. Both effects favour a rapid ^build

up of the boundary charge, ^{as shown in} figures ³

and 5. These properties will be studied for an n type bicrystal. Similar effects hold for _p type specimens,

but with an accumulation instead of an inversion layer.

746

Fig. ^5.

(a) ^The occupation ^{ratio as a function of} temperature in a p type bicrystal (doping ^{ratio 1015} at. cm - 3, ^tilt angle : 2°).

The boundary ^is electrically ^{neutral at} To - ^{140 K.} (b) The barrier

height ^{as a} function of temperature. (c) ^{The hole} density po in the

boundary plane, ^{as a} function of temperature. ^At To, po

N 8’ ^the doping ^ratio.

Solving equation (5) for the Fermi level gives :

Equation (24) ^{is a statement of the} equality ^{of the}

Fermi level with the chemical potential of the boun-

dary electrons :

Eo ^{is the} binding energy of ^an additional electron and - k Log f B/( 1 - f B) ^the corresponding change

of entropy of the electrons in the boundary ^states.

The term, - q V o represents ^the part ^{of the chemical} potential associated with the electrostatic interactions.

As a consequence of these interactions, changes ⁱⁿ

the boundary charge ^{result in} changes in the electro- static _energy of the bicrystal ^{and the} entropy ^{of the} electrons in the screening layer.

Since fB ^{is very close to} 1/2, ^the quantity

is _very small as compared with the other terms in

equation (24). ^Let We(6) be the electrostatic _energy per unit boundary ^area. According ^to appendix ^C,

the change ⁱⁿ W,,(a) ^{for each} additional electron is

given by :

where _po is the screening charge density ^{in the boun-} dary plane. ^The quantity - q(oWe/oa)T ^{will be}

evaluated in two limiting ^cases, (a) ^{when the} tempe-

rature is much lower and (b) ^much higher ^than T;.

(a) T Ti (no ^inversion layer). According ^to equation (C. 8),

The effect of varying ^the boundary charge ^{is shown}

in figure ^{6a : an} electron is transferred from A (the

limit of the depleted layer) ^to B, ^the boundary plane,

so that We ^is changed by - q Vo, the work done against

the repulsive electrostatic forces.

An approximate ^{form of the} equilibrium ^condition

is obtained by eliminating q Vo ^from equations (24)

and (26), ^{as follows :}

where fa is deduced from equation (27) ⁱⁿ conjunction

with equations (4) ^and (25).

An equation ^{similar to} (27) ^{has been} given by

Read [14] ^{for a} single dislocation (« minimum energy

approximation »). ^This equation ^holds only ^when

there is no inversion layer : ^the entropy stored in the

depleted layer ^is nearly independent ^{of the} boundary charge, and therefore the equilibrium ^condition

involves the variation of the electrostatic _energy alone.

(b) T >> Ti (inversion layer). Varying ^the boundary charge ^now amounts to transferring electrons from C

at the top ^of the inversion layer, ^{to B} (Fig. 6b), ^{a local} rearrangement ^as is discussed in appendix C. Therefore the electrostatic _energy is not expected ^to depend very much on the boundary charge. ^{However the} entropy should have a strong ^{variation as new holes are}

Fig. ^6.

The electron transfer corresponding ^{to a} variation of

the boundary charge. (a) No inversion layer : ^{A - B.} (b) ^Inversion

layer : ^{C -+ B.}

added to the inversion layer. ^{We shall} consider the

case of a strong ^inversion layer, ^{for which}

According ^to equation (C 10),

Transferring ^an electron from C to B varies the internal _energy of the bicrystal by ^{an amount :}

Let 65 be the corresponding change ⁱⁿ entropy : ^: 6S consists of the entropy change ^{of the} boundary electrons, together with that of the holes in the screen-

ing layer. ^At equilibrium ^{the free} energy of the bicrystal

is minimized with respect ^{to the} boundary charge ^and

therefore it follows that :

Neglecting ^the change in electrostatic _energy and the

change ⁱⁿ entropy ^{of the} boundary ^electrons gives :

A new form of the equilibrium ^condition is obtained

by eliminating Eo ^from equations (24) ^and (31) ^so

that :

The quantity - q Vo ^{is now related to} the variation of entropy of holes in the screening layer rather than the increase in electrostatic _energy.

The _occupancy of the boundary ^states is found from

equations (4), (20), (25) ^and (28) : ^the dependence ^on temperature ^{is much} stronger ^{than in case} (a). ^{On the}

other hand, fB ^no longer depends ^{on the} doping ratio,

which it to be expected ^as the electrons in the boun-

dary ^states and the inversion layer ^form practically

a closed system ^with respect ^{to the rest of the} bicrystal.

6. Conclusion.

The equilibrium properties ^{of a}

tilt boundary have been discussed, assuming ^the boundary ^{states to form a} partially filled band with

a neutral level located in the lower half of the _energy gap. The density ^of boundary ^states is estimated to be about 1013-1014 cm-’ for a low angle boundary ^of

a few degrees.

In addition to the electrostatic _energy, the entropy stored in the screening layer ^has been shown to play

a major ^{role in} determining ^the occupancy of the

boundary ^states. Both factors result in the boundary remaining practically ^{neutral, with a net} charge ^of typically ^{1011-1012 electrons cm- 2.}

The barrier potential ^is calculated to be a few tenths of a volt. The thickness of the screening layer ^{is about}

one micron for a moderately doped (1015 cm - 3) ^bi- crystal. ^The properties ^{of the} screening layer depend

on the type ^of semiconductor and show a strong depen-

dence on the temperature ^{and the} doping ^{ratio. In n}

type germanium ^a depleted layer ^{is found at low tem-} peratures, ^{with a} gradual transition towards an inver- sion layer ^at higher temperatures.

A parallel evolution takes place ⁱⁿ p type germa-

nium, ^{with a} depleted layer changing ^{into an accumu-}

lation layer, ^{as the} temperature is increased through

the neutral temperature.

In this paper the boundary ^{states have been assum-}

ed to have the same energy for the sake of simplicity

Similar results are expected ^{for more} complicated

band structures, since the equilibrium properties ^of

the boundary depend only ^{on the} density ^{of states near}

the neutral level.

The effects of impurity segregation ^{have not been}

considered. Segregated impurities ^have apparently

been observed near the core of boundary dislocations

by high resolution electron microscopy [16]. ^However

little is known at present concerning the effect of these impurities ^on electrical properties.

Experiments ^are in progress ^to examine the most

important features of this model, namely :

1) the existence of a neutral temperature ⁱⁿ p type

germanium ;

2) the formation of an inversion layer ^{in n} type germanium (eq. (23));

and 3) ^the dependence of the barrier potential ^on temperature (Eq. (18)).

Acknowledgments.

^The author is indebted to G. Faivre, ^P. Haasen, J. Labbe and G. Saada for many useful discussions.

APPENDIX A

Including ^the screening charge density (14) ⁱⁿ

Poisson’s equation (3) gives :

The electrostatic _energy density ^{is found} by multiply- ing both sides in equation (A . I) by d V/dx ^{and inte-} grating, ^which gives :

The constant C is found by setting ^{x to a value much} larger than the thickness of the screening layer, ^where both we ^{and V are zero. Therefore}

and finally :

748

The value of _w, in the boundary plane ^{is a} particular

case (see text), ^{as follows :}

P, Nd, ^{n° and} nd° ^are negligible ⁱⁿ comparison ^to N, the latter being practically equal ^to Nd (see Eqs. (11), (12), (13), (21) ^and (22)). Therefore :

Two limiting ^{forms of} equation (A. 5) ^{are used in} appendix ^{C. When} po Nd (no ^inversion layer),

then :

The second form applies ^when po >> ( - qVolkT) Nd,

so that :

APPENDIX B

Approximate solution of the equilibrium equations

for an n type bicrystal.

^An approximate ^solution

is given ^{for the case in which} A ^is very close ^to 1/2.

Solving equation (5) ^{for -} qVo gives :

According ^to (4),

Assuming fB - 1/2 ^{means that :}

Equation (B. 1) gives ^to first order in a :

According ^to (9) ^{the hole} density ^{in the} boundary plane

is :

giving ^to first order in a :

From (A. 5), (B. 4), (B. 6), (12) ^and (15) ^one may write :

According ^to (B. 3) ^the quantities involving ^{a on the} right hand side of (B. 7) ^are negligibly small, ^and therefore,

Equation (B. 8) gives ^{6 as a function of} temperature and the doping ^ratio. Lastly, fB ^{can be} deduced from

a by ^{means of} (B. 2). The conditions of validity ^of

this solution are discussed below.

Equation (B. 8) ^{shows that the} boundary charge ^is independent ^of N,, i.e. the misorientation angle ^0.

Therefore, ^{the condition} (B. 3) is satisfied if 0 is

large enough. ^Numerical computation ^{shows that for} 0 Z 10, fB ^{remains very close to} 1/2 throughout ^the

extrinsic temperature range and over a wide _range of

doping ^ratios (between ^{1 O 13 and 1018} cm,-3). ^For

smaller 0, fB may deviate noticeably ^from 1/2 ^{and the}

solution is no longer satisfactory. ^{Moreover the} boundary ^{can no} longer ^be considered as uniformly charged, ^because of the increased spacing ^between

dislocations.

APPENDIX C

The electrostatic energy stored in ^{an n} type bicrystal.

- Let We( (J) be the electrostatic _energy stored in the

bicrystal per unit boundary ^{area. We} shall calculate the variation - q(O We/0 (J)T ^{when the} boundary charge

is varied by ^{one electron} at constant temperature. ^We shall also discuss the consequent modifications in the

screening charge distribution. Similar considerations

apply ^{to a} p type bicrystal.

1. THE VARIATION OF THE ELECTROSTATIC ENERGY.

- Let us suppose Poisson’s equation is solved with the boundary charge ⁷ (Fig. 7a). According ^{to Cou-}

lomb’s theorem the electric field at the origin ^is u/2 ^8.

Consider the point ^M (abscissa bx) where the electric field is ( 6 ⁺ bQ)/2 ^{s :} clearly ^the part ^{of the curve to} the right ^{of M} describes the variation ^{of the} _potential

for the boundary charge ^{Q + 6J, the} plane ^{of the} boundary being ^{shifted to the} point of absissa bx.

The relation between bx and 6J is found by ^consider- ing ^the screening charge distribution (Fig. 7b) : according ^to Gauss’s theorem the screening charge ⁱⁿ

the shaded area is one half of ba

Figure 7c shows the density of electrostatic _energy

as a function of x. The variation of We is twice that of the electrostatic _energy under the shaded _area, as

follows :

First order expansions ^of (C. 1) ^and (C. 2) may be used for small ba, ^as follows :

and :

Fig. ^7.

(a) The electrostatic potential ^{as a function} of distance from the boundary plane. (b) ^The screening charge distribution.

(c) The electrostatic energy density. ^The figure is drawn for ba > 0.

For a negative ^{variation M} is located on the dotted part ^{of the curve} (Fig. 7a) extending ^{to x 0.}

where _po and a’18 ^{s are} respectively ^the screening charge density ^{and the} electrostatic _energy density

in the boundary plane. Combining (C. 3) ^and (C. 4)

gives :

Therefore _varying ^the boundary charge by ^{one elec-}

tron results in an increase of the electrostatic energy of the bicrystal by ^{an amount :}

Homogeneous doping ^{is a} necessary condition for the validity ^of (C. 6), ^as translational invariance has been assumed for Poisson’s equation.

2. THE RELATED CHANGES IN THE SCREENING LAYER.

-

We shall consider two cases, (a) ^{when the} tempe-

rature is much lower and (b) ^much higher ^than T;.

a) ^T 1 (no ^inversion layer). ^{In this case} Po ’" qNd’

According ^to (C. 3) ^the width of the depleted layer ^is changed by :

Therefore varying ^the boundary charge ^{amounts to} transferring ^some electrons between the conduction band and the boundary states, ^as is shown in figure ^6a.

According ^to (C. 6) ^the change ⁱⁿ We per electron is a’/8 ENd which when combined with (A. 6) gives :

b) T > Ti (inversion layer). ^Now Po ’" qpo. Vary- ing ^the boundary charge results in a change ^{of the}

width of the inversion layer by ^{an amount :}

The electron transfer takes place between the valence band and the boundary ^{states as} is shown in figure ^6b.

The variation of We will be found in the ^case where po » ( - qVolkT) Nd. ^Then a’/8 ^{s -} kT po (A. 7) ^and

therefore :

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