- Email: [email protected]
Semiconductor surfaces and the electrical double layer
ADVASCES IN COLLOID AND IXTERFht:E SCIESCE Elsevier Pubhshmg Company, hmsterdam - Printed in The Netherlands
Semiconductor
PktCips Rrsearck
List
(1)
of
symbols
Surfaces
Laboratones,
and the Electrical
S. V. Pkzlrps’
Gloeitampe,lfubriekeH.
319
Introduction ............ Outline ........ (1.9) The bulk of a semlcoaductor._&t (I 3) Physical reasons for the existence
....
....
notions. k;m&cai of a double layer.
.
\:alud Surface
Potential distributions Potentials . . _ .................. (3-l) The chenucal potential of the electrons. The band scheme (2.2) \Vork function and photoelectrrc threshold. Experrments
(3) The (3.1) (3.3) (3.3)
....
Capacitan e equatwns &d&s Experiments ........................
of ma&&de
:
:
:
. :
:
(3.5)
Equations
of state. Xdsorptron
References
. .
: : 1 1 : .....
.
: :
:
:
.... 1 :
: %bl . 286 291 291 . 293
......... 1 :
. !i%Jz
:
1 :
. 397 1 * . 397 298
-e.l _.
..... : :
: : :
:
301 : .. 301 305
Isotherms
308 308
309 ...
314 311
....
316
.................
Methods used in colloid chemtstry and appltcable (6.1) Electrokinetw phenomena ........................ (6-Z) Contact angles ............
Acknowledgement
.
states
(5) The integral j3_ nd*y,. Adsorptron equations ................. (5.1) Adsorption and double-layer parameters . ................ ..................... (3.2) Thermodynamw functions (5.3) Comparison with the “first method” of XVerwey and Overbeek (5.4) Expressions for Jz arly~, . Its relative magnitude ............. (6)
. .
.........
space charge. . _ Gnuy layer or space ch&e_la;_e; : : 1 : : 1 : 1 : 1 : Simplafying assumptions. . 1 .................... The Por~on-lloltzmann equation. Three approrlmatlons
(4) The capacltazce (4.1) (4.3)
The :VcthcrKands
Eindhoum.
..............................
(1.1)
(3)
Double Layer
to semiconductor ................
problems
...
325 335 399 330
..............................
..................................
330
Advan.
Colloid Interface
Sci., 1 (1967) 277-333
279 LIST OF SYMBOLS
The right hand column gives the place of first appearance in the text according to page number and, where appropriate. number of equation {in brackets) and Figure number. A A
a l-3 b
c CH, CG
c
%C
C *Y
Cion C Lb-p.
co D E=A Ee &----EY
ED -EF E cal
Eh
El? e
eF FS 271,PI AFtZ! AF
acceptor atom interfacial area, surface area Van der \%raaIsparameter accounting for intermolecular attraction (ct > 0) or repullsion {U < 0) second virial coeflitient Van der 1Vaals parameter accounting for excluded volumes of molecules differen tiai capacitance of double layer differential capacitances of ltiehnholtz and Gouy layer differential capacitance of space charge layer differential capacitance of a layer of surface states differential capacitance of a layer of ionic groups adhering to a solid concentration in dilute soiutions contact potential difference standard value of ~~~n~entrati~~n
285 309 318 319
donor atom energy level of electrons in acceptor states bottom of conduction band forbidden band gap (considered temperature-indepeudent in this paper; in fact, Ec--Ev = 0.67 V at room temp. and 0.35 V at a~s~~ut~ zero, see alany’s hook p. 37) energy level of electrons in donor states Fermi energy, or efectrochemical potential of electrons potential difference in galvanic cell containing a calomel electrode potential difference in gakanic cell containing a hydrogen electrode top of valence band charge of proton (- 4.80 x 10-l@ e.s.u.) electron Helmholtz free energy (often if in American literature) Heimholtz free energy of interfacial phase Helmholtz free energy of bulk phases I, If (Helmholtz) free energy of double layer per unit area =F - Fo; difference of two Helmholtz free energies, Fatyo#OandFoaty~===O Advan.
Co&id
Intnvfacs
Sci..
318
302 304 302 303, 302 317 296 31s 285 292 28-4
2&1 292 29 I 290 305 284 285 285 309 309 313 310 1 (1967)277-333
I.IS-l-
280
G I;6 G*, G”
AC b” N
h h+ i
k I, fJfi Wi)o 7JLe*, 9JzJJ*
7JCe NA xc
7J 7t ni
‘Jj P
P PO --Q T
Gibbs function (often F in American literature) tiibbs function of interfacial phase Gibbs function of bulk phases I, II = GG,: difference of two Grbbs functions. yo # 0 and G, at ye = 0 factor appearing in Fermi-Dirac statistics magnetic field strength
309
Coiloid
SYXIROLS
(53) (53) (53)
G at
Planck’s constant (=: 6.62 x 10-a’ ergsec) hole current Holtzmann’s constant (zz 1.3S x IO-l8 erg.deg-l) Debye-Hiickel length (semiconductor notation) number of particles of component i number of particles of component i at ye = 0 effective masses of electrons and (heavy) holes respectively. For “heavy” and “light” holes, see textbooks. electron mass (z 9.11 x 10-2a g) concentration of acceptor atoms = 2~1~~~. number of states per cma available to conduction electrons concentration of donor atoms number per unit area of adsorption sites (Langmuir theory) concentration of conduction electrons coefficient in Freundlich isotherm equation intrinsic concentration of conduction electrons concentration of ions j concentration of holes pressure upon system standard value of pressure charge of an electron (solid state notation) absolute temperature El)/kT in the bulk of a semiconductor (EF(EF-- El)/kT at the surface of a semiconductor volumes of bulk phases I and IT applied potential difference potential difference across oxide layer liquid volume transported per unit of time work function of a metal and of a semiconductor distance from interface or surface exponentials related to the degree of occupation of acceptor (A) or donor (D) surface states at potential tyo Advan.
OF
310 291 323 291 285 326 285 285 310 310
(59) (12) Fig.
15
(9) (2) ( 129)
(61 (6) (59) (59)
291 291 295
(26)
291
(1-I)
295
(25)
318
(102)
2S-l 318 285 285
(101) (5)
2S-4 309 318 293 2S5 299 299
(1) (7) (1) (55) Fig. 4 (6)
(55) 329 295 326 296 293 303
Fig. 6c ( 129) Fig. 7 Fig. 4 (49)
Inferface SC<.. I (1967) 277-333
LIST
OF SYJlbOLS
osygen y
281
to germanium
ratio in c&de con~pouncl GeO,
bw-~
299
partitifjn function ijQ(ei]krr) vnlency of cliarged c0mp0nent
301,328 292 299 285
j
contact angle degree of ionization in the adsorbed state degree of ioni.*ation in the adsorbed state at vt, = 0 degree of dissoc;&on of 02 molecules at interface degree of ionization of 0 atoms at interfrtce factor accounting for electron spin number of acceptor (A) or donor (0) surface states per cmJ_ number of occupied srcceptor surfxe states per cm?
329 321 32L 322 3!!2 291
number of empty donor surface states per cm2 number of excess mo,lecules (ions) 0,f component i per unit area at the interface or at the surface standard value of the number per unit area nf esc’e+s molecules (ions) at the interface or at the surface number of excess molecules or ions per unit area at the interface or at the surface if ~0 = 0 interfacial tension, surface tension distance from pfane x = 0 where a surface charge cq is located (sometimes identified with thickness of Helmhcftz Iayer) dielectric constant dielectric constant in medium of Helmholtz layer dielectric constant in medium between planes x = 0 and x = S zeta-potential (in eiectrokinetic phenomena) viscosity
292
electrochemical potential of component j electrochemical potential of component j at yo = 0 degree of coverage (in Langmuir”s theory) Debye-Hlickel lehgth (ionic solution notation) specific conductivity chemical potential of component i chemical potential of component i at fyo = 0 chemicaf part of ekctrochemicai potential of component j
A&an.
Coltoid
Zderjacc
Sci.,
292 292
312 317 31s 309
314 265 304 315 323 326 311 311 318 283 326 310 310 311
1 (1967)
277-333
LIST OF SYMBOLS
chemical part of electrochemical potential of compnent j in the adsorbed state chemical part of electrochemical potential of component j in the adsorbed state at ~JQ= 0 chemical potential of adsorbed particles, considered as a function of the amount adsorbed chemical potential of considered component in a bulk phase as a function of concentration or pressure standard value of chemical potential of adsorbed particles at given T and given amount adsorbed standard x*alue of chemical potential of considered component in the bulk at given T and gix*en concentration ehxtrochemical potential of electrons or Fermi energy electrochemical potentintof holes or minus the Fermi energy interfacral pressure. surface pressure charge density conductance (of cylindrical Ge single crystal) interfaciat charge, surface charge; often expressed in this paper in number of electronic charges per cmz value of the space charge per unit area value of the charge in surface states, per unit area value of the charge in ionized layer, per unit area relaxation time (51ow states) energy parameter, (Ep - Et} in the bulk energy parameter, (EF - El) at the surface photoelectric threshold of a semiconductor chi- or Lange-potential; potential difference originating from a layer of polarised atoms or molecules electrostatic potential in space charge or Gouy layer with respect to the bulk of the phase (J = oo) electrostatic potential at surface or interface (.,, = 0) with respect to the bulk of the semiconductor phase (x ?= w) potential drop across Helmholtz + Gouy layer (or Stem + Gouy layer) total potential drop across double layer potential difference between the plane x = 6 and the plane x = 00 frequency, cycles per set AhaH.
Colloid
TnlerfucslSci.,
311 311 317 317 317
318 291 291 317 298 288 286 299 301 301 330 293 293
296 295 293
293
302 302 30-L 305 1 (1967) 277-333
~~~i~onduc~or Surfaces
and the Eleetricaf
Double Layer
(1) Xntroduction
Electric:al double layers are found at interfaces and surface5;. Speciaf attention will be given fo semiconductor surfaces, and to interfaces between semiconduetnrs and an a~pcous electrolyte. The most extensively studied surfaces and interfaces are those in ~rhich germanium participates, but properties of systems with other semiconductors (CciS, ZnQ, 5%) wilt be reported, EIectritrtl properties pertinent to semiconductor systems wi!t he comprecl with “cl;~s4~;11” esamptes of double layer systems, notably those at the f-fg-acl. sofn. interface, the AgI-aq. soln. interface, the doubfe layer-~ formed by ionized surfaetant molecules at a water surface or at a water-oil interface, and also, with properties of ionic solutions, In this respect it is useful to note that it is possible to distinguish two groups of workers in the field of double-layer systems_ The first group is mainly concerned with the “classical” systems mentioned here, and the second li;roup is concerned with systems in which semiconductors play a preponderant role. This paper is an attempt towards a more or less unified view of aii the systems, although it is primarily written ft)r students of the “classicaY’ systems, An electrical double iayer is formed by the unequal distribution of charge carriers, which reflects the physical and chemical discontinuities at the interface neutral system. The charge carriers with which we (surface). It is an electrically are concerned here should be mobile, They are the conduction electrons and holesl~~ in a semiconductor, the conduction electrons in a metat, and the ions in 3~1 aqueous solution. Apart from contribution due ta mobile-charge carriers, the molecules of the main constituents at the interface can take part in the doublelayer system. They form a moIecular condenser. AR important evampfe is that of water molecules present at the water surface or at the water-AgI or the: water-Hg interfaces, Hero the (polar} water molecules are oriented with respect to the plane of the interface, the orientation being different for thedifferent cases. For thewaterHg interface, the water molecules may have turned their positive side to the mercurys, in the ease of the water-AgI interface, the negative side may be4 turned Advan.
Co&id
Inter/ace
Ssi.,
1 (1967)
217-333
284
X.
J. SPARNAAT,
SEJIICONDUCTOR
SURl=ACES
to the AgI. Concerning the water surface, it is as yet uncertain which orientation is preferred by the water molecules5. 1.2
THE
BULK
OF
A
SEXICOXDUCTOR,
FIRST
NUXERICAL
XOTIOSS,
VALUES
Before proceeding with double-iayer systems some numerical values of pertinent quantities will be mentioned and some definitions will be given. \Vhere semiconductors are inx’olved, we refer to germanium. Thusin apure-Gesinglecrystal at room temperature (3OO”K), about 2.-l x 1013 Ge atoms per cm3 are “ionized”, i.e. a valence electron of such a Ge-atom is brought into a state of higher energy and is, to a certain extent, free to move. This valence electron thus has become a conduction electron. The minimum difference of the energy levels of the conduction electrons and the valence electrons is usually denoted as E, - E,., where E, is the (minimum) energy of a conduction electron and E, is the (maximum) energy E, is about 0.67 V. Quantum of a valence electron. For Ge the value of Ee theory requires that the electrons cannot occupy an energy level between Ec and Ev. The energy distance Ee - E, is therefore referred to as the forbidden energy gap. At the ionized Ge-atoms, there is an electron vacancy. This is called a hole. A hole is mobile because a valence electron of a neighbouring atom can occupy the vacancy. According to views which are now standardle”, a hole can be considered as a mobile positive charge with a positive effective mass. For Ge single crystals at 3OO”K, the drift mobility* ,D)&of the conductivity electrons is 3800 cm”/Vsec, and that of holes py, is 1800 cm’f\‘sec. lt is interesting to note that these values are about 107 times higher than those of ionic mobihties in aqueous solutions. The case just described is that of intrinsic conductivity. The concentrations of conduction electrons and of holes (denoted respecti\-ely by n and by p) are equal: n=p
(11
If the crystal contains impurities then as a general rule n # ~5.Thus if pentavalent impurity atoms (P, As) are present at lattice sites in the Ge-crystal, then n > 9 (*r-type conductivity). Four electrons of the pentavalent atoms are required for chemical satnration, the fifth has become a conduction electron. The pentavalent atom is called a donor atom, and has obtained a positive charge, lt is fixed to the lattice position. If trivalent impurity atoms (B, Al) are present on lattice sites, then TV< p (P-type conductivity), and there are immobCized negative charges. The technology trivalent impurity atoms are acceptor atoms. In semiconductor standard techniques are now available6 (for example diffusion at elevated tempera* The drift mobility is measured by conductance measurements. This may be distinguished from the Hall mobdlty. If a magnetic field is applied normal to the direction of the electric field. the deflection of the movement of the charge carriers leads to a potential difference. This Ha11 effect, as it is called, is often used to determine the concentration and the mobility of the charge carriers. The Hall mobihty differs somewhat from the drift mobdity’*s. Advaa.
Colloid
Interface
Sci.,
1 (1967)
277433
ISTRODUCTIOS
265
tures) for the preparation of crystals with almost any desired \*alue of rr-type or p-type conducti\-ity. The required impurity content is u~uall_v \-cry small. A Gecrystal with a conductivit_v (tr-type or P-type) of 0.1 R cm contains about 3 X lOI impurityatomspercm3oralmost 1 ppm. (A pure crystal at 3OO’K has a conductivity of about 46 Q cm and contains 2.4 x 1013 impurity atoms per ems). A result of solid-state theory is that the ionization reactions of Ge and of the impurity atoms can be considered as chemical equilibria, which can be treatecl in the same way as those occurring in ionic solutions for e-uample. Thus:
and
Ge + Gef + eD 7 D’ + eA % A-
+
or 0 s
h+ +
e-
12) (3)
h+
(1) where e- denotes an electron, h+ a hole. D a donor atom and A an acceptor atom. \Ve consider in this paper fully-ionized donor and acceptor atoms. Furthermore: n x p =
constant
=
~12
(5)
where ni is the intrinsic concentration of conduction electrons and holes;. (Thus eq. (1) could be written as PI = p = 9Li) Eq. (5) is only correct if the temperature and the influence of light are held constant. The value of ILLcan be increased by thermal anti by photoactivntecl stimulation of the ionization of the Ce-atoms in the crystal. Then new electron-hole pairs are created. Conversely, a lowering of the temperature and/or a decrease of the mtensity of Illumination leads to recombination of holes and electrons_ The recombination process is a complicated one nncl requires specific recombination centres- impurity atoms ancl imperfet tions in tiw bulk serbweas such and so too does the surface of the cry&al. For our purposes it is important to note the latter. The conductance of a semiconductor sample increases with increasing temperature (although the mobitity of the charge carriers decreases). This was first noted by Faratlay?m* on A&. The conduction electrons and the holes are in thermal movement just as are the ions in a solution, and just as in ionic solutions, it is possible to define a DebyeHEeke length in semiconductors_ This is often written as L in semiconductor literature:
Where E is the static dielectric constant (for Ge. E = 16). k is the Boltzmann constant, T is the absolute temperature, and e is the charge of an electron (e = 4.8 x lo-10 e.s.u.). At room temperature kT = 4.1 x 10-l” erg and L becomes about 2 micron (2 x 10-a cm). The definition of the Debye-Hiickel length for ionic solutions (usually denoted by z-1) is somewhat different (7) Advatr. Colbid Interfucc Sci.. 1 (1967) 277-333
Jf. J. SPARSAAY,
2236
SE?4ICOSDUCTOR
SURFACES
where the summation is over all the ionic species present in the solution, zj being the vakncy for ion j, and f&J its concentration. For a 10-3 tV l-1 electrolyte the value of the summation is X2 x 1017 cm-3 and. at 3oO”IC. x-1 becomes 10-S cm (for water, E = SO). On the whole, even in dilute aqueous solutions, the DebyeHUckel length is several orders of magnitude smaller than that in semiconductors. 1.3
PNYSICAE.
SURFACE
REASONS
FOR
THE,
RSISTESCE
OF
A
DOUBLE
LAYER.
STATES
There are two physical reasons far the existence of double layers. The first is the preference that mobile charge carriers may ~zhowfor one of the two substances bordering the interfczce, Three examples will be given: (a) 2% %kte@zce be&z-em IWO tiifirmtt metds. Here the electmns have different afCnities for the two different metals. The obsertqation and the interpretations of the phenomena concerning metal-metal interfaces have a Lang history, starting with the work of Gaivanig and of YoltatQ, IVe shall only mention here that it was probably Welmholtzll who introduced the name “electrical double layer” in 1853, assuming that at the metal-metal interface a very thin ~~osit~vely-eharge~~ layer was facing a everythin negatively-charged layer. As will be pointed out below. most double-layer systems have not as simple a structure as suggested by this view and the name “efectrical double Xayer” must now be considered as somewhat misleading. (6) A ~~?~~~~u~~~i&~i~~ s~+t5 cvystaG wkid.. catt be &vi&d into two rlrgims. One region has #-type conductivity, the other region has P-type conductivity. II’here the two regions touch, a double layer is formed. Electrons of the pc-type region diffuse somewhat into the j-type region (and partly recombine), and holes move to the n-type region. The dictible layer is quite diffuse, not only because the conduction electrons and the electrons are in thermal movement, but also because the transition from It-type to P-type conductivity regions, as determined by the donor and acceptor concentrations, is not so abrupt as for example the transition from one metal to the other in example (u). (c) A membrcrnesejkxratlng ho solutions f an3 II of th same salt CA, tdtidt we asstrme
compZei%Jy ionized,
ad
wlticR have
&+erent
ctmce~ttrations
cz and
en
The membrane is assumed to be permeable only to ions C+, These diffuse across the membrane from T to IL, Then f contains an excess negative charge formed by A-, and 11 contairzs an excess positive charge. Thus a double layer is again formed which again is of a diffuse character. The latter two examples are closely related to the Donnan equiXibrium_ fcx >
c&.
The second reason for the existence of a double layer is the adsorption of mobile charge carriers at the interface or surface, which then assumes an interfaz;irtlcharge (surface charge). This is denoted by CFand is expressed per unit area. R&an.
Co&M
Inlerfucc §cci., 1 (1967) 277-333
287
IXTRODUCTION
A charge -u should be present nearby, and should make the double-layer system electrically neutral. The adsorption of neutral species can aho lead to the formation of a double-layer system. Owing to the adsorptive forces, the adsorbed molecules or atoms can be ionized, thus leading to a charged surface, with the nentraliaing charges nearby. There are numerous eramp!es of double-layer systems ia which the arlsorption of ions is preponderant, notably the “classical” sybtems and the surftictant systems mentioned in the first paragraph. Examples of neutral molecules which are ionized after adsorption are those of O-, adsorbed on Ge and Si, on CdS, aA on ZnO. A number of polar gaseous ambients probably hehave in the same way on one or more of these adsorbents. A special example i5 that of the adsorption of electrons (or holes) at a semiconductor surface. The atoms at the surface have, as a general rule, a different affinity for electrons than those in the bulk. This was first pointed out by Tammr~ in 1932. These atoms, sites for charge storage, are the surface states. Tamm considered in his theory an ideal crystal bordering vacuum. States found under t’xperimental circumstances which approach this ideal case, are called ‘famm states, Another approach to the theory of this ideal ca+e was given by Shockleyl~ in 1939. The reasons why the electrons can show different affinities have been the subject of much theoretical work 14, but up till now the problems here ha\penot’” been solved. Surface states are often conceived of as adsorption sites for electrons or for holes. If electrons can he adsorbed. the states are acceptor states. if holes can be adsorbed (i.e. if valence electrons can be repelled) the states are donor states. It is not impossible that one single state can act both as a donor state and as an acceptor state. Acceptor states have a negative charge when occupied. Donor states have a positive charge when empty. The occupancy of these “adsorption sites,” by electrons and holes is calculated by means of Fermi statistics and the same equations are obtained as for the occupaccy of adsorption sites in Langmuir’s well-known theory (1917). This theory is in turn similar to that of Stern (192-f) for ionic adsorption (see ref. 27a). The occupancy of a surface can be varied esperimentally by placing a metal electrode parallel to the surface of a (grounded) semiconductor crystalandapplying an electrical potentiaf difference between the metal electrode and the semiconductor, Thus, if the metal electrode is made positive, electrons are transported to the surface of the semiconductor. Some of them are then in (acceptor) surface states. The others either have recombined with holes or are in thermal movement and &x-m a diffuse layer bordering the plane of the surface states. These electrons participate in the electrical conductance of the sample. Consequently a change of the electrical conductance of the semiconductor can be measured as the result of the application of the transverse field. This change is called the field effect. If it is small, a relatively large fraction of the electrons will be,in the surface states. The semiconductor behaves like a metal. If it is large, there WI! be few electrons in these Advaaa. Colloid fnterface
Sci.. 1 (1967) 277-333
M. J_ SI’hRNAAY.
Fig_ 1 Equipment crystallsb.
fcr merisuring conducrance
SE3%IICOE;I3WCTOR
SURFACES
and field effect of a thin cy%ndricaS Ge singte
states. Fig,. 1 showz the experimentaf arrangement, and Fig. 2 gives some results_
From the fidd effect the denftit~- of states can be estimated. Tf~us, cleaned Gel5 and 5316s-m-faces in vac--tum have about 101s states per cm%_This is the same density as
suggested bj- TZrmm’Y theory in \GGch every atom at the surface (there are ahout 101s atoms per crnl at the s&face) mav behat-e as a state. There are both donor and acceptor states, but acceptor states art: (weakly,‘1st~dominant. Handlerls* was the first to carry out this type of experimen t. He found a stroq dominance of acceptor states at the Ge surface.
i !
Now
r
gaseous
impurities,
such LS CO, even in quite small
i L-J-
a>
I
Fig. 2. Fieid effect; influence ~5 Fxygen gas a) high vsr*.um (c 10-S Torr): h\ n-_iddIeregion. oxygen pressure - 104 Tow; c) sigh presswe, about ii.01 Torr. The vertical scale shows the relative change of the conducta.nce_ Tfre horizontat scale shows the salues of the electrostatic potentialdifference between Pt ekctrode and Ge wire, AXso given is a tfme scis$e,showing that in region c) (high oxygen presstre) slow states are formed. A&an.
Colltid Interface Sci., 1 (1967) 277-333
289
ISTRODUCTIOX
quantities,
git-e the states an enhanced
difference
may
at least partly
acceptor
be found
character.
in the methods
Thus,
the origin of
of purifying
the
the surface.
Just as in Langmuir’s theory an “adsorption energy” can be defined. Estimatesbf such adsorption energies for the cases considered here will not be given, since these estimates
are little more than guess work. The states are “fa&”
that the time they need to adapt themselves
to changed
states. This means
eIectrica1 circumlstances
small. less than IO-~ sec. If s.uy*gen is allowed to react with the germanium at room temperature,
oridcs
are formeri. The density
of the tast states
is
vr*rfnce
derreaass,
finally to about NW per cm=, depending on the final oxygen pressure, tird “slow &ate5 ” are formed with relaxation times of the order of minutes. The o&latior~, as carried out in the manner described, w&l-defined
relationship
gas pressure.
between
is an irrever=ible process. There is therefore no the
In the experiments,
degrtte
osidation
of
of the surface
the gas pressures were in the range
xnd the
lc)-s-10-e
torr, and at the latter pressure there are about 101s fast states per ems. If the ouidation has taken place through an etchant, for example ‘“CY--1” (a mixture of 25 parts 6Ou/, HSOa, Bre),
15 parts
then the density
100 y/u CIIsCOUH, of fast states
states per cm” with relaxation
12 parts 3Es7& HF.
and 0.2 parts lOQy$
is still lower and there are about
times extending
1Ota slow
to hours, the whole situation
being
leas well-defined from a molecuhtr puint of view than in the case of careful gaseous oxidation of II cleaned swface. Htzxvex~er,ON ing to the much greatez- experimental accessibility
of etched
surfaces.
thctse were the first to be investigatecl.
theoretical concepts were used in 1947 hy anomalously low field effctcts found at etch4 suggested
states
layer of, say,
IO0 A (10-g
the chemical
nature of the semiconductr_r,
surface.
finrdzenr’ Ior the interpretation of Si surfct,-es. Barcleen and 13rattainia
in 1953 the moclel of an oxiciidecl semiconductor
layer of fajt
at the interfwe
between
cm) thick delxnding
In the case of etched Ge surface,
ems and about tionedabove,
101s slow states per ~4,
about lO0A.
Aspects
Tamm’s
surface.
semiconductor on the twchod
consisting
and oxide,
of a
zn oxide
of oxidation
and on
and a layer of slow states on the aside to lOra--IOll fast states per
this amounts the oside
layer
thickness
being,
as men-
of the potentials used in this model will be dis-
cussed in the next section, Very small densities nium stirface is brought iced, The anodic
of (fast) sn_f, _ states hove been iound when a germar +-e
into contact
reaction
with aqueous
leads to the formation
reaction with water, Is dissolved.
The remaining
electrolytes
andisslightlyanod-
of a surface aside,
G=-II@
interface,
which,
after
which naturally
concdns excess oxygen atoms, is almost “ideal” (or “perfe
Ge electrode,
because it is easy to increase the anodic current for
whereas there is a limiting current density (of about ImA/
cma) for n-type Ge-efectrodes (see Fig_ 3). Thus, holes are transported from the bulk of the germanium
to the surface, or rather, valence Advan.
Cdioid
efectrons are transported
iaterfate
Ssi..
1 (1967)
277-333
?r!. J_ SI’ARNAAY,
290 from the surface initiates
the
Garrettzl,
to the bulk,
dissolution
have
given by the American
Fig. 3a. Xnod~ aq.-soln. (pLI.
=
of
investigated
leaving
a broken
oxidized this
workers
chemLa1
compounds.
case.
SEMICOSDUCTOR
that the anodic
bond at the surface.
Tumer%o,
Gerischer
et aL.l9a
reaction
density vs. applied potential (ref. electrode, I)-calomel electrode (from ref. 20).
SURFACES
and
and
Brattain
inferred
the data
from
characteristics
calomet
This
eh-ctrode)
should
be
in the cell
Ce-
Pig. 3b. Model of the anodlc dissolution reaction: a) formatlon of oxidized compound intxface; b) further oxidation foIloww1 by dissolution of a germanic mxd**~.‘Q. According to Gerischer13=. A, + le = 1.1 1.8.
affected
by the addition
the solution. transfer
of electrons
mechanism
Other
Most
example,
compounds
(for example
mechanism
one state
to another.
can be expected
Cesf-Ce4fsulphate)
in a redox This
to affect
system
mechanism,
each other
requires
to the
and the hole
at the interface.
to be the case.
experiments, and surface
attention
because
from
just described
This appeared
potentials
of redox
The reduction-oxidation
at the
will
notably
be paid in what
these are more closely questions
those
recombination.
concerning
related surface
concerning
follows
eIectron
emission,
contact
the general picture outlined here.
confirm
to questions
to the “classical”
concerning dcuble
layers
potentials, than,
for
recombinationAdvan.
Colloid Interface
Sci.,
1 (lb’i7)
277-333
POTESTIAL
DISTRIBUTIOSS.
POTESTIALS
291
(2) Potential distributions. Potentials 2.1
THE
CHEMICAL
POTESTIAL
OF
THE
ELEGTROXS.
THE
SASD
SCHEJIE
The conduction electrons will be con&lered as analogous to ideal gas atoms. Two differences compared with ideal gas atoms ~111 be taken into account. First, there is the influence of the crystal lattice. As discussed in textbookslJ on solid state physics, this can be satisfactorily accountecl for by the introduction of an effective mass, MC,*, instead of the real mass 2~~. For germanium, we* z $ w,. Secondly, electrons have two spin states. Considering these two differences, the chemical potential, pe, of the electrons can be written ast: w = E, + kT ln f~rA,~ (8) where E, is the (minimum) energy of the conduction electrons and where fi, = h (2% wc’,kT)- i (9) Here h is Plan&s constant. The two spin states of the electrons are accounted for in eq. (S) by the factor _Sin the logarithmic term. The equatron of the chemical potential of the holes, ph. is the symmetrical counterpart of that of the electrons* CL,,= -E, _t kT In &&A,,~ (10) where E,, is the (maximum) energy of the valence elecrrons and where Ah is simrlar to rle except that wte* is replaced by the effective mass p)ch*of the holes. For germanium, ML~* z + me. It is a parameter with a positive numerical value. The equilibrium situation, given by eq. (2). requires: re tCLh=O (11) The chemical potential, pc,, is usually calLed the Fermi energy and denoted as Eg. The reason is that the electrons obey Fermi-DIrac statistics, which is concerned with the number of way 3 in which M particles can occupy LV quantum states with the same energy. This number, say g, is: IV! g= (12) ts! (N 9L)! B” where /2 accounts for the electron spins. If ILUre are no paired electrons at T = O°K at the considered energy levels, then #4 = f. If all electrons are paired. then /3 = 2. For n $ N this becomes: g = NA (n!)-’ fin (13) This corresponds to Boltzmann statistics. The condition n g N is often fulfilled: The p,;spression ZAe3. denoted as N, in solid state physics, can be identified with the number of states per cm3 available for tc tonduction electrons (with p = 2) : 211,~9 = NC z @.8 x lOID cm-3 whereas +zis of the order of lo*7 cm+ at the most. t
There
should
be no confusion
between
the notation Advan.
p here
(14)
and iCs use elsewhere
Colloid
Ittterfuce
Sri..
for mobility.
I (1967)
277-833
%I, J_ SPARNA.&S,
292
SEMXCOSDUCTOR
SURFACES
With the aid of eq. (131, it is not difficult to derive eq. (8) by means of the methods of statistical thermodynamics. The partition function Z of it eiectrons, ah having the energy E,, becomes:
Here the assumption The Helmholtz
F
*c
-
exp
Nen (n!)-1
Z=L:
Ee
ICT
is made that E, does not depend on the temperature.
free energy F is
=-kTln2
IIf3
and the chemical potential is
fi~Eps=
117)
After tvriting In tz! = rtln 7t - n (Stirhng’s approximation), eq. (S) is easily obtained. Similar reasoning can be applied to the holes. For surface states the situation is different because it is improbable that the Boltzmann approximation can be used. If there are I’r acceptor surface states per cm” and re are occupied by an electron, then by using eq. (12) instead of eq. (I?);:, unefinz&Iyobtains:
where E& is the “adsorption {free} energy”. Each neutral (i.e. empty) state is assumed to have one unpaired electron. Each charged state is assumed to haxe two paired electrons. Thus /3 = 2. For I’D donor states per cm9- and rr, holes, one has in a similar way [here the energy Ea is now replaced by ED. \Ve assume that the neutraf (filled) state has one electron which can occupy two spin states: /$ = +] rh
rrJ
=
1 + 2 exp -
(19)
ED--F
(
kT
>
The same chemical potential EF appears in eq_(lQ) and in eq.(18), because it is assumed that the system is in eqtrilibri~_;m*_ If* returning now to the conduction electrons in a semiconductor, EC, EY and EF are plotted against distance from the surface, the band scheme is obtained (Fig. 4). Disregarding the individuality of the atoms of the crystal lattice, this consists, in the bulk of the crystal, of three paralfel horizontal lines. Of course there are more energy levels in the crystal. These would be represented by hofizontal lines in the picture higher than EC and lower than E, (the name “band scheme” is then elucidated), Moreover, in impure crystals, energy levels arising from the presence of * “Quasi Fermi” This is the case
energies
st the surface are sometimes
when li~;ntis shining on the surface.
introdueed
if there is no equilibrium.
A&JCZSL Colloid Inlevfaca SC&, 1 (1967) 277-333
POTEXTIAL
DISTRIBUTIOSS.
293
POTESTIALS
donor and acceptor impurity atoms would be indicated by short horizontal lines representing levels in the forbidden energy gap. Their position with respect to distance would indicate the position of these atoms. h pentavalent -4s impurity in the Ge-lattice results in a donor level close to E,. Little energy is gained if a conduction electron “falls” into a state determined by an ionized As-atom, and it has already been assumed that ionization of such a donor was complete. Similar considerations apply to the trivalent B impurity which provides an acceptor level close to E,_ However, we shall not consider such energy leveis because we are interested
+ es ,/- _-_--____ $3 ---_--_ +r/
(n-type chttiv~ty)
EF
6
E r
~__*Ep\
W=U
____-
I
Dlstaxe
x -
X=0 Fig. 4. Band scheme.
Symbols
explained
in text.
only in the band scheme close to the surface. Just as in an aqueous solution, close to its surface or the interface with other substances, there will be a non-zero electrostatic potential, y, whose value depends on the distance x from the interface (surface). The energy EC of the conduction electrons then contains a contribution -ey, where --e is the charge of an electron, and so does the energy Ev, the energy of a valence electron. In Fig. -4 the symbol q is used for the charge of an electron. For this reason the bands near an interface or surface are, as a general rule, curved. Owing to the negative sign of the charge of an electron, if the bands are bent upwards, the electrostatic potential is bent downward and vice versa. Eqs.(S) and (10) can be written as: Ep
-
=
pe
Ep =
=
Ee
ph =
bulk
-
-
ey
+
E, bulk +
kT ln +1&s
G33)
ey +
(31)
kT h f&/1$
In the bulk of the semiconductor (x = 00) the electrostatic potential y = 0 and also d+lx = 0. It is seen that for any value of y the product n x p has the same value, given by:
Taking A, = An =c- 4 and denoting 90 = Ep - Es bulk. where El bulkis the energy vafue just helfway between Ee and E,.in the bulk, or also: where El is the Fermi energy ior intrinsic material, one obtains for >z and $:
tt = ttt erp
Qlb k2.
eyt exp kT
ni = 2/l-”
exp
-
(Ee 2kf
E.4
t-Z3 ,
Me&d
bkculor denser
‘-L=
-
Li m_m- 88
_e3
-
m
Fig. 5. Comparison between Gouy and spacecharge:a) liquid. Circles surrounding(+-)or (-) signs represent ions: b) semiconductor. Squares surrounding (+) signs represent charged donor atoms, fixed in thelattice. Open (+) signs represent holes. Open (-} signs represent conduction ektrons. The surface charge is chosen negative (bands curved downward). and this is effected by meazrs of afczptor states. The Iquid corresponds to an ‘“intrinsic’” semiconductor.
For n-type material the Fermi energy is in the upper half of the energy gap, i.e. q% > 0, and for #-type material it is in the ~owferhdf. The case qb = 0 (intrinsic conductivity) corresponds to that of l-1 ionic solutions (see Fig. 5). Since we have
POTESTIXL
DISTRIRUTIOSS.
assumed
full ionization
POTESTIALS
rf donor
ND and A*_%respectl\.el~.
these
concentrations
Flat
surfaces
2%
and
placed so as to coincide
their concentrations,
as:
being independent
and interfaces
impurities,
ilCCeptOr
can be written
c>f the electro5taric
Gil
be consldered
only
potential.
and the plane x =
with the plane of the fast surf-ace states.
0 Is
Here, the electro-
static potential is denoted b_v PO_ The energy of the electrons in the surface states thus contains a contribution --Ed+ It is assumed that the outermost layer of atoms is polarized. potential
Then,
jump.
interfaces.
just
as at the surface
the x-potential,
If there
of other,
is present.
is an oxide
layer
Similar
solid
or liquid.
potential
jumps
materials.
a
also occur at
OS in the Rardeen-~rattai11~ttaixl model,
the in-
---_ --Jr+-
T +!w
0
---XT
----$2 _-_’ ____ V&i
_ _-- -_
3i_--__ *
cj
xE
Fig. 6. Bardeen-&attain model’8. a) Semiconductor covered with an oxide layer. An electrode for measuring field effects is also shown. An electric current J II Indicated. The charge in the states is modified by the applied potential and so is the charge CT*, sf the space charge region. The modification of the value of the space charge leads to the field effect. b) Energy Itwels. e) Potentials.
fluence
of the slow
surface
should
JV,
=
state;and
~&-OX
d-
se
where xseWox and x,,~-,,~~ oxide
interface
difference
the molecular
also be considered bulk
-
condenser
eYo
+
Xox-vae -
refer to layers of polarized
and to the outer
across the oxide
at the outermost,
(see Fig. 6). The work function
oxide
surface.
IV,,
EF -i- VOX atoms
oxide,
is (27)
at the semiconductor-
respectively.
VOX is the potential
layer. -4dvaa.
Colroid
later/ace
Sci..
1 (1967)
277-333
296
M.
J. SPARNAAY,
SEMICONDUCTOR
If no oxide is present [for example because the semiconductor been thoroughly cleaned). eq.(27) reduces to: rv,, =
x + E~t,ur~-eyo-E~
SURFACES
surface has (27a)
The work function is defined as the difference between the electrostatic potential just outside the crystal and the chemical potential of the electrons inside the crystala=_ The value of IvSc will vary from face to faee of the crystal and these variations can be due to the difference of the X-potential and those of yo when going from one face to another. If two semiconductors. or a semiconductor and a metal, are electrically connected, equilibrium requires that the chemical potential of the electrons (the Fermi energy) has the same value in both materials. The difference between the two work functions of the interconnected materials is the Volta-potential or the contact potential difference, 4c.p. (see Fig. 7). This potential difference can be measured if one of the materials is set in vibration. An alternating current then flows.
Fig.
7. Potentials
plained
in text.
pertinent
to metal
surfaces
and
to semiconductor
surfaces.
Symbols
es-
A variable potential difference is usually applied to the circuit which makes this alternating current zero. The value of the variable potential difference gives the value of the 4c.p. This is the Kelvin or vibrating plate method, which is used for a wide variety of surfaces of semiconductors and metals, solids and liquids. The photoelectric threshold, vse differs from the work function in that it should be counted from the edge of the valence band (see Fig. 7j. q&xc=
W*e --
EF +
Ev surface
WI
incident on the surface penetrates in many materials Light of frequencies 2 h-+,, to a depth of several atomic diameters and leads to the excitation of valence electrons. A fraction of them leaves the crystal and is measured as the photocurrent. The photoeIectric threshold is given by 10 = h-rpse. Since the penetration depth is, in a number of cases, an order of magnitude smaller than the depth over which the bands are curved (this depth is of the order of the Debye-Htickel length in the material), the values obtained in these cases are characteristic of potential values at the surface. Advan.
Colloid Interface Sci.. 1 (1967) 277-333
THE
SPACE
CHARGE
297
In metals the difference between vae and Ws,, disappears, at least in principle. There is no energy gap and the concentration of conductivity electrons is of the order of 103z cm-a. Fowlerfa calculated the photoemission of these electrons as a function of the photon energy, the photoelectric threshold appearing as a paraThe \*alue of 2w.p. meter in this theory. “Fowler plots” provide a value of T1-urerar. between the metal and a semiconductor then gi\*es a value of Il’ser,~tCOndUCtOr. This value can be compared with the value of r-faeot thesame semiconductor surface. R-v means of eq.(ZS) a value of EF - EY lur~aee is thus obtained_ Theexperiments can be carried out on various samples of the semiconductor with different impurity concentrations. The balk x*alue of Ep - E, can be varied by varying the impurity concentration_ As pointed out. measurements of qsc and IV,, provide srlrJnce values of E,. Thorough investigations of the case of a clean Si surface have shown that EFthe srtrfuce value of E B - E, varies much less than the balk value. Thus the value of elyo varies almost as strongly as the bulk value of EF - E,. This means that the space charge in highly doped p-type material is negative, this negative space charge being compensated by excess positive charges residing in the surface states (empty donors). In highly doped tl-type material the space charge is positive and there is an excess negative charge in the surface states (occulted acceptors). of the energy distance EF Ev at the surface can Such a “stabilization” evidently take place only if the den&v of states is high. It should be large compared with the number per cmf of charge carriers in the space charge, which 1s limit of the density of states to usually less than $01” per cma. Therefore a lok%eer have the required stabilizing effect will be at least 101s per cm2. In fact, for cleaned Si surfaces in high vacuum (pressures less than 10 -9 torr), state densities (donors and acceptors) of about 101s per cmz have been reportedz’=ss. From field effect measurements even higher values have been foundl6. For metal-semiconductor interfaces Mead--5a formulated a remarkable rule. based on numerous experimental data concerning a wade x-ariety of combinations of metals and semiconductors brought into intimate contact. According to this rule the Fermi energy at the interface is stabilized in such a way that (E, - EF) =: ~(EF - EY). It should be stressed that a large density of states such as is necessary here does not imply a large density of states at the clean surface of the pertinent semiconductor. This is pointed out by Van Laar and Scheerssb for the case of (cesiated) GaAs.
(3) The space charge 3.1
GOUY
LAYER
OR
SPACE
CHARGE
LAYER
Gouy*-6 was first to give a quantitative treatment of the space charge problem. He appIied his results to the Hg-aqueous solution interface. Gouy’s theory Advaw. Colloid Znfsrfacts Sci.,
1 (1967)
277533
31. J. SPhRh’hhY,
298
SC3lICOSDt*CTOR
SURFACES
has now become standard. The space charge in ionic solutions bordering a solid, a
liquid or a vapour is now called the Gouy layer, or, less frequently,
the Gouy-
Chapman26a laver, Gouy’s theory has also been applied to the problem of the distri_ bution of ions around charged hydrophobic collo;dal particles and it has been extended to the case of two particles at a distance which is of the order of the DebyeHEckel length and less. More recently (see Fig. 5b). Gouy’s theory has been applied to space charge problems in semiconductors although it was presented there as a new theory. In both cases (semiconductors and ionic solutions) the Boltzmann eqs. (23) and (24) (and eqs. (25) and (26)) should essentially be combined wit?1 the Poisson equation: div (c grady)
=
-
42~
(29)
where 9 is the charge density at the position considered. This is written as: e =e(fi3.2
It +
SIJIPLIFYING
f\‘n -
n’,)
(39)
ASSUJIPTIOXS
1e consider the charge carriers as point charges. This is certainly a better approsimation here than for ions in an aqueous solution which are surrounded by
hydration shells_ Furthermore it may be recalled that the concentrations p. 12,f\‘D and LVX indicate average values. This makes the potential y in eqs. (23) and (24) an average potential. In contrast, the potentia? appearing in the Boltzmann equations
is the potential of the average force, grady [=(kT/lre) gradlt for electrons; = (kTf+) grad+ for holes] as pointed out long ago by Onsager”*. x+Ireidentify these potentials. In doing so, fluctuations terms are neglected, as can be shown by a statistical analysis_ The problem has been thoroughly studied for ionic solutions (ref. 28,!29). The result is that the potential calculated when neglecting thefluctuation terms approaches the zero value somewhat “too rapidly” as the distance is increased, the whole effect amounting to a few percent in the final ,numerical values of double-layer parameters. We assume the same to be true for semiconductors. Finally we recall that eqs. (23) and (24) are Boltzmann approsimations of an analysis based on Fermi statistics and that this is justified if - (EF - &) s kT and (EF - EY) 9 kT. Special numerical examples with Fermi equations instead of Boltzmann equations were worked out by Seiwatz and Greenso, who also took into account possible effects of incomplete ionization of donors and acceptors. Another
correction which may become of importance is that which arises from a possible quantization of the movements of the conductivity electrons in the space charge. This quantization may occur if the value of ~JUvaries much more than kT over an interatomic &stance, irrespective of the values of EF - Ee or Ep - E,. This interesting problem is as yet far from solvedsr. Neglecting the possible corrections enumerated here we shall use a simplified form of the Poisson equation: divgradv=-4n$
(31) Aduaa_ Colloid Interface
Sci..
1 (1967)
277-333
THE
SPACE
299
CHARGE
Thus, it is assumed that the dielectric constant E is inciependent of the field strength, This is certainly a good approximation for most, if not all, semicouductor crystals in which we are interested. For aqueous solutions, in which the influence of the field strength upon E is likely to be greater, the effect can usuaIIy be neglected=, For the flat-plate case the (Laplace) operator, div grad, becomes dZ/cW. Any plane .Y = constant, is an equipotentlal plane.. For x # 0 it can be shown that this assumption is consistant with the identification of the ax*erage potential with the potential of the average force. For x = 0 it means that the surface charge is assumed to be smeared out, i.e. that “LTiscrcteness-of-charge” effects are neglected. It is difficult to give more than a quantitative estimate of the consequences of the equipotential assumption, for one thing because it is not known how the charges on the surface are distributed. I(ryIo%~33showed that in the Stern layer the potential should decay faster to zero with increasin,= x than would be the case assuming a smeared-out sumface charge. \Ve return below to di&reteness-of-charge effects (section 6). For the present, however, we assume a smeared-out surface charge. 3.3
THE
POXSSOX-BOLTZMAXX
EftUATfOX.
THREE
APPROSI.UATIOSS
The Poisson-Boltzmann equation that fits our pcrpose is obtained by combining eq. (31) arxl eq_ (30). Eq. (30) is rewritten as follows: e=
-2
e8zi
5 ey (siair ‘Fb kT -
siuh
qb kT )
(32)
We shall use the following notation:
the Poisson-Boltzmann
equation reads:
=
sinh 2(b)
L-s(sinh(ub
+ r) -
The surface charge, compensating is given by: E dW use = 4n ( -Z-Z=O=>
(33)
the total charge per unit area of the space charge, ekT one
dy ( d.r ) xiliio
(34)
eq. (33) should be integrated_ The integraIn order to obtain an expression for a Ottr tion can be carried out by means of: _
Then the root must be taken. The positive r-f negative. Then:
is chosen if the space charge is
31. J. SPARS,4AY,
4tkTL ,
use = *
42
Kingston and Neustadter numerical function
values
[cash (zzz-,-j- Z) -
3-1 have
SURFACES
SEMICOXDUCTOR
cash zzb-
Z sihn z&j*
given ubCas a function 6f zzSand Zzband calculated
(see Fig. 8). The second
of x. will not be undertaken
integration,
here. Instead,
which
should
provide
y as a
we give three useful approuima-
Cons of eq. (35).
-16
-12
-8
-4
0
f6
Accomuiotlon layer
Fig. 8. Value of the space charge use as a function of u J and ub_ The function ZQ,bSc. can be obtained from eq. 35 and is i; = &2 [cash (ut, f =) -cash t = itstl,,
F_ proportional to z sinh ub]: hmwhere
(a) The Mot&Schottky apjmrimation 35a36. This is obtained for 1Cb 9 1 and “-type material, one 3 4 0, or for -#lb 3 1 and z 9 0. Thus, for strongly-doped has; &
bee
=
(
ND kT t [eL > 2Jz
,I -
114
For large negative z-values. cr,c is proportional to lzli. Thus we have here the case of an n-type semiconductor with the bands bent up quite strongly. It should
be noted
that
a Debye-Hiickel
length
can be defined
here by using ND
instead of ~1. (b) Both tib mrd E me smatC, positive OYnegalive. Then: Qac =
e 4nt
(37)
w0
and zzbhas altogether disappeared i.e. only a term containing #b’ would have been Advaw
Colioid I~~terfuce Sci.. I (1967) 277-333
THE
301
ChPACITASCE
present_ Eq. (37) is the same as that obtained ersf?a .C_
For jq >
for small potentials in Gouy lay-
1, this simplifies to:
where ose > 0 for z > 0 and czc ( 0 for J -=z 0. These three approximations fill prove to be ver>* useful in the following sections, where the differential capacity, 3a/3ye, and the free energy of the double layer, - Jce adye. are considered. *
(4) The capacitance 1.1
C.4PACITASCI:
EQL’rlTIONS.
ORDEIPS
OF
JIAGSZTUIJE
Capacitance equations xvi11 be given here for a semiconductor-aq -soln. interface rather than for a semiconductor surface, capacitance measurements m the latter being relatively mnccurate. Knowledge concerning dr_v surfaces has mainly been obtained by other methods (see previous sections). Lvhereas knowledge of semiconductor-aq.-soin. intertaces is to a fairly (see, however, ref. 63) large extent due to capacitance measurements. The capacitance of such an interface can usually (prox4ding electrical leakclgc across the interface is not preponderant) be measured if it is made part of a cell in which, apart from the semiconductor electrode, the other electrode has ;L large surface area, thus making the capacitance of this electrode large. A simple model of the interface will be considered, in which it is assumed that the charge at the semiconductor side of the interface can be written as 0 =
orlc +
CSs +
oion
(-10)
where otscis the contribution of the space charge, asa that of the surface states and where oren is the contribution of an ionized layer which arises from the chemical interaction between water and the semiconductor. For the (anodized) Ge-aq.-sohl. interface this layer is composed of ionized HeGeO,+l groups. Here _1cis a number which is 2 if the germanic acid is an independent molecule. However. we consider a germanic acid group which is still chemically bound to the Ge crystal and it is likely that x is smaller than 2. It is possible that this “ionized la_ver” can be identified with a layer of “slow states”, but, since little is known of the physical nature of slow states. we prefer in our simple model to use the concept of an ionized layer. ~dvan.
Colloid
lntcrface
Sci., 1 (1967) 277-333
bf. J. SPARNAAY,
302
SEMICONDUCTOR
SURX’ACES
At the sofution side of the interface the charge o is compensated by the charge of a Gouy layer and a charge in the Stern layer, which is formed by specifically adsorbed ions. The potential drop Vtot across the interface will be written as: ytot =
yo +
g +
yso1n
(41)
where T,UO has the same meaning as before, x is the potential drop across the molecular condenser at the interface and vaoln is the potential drop across the solution (Stem layer and Gouy layer)_ Following the same method as that indicated by Dewald37, the capacitance C is written as (42) where, assuming that 2 remains constant if ytot is varied, WY0
=
I--
‘;)ytot
ayso1n
(43)
bytot
The foltowing differential capacitances are defined:
and
Csoln =
aa dqJso1 II
Tt is not difficult to obtain the following expression for the capacitance: c-1
=
C&n-- 1 +
(CM2 5
C&1
(1 -
CionCsoln-1)
Orders of magnitude are now given. From capacitance measurements of the Hg-aq.-soIn. and the AgI-aq.-soln. interfaeess’a, it has been found that C Boln is at least of the order of 10 FF cm-‘, depending on the electrolyte concentration_ For C=+ theoretical curves have been given by Bohnenkamp and Engell 19. These are reproduced in Fig. 9. The curves show minima just as do the capacitance-potential curves of the Gouy layer. The numerical value of the capacitance C sc in Fig. 9 is of the order of 0-l FF cm-l. It shouId be emphasized that this is a much lower value than that of Csoln. Indeed in many cases one has: CfJc% Csoln
(45)
Concerning C Bc, consider the three approximations given for ose at the end of the previous section. a) The Molt-Sclwttky zzpproximation. Differentiation of eq. (36) with respect to ‘190and taking z large negative, gives:
Advan.
ColIoid Interface
Sci., 1 (1967)
277333
THE CAPACITANCE
’
Fig. 9 Theoretical capacitance-potentid curve for Ge-soln. sidered. From ref 38 (Bohnenkamp and Engelt).
b) c
Low 2x?=
irx@rit_y & 4xL
cowetrtration
interfaces.
arrd low ~ote-nfial.
303
So surface states con-
One snnpl~ has
which is the same equation as that obtained for the low-potential approximation of the Gouy capacity. Only the Debye-Htickel length x-1 is replaced by its equivalent L for semiconductors. Eq. (47) in particular illustrates why the inequality (43) is -not surprising: in many instances L S z-1. c) Low iwq5zfrity coxcedration and high ~0tehkI. From eq. (39). one has:
c
SC
=
-
e Qsc
2kT
Concerning surface states, let us consider one set of donor states and one set of acceptor states. From eqs. (18) and (19) and differentiation with respect to ~0, one obtains:
c
ES
=
- tT
where .%A
=
2 exp.
zD
=
2-l
t
e
r*.xa
(1 + .ZAJ2
(491
1
exp
kT and EA and ED are the energies of the state at yo = 0, Plots of Css against ~0 should show maxima at eye = EA Advas
Culioid
EF -j- kT
ln 2 _
Interface Sci.. 1 (1967) 277-333
51. J. SPARSBAY,
SEMICO?r’DUCTOR
SURFACES
andateyo= ED-EE~kT In 2. For very small x and very large x, Boltzmann approximations are valid. On the whole, CSSis of the order of 0.1 PI? cm-l, if there are about 1010 states/cm”. Its value is of the same order of magnitude as Csoln if the surface-state density is about lOl%m-2’. Concerning the contribution of the ionized layer, Iet us consider the case of weak ionization. As derived by Payens 3% for the case of weakly-ionized monolayers at the wnter*il or water-air interface. the surface charge oion is proportional to esp eyttioln/kT. Then:
c
e iOR
=
k?’
uion
As we shall see in section 5, the case of adsorbed gases, which are weakly ionized, is similar to the case considered by Payens;. In fact. the cases of weakly-ionized monolayers, weakly-ionized groups chemically bound at the surface, weakly-ionized gases on the surface and charged surface states have a common theoretical b&as&. In view ot eq_ (14). it is useful to compare values of Cion with those of Csoln. If crion is providecl by IO’“- charged groups/cm”, the same order of magnitude is obtained for Cion and Csoln_ IVe conclude that the measured capacitance C is often determined by the value of CS, or of (Cse + C,,). Finally we note that the contribution Csoln may be analysed in a way analogous to that followed by Dewald. Thus, if it is assumed that the charge u of a double-layer system is determined by the charge in the Gouy layer only. and if it is furthermore assumed that the thermal movements can bring the ions to a distance from the charged wall, which is not shorter than a certain value 6. then C-1 can be written a@: cion-1
=
3WbOln
=7i
30
3 (Y8oln aa
3w
W)
=
co-1
+
en-r
(51)
where yb is the potential difference between that of the plane -2:= 6 and that of the bulk of the solution (x = co). Co is the Gouy capacity and CH is the Helmholtz capacity. This is often written as .&&~a where EII is the dielectric constant in the region between the planes n = 0 and x = 6. Eq. (51) corresponds to eq. (44) in the absence of surface states and ignoring the contribution Clan_ If now. apart from the Gouy layer, specifically adsorbed ions (charge da,-&per cm=) are present. i.e. if a Stem layer is present, eq. (51) should be modified. The charge in the Gouy layer is now no longer determined by Q, but by (0 -
dads)-
Then instead of Co-l
Wd = 3u
one has:
3 (a -
3yd 3 (0 -
uads)
-
'hds)
3a
-
cc-1
(52)
It is assumed here that the specific adsorption has taken place at the plane x = 8. This is a simplification. Often the adsorption takes place between x sz -&a and x z b (see ref. 39a, and b). Advan. Colloid InferfaceSci..I (1967) 277-333
303
THE CAPACITASCE 4.2
ESPERIMESTS
3feasurements
Bohnenkamp current
of the capacitance
and Engell IQ. Their
density
being
of Ge electrodes were first carried out by
electrodes
10 p_?(Lcm-* - or
were
alightly
anodized,
the
anodic
more_ They obtained curves which to a large
extent could be explained by space charge theory aione (Fig. 10). Thus, the curse
7
600
Fig. f0. Experimental capacitance-poteatiatl curves fur Ge-soln. interface. Ge-electrode anodized_ Solution, 1 X KOH. Eh potential referring to hydrogen electrode. Frequency range indicated. From ref. 19 (Bohnenkamp and Engell).
capacitance versus applied potential was found to have a minimum. The capacitance minimum was of the right order of magnitude to be expected from space charge theory. This indicates that under these circumstances surface states were absent. They also found a frequency effect at frequencies ranging from 10y cycles/ set to lo6 cycles/set. This may be ascribed to a decreased participation of the ionized layer in the charge transfer at increased frequency. This would lead to an the time effects apparent decrease of Cr,, at increased frequencies. Alternatively may be ascribed to time effects of the X-potential. It does not seem probable that surface roughness plays a role here- Frequency effects due to surface roughness can be expected to occur at frequencies between zero and 10s cyclesjsec as pointed out by De Levie40. If, prior to the measurements,
the Ge electrode is cathodized, Adva~~ Celloid Interface
rather different
Sci., 1 (1967) 277-333
306
3%. J.
SPAKNAAY,
SEbIICOKDUCTOR
SURFACES
capacitance values are found. Thus, Efimov and ErusaIimchik4r, applying a cathodic current density of 3 mA cm-e, found capacitances (in acid solutions) of about 10~ F cm-s, the capacitance-potential curve showing a shallow minimum accompanied by “shouIdcrs” (see Fig. 11). Here surface states will be present, their
60-
&I--
20 -
Fig. Il. Experimental capacitance-potential curves for Ge-soln. interf;sces. and Erusalimchtk). dized. Freq. 200 cycIes/sec From ref. 4 1 (Efimov
densitybeingof
the order of
Ge-eiectrode
catho-
The frequency used was low (200 cycles/set). Therefore surface roughness and electrical leakage might h;we played a role here. (A method for obtaining “state-free” interfaces by means of a cathodic treatment, was recentl+a reported). At the same pH value, the position of the capacitance _ minimum in the “cathodized” case is about 500 rnv more negative than in the “anodized” case. These two cases may be seen as two extremes~r. This is in agreement 44th the findings of &attain and Roddyao, who measured capacitances as a function-of the time elapsed after switching off the anodic current, applied during 1011 cm-z.
the measurements. It appeared that the potential of the capacitance minimum shifted about 200 mV in the negative direction. This may be interpreted as a shift of the X-potential combined with a change of the charge of the ionized layer and/or a change in the charge of a Stern layer. These effects are probably also related to the crystallographic orientation at the interface, because different results were found for different crystal planes of the Ge electrodes. The low-frequency objection was raised by Krotova and Pleskov4s against the measurements of Boddy and Brattain. However, the latter authors found little differencedzb between data obtained bv d-c. and by low-frequency a.c. measurements, at least for neutral 10-r M &SO4 solutions. Krotova and Pleskov, working with 10-Z M NaOH solutions, reported the existence of 1012 surface states per cma. Therefore the disagreement must probably be attributed to different densities of surface states. Advun.
Colioid
Intevface Sci., I (1967) 2774333
THE
307
CAPACITAXCE
At extreme pH values, fast states are rapidly formed and, as reported by Bohnenkamp and Engell, a more or less state-free interface can be maintained only for a few minutes after switching off the anodic current. If, however, Hz02 is added to the solution, the interface is stabilized. This was found by Brouweres. A full interpretation of Brouwer’s results, which is at variance with the interpretation in refs. 19, 42 and 46, will be published. The surface states formed upon cathodic polarization may be due to germanium hydride complexes, or to adsorbed cations. Thus, BoddJ* and Brattain”” concluded from the increased capacitance x-alues after Cu’+ addition to the solution that surfac= states due to coPper are formed. State formation at low pH values can probably not be explained along these lines, because at low pH, eopper is desorbed. Ions of less-noble metals are desorbed even at higher pH values. Other methods, such as the measurement of surface recombination velocity (Pieskov rt 11.458, Ham-ey45b) are considered in reviews by Cerischerl9 and. most recently, by BoddylG. The role of the pH, which seems to be more related to that played in colloid chemistry, mil! be discussed in section 6. For the ZnO-aq.-soln. (pH = 8.5) interface. Dewald found from capacitance
Fig. 12. (Capacrtance)-2 plotted against potential for %&borate buffer soln. (pi-f, 8 5). Straight lines indicate validity of Moot+Schottky approach in this region. From ref. 37 (Dewald), Rdvna.
Colloid Inter$zrc Sci..
I (1967)
277-333
308
M. j. SPARNAA’S-,
measurements
that
in carefully
purified
surface states were almost completely
borate-buffered
SEMICONDUCTOR
1 X
KC1
SURFACES
solutions
fast
lacking, their density being less than ID* per
cm*. He used ZnO crystals with impurity concentrations leading to bulk electron densities up to 101* per cm 3. At this density an unexplained anomaly occurred. Apart from this anomaly, ZnO is particularly suitable for these measurements, because the impurity content can readily be varied and, since the energy gap is about 3 V, a wifie range of potential variations can be covered by the experiments. approxiThus, Dewafd found that over a range of nearly 3 V the Mott-Schottky mation applied (see Fig. 12) _ Measurements were also carriedout in the region where the Poisson-Bultzmann erjuation should be replaced by a Poisson-Fermi Here also gcod agreement with space charge theory was obtained.
(5)
The integral J**~dy+
3.1
ADSORPTICIS
AKD
equation.
Adsorption equations DOUBLE-LAYER
PARAMETERS
The adsorption of neutral substances may affect the value of the parameters
of the double layers at the interfaces
or surfaces where the adsorption
has taken
place. In principle it is immaterial whether these substances +a.reionized zx nonionized prior to the adsorption. 1Ve illustrate this statement by two esamples which can be considered as estremes and by a third example \vhich is intermediate. Firstly, in the case of a colloidal solution of Agl particles, the addition of a completely ionized sotution of Agh’Os results in the adsorption of Ag+ ions, which makes the charge at the AgI-aq,-soln. interface more positive and the surrounding solution more negative. Secondly, the addition of the completely non-ionized system often leads to charge transfer, the a+ oxygen z;“= to a semiconductor-gas sorbed oxygen being negative and the semiconductor becoming positive. A third example may be added: the addition of a partly ionized long-chain fatty acid HA to an oil-aqueous
solution interface leads to the adsorption
of non-ionized
mole-
cules HA together with ions A- at the interface. The degree of ionization at the interface usually differs from that of the non-adsorbed acid. This example may therefore ‘be considered as intermediate between the other two. These three systems: (u) the AgI-aq.-soln. system, (b) the oil-HA-aq.-soln. system, (c) will often adsorbing interface,
the semiconductor-& gas system, serve to illustrate the considerations presented in this section. -Other substances may only indirectly affect the charge distribution at the for example because they have a special dipole orientation. These will not
be considered.
Knowledge of the adsorption isotherm of ionizing agents should yield inA&an,
Gdt5hi
fnlerfacc Sci..
1 (1967)
277-333
ADSORPTIOS
309
EQUATIOSS
formation concerning the charge redistribution at the interface. Such isotherms. are considered here from a thermotogether with appropriate equations of state, dynamic point of view. 5.2
THERMODYSXMLC
FUSCTIOSS
An electrically-neutral system will be considered which consists of two bulk phases and an interfacial phase. each phase being electrically neutral (see, however, subsection 5.3). The reason for the introctuction of a separate interfacial phase is our use of the interfacial tension as an independent variable. The int rfacial tension is a characteristic of the interracial phase. Our treatment bears a close resemblance to the “first method” of \‘erweyp7d and Overbeek (V.O.)“e.d and to the method of Frenkel 4Qfor the determinatirx~ of the free energy of the double layer. This 141 be discussed in section 5 3. Charge transfer from one phase to another is assumed to be reversible. Thus we preclucle here a system containing an ideally polnrizahle interface such as the Hg-aq.-soln. system. \Ve follo\v Gibb5a8, who detined the interfacial phn+e by a subtraction procedure. First the two bulk phasea were taken to be homogeneous right up to a hypothetical dividing surface. Gibbs showed that the choice of the location of this surface is allowed to be arbitrary, at least for planar interface-. Second, the phys~cat situation, Le. the heterogeneity of the interfac e, was considered. The thermodynamic quantities were then defined by subtraction. The cluantlties pertaining to the interfacial phase are then excess quantities. The locatiw of the Gibbs dividing surface is usually chosen such :LS to make the excess mash of one ot the main constituents a bulk phase) zero. The Gibbs function of the interface is: G” =
G -
(G1 +
where the superscripts
Gl[)
(53)
s, I and II pertain
to the interfacial
The quantity without phases, respectively. whole. Similarly one has for the Helmholtz Fs =
F -
(P
+
(of
superscript
phase and the two bulk
pertains
to the system as a
fret: energy:
PI)
(34
For bulk phases the relation between the f_;ibbs function and the Helmholtz free energy is given by: Jx =
Cl _
plYI;
FIX =
GII _
&l/II
(55)
where p is the pressure upon the system, and Vi and I/lx are the volumes. Only if the interface is curved are the pressures in phases I and II different. Forinterfacial phases one has: I;” =cP$_yA
(56)
where y is the interfacial tension and A the interfacial area. Thus the Helmholtz free energy of the whole system can be uiritten as: Advan.
Colloid
Xntcrfaca
Sci.,
1 (1967)
277433
.M. J.
310 F =
G
SPARNAAY,
SEBIICOSDUCTOR
SURFACES
--_pV3-~A
(57)
where V = Vr + VII_ The interfacial phase in Gibbs’ sense has no volume of its own. Assume now that at a certain composition of the system the double layer potential is zero, and that the reversible addition of a neutral substance leads to a non-zero value ps. For each of these two compositions of the s_vstem an equation similar to eq. (57) can be written down. Subtracting
AF=
these two equations
gives:
AyA
AG-AapV+
(5%
where AF is the difference between two Helmholtz free energies, to be denoted as F - Fo, and similarly for the other quantities. In pnrticular, the difference AC can be written as: 4 G =
G -
where ,ur and @I),,
G, =
C plttrl -
r: (&&z~)~
are the chemical
potentials
of the neutral components
(59) i, pai
and (PM&, being the number of molecules of these components in the system. Thermodynamic equilibrium requires a uniform value of pi in each phase and, if reversible transport is possible between two phases, ~1 should have the same value in these two phases. Characteristics of the double layer do not appear in the sum X ~_LS $>rrand C @t)O(ll~t)O. Thus, considering by way of example the double layer at the AgI-aq.soln. interface, we can vary the potential ye by the addition of the neutral salt AgNOs or of KT. Tf the only change taking place in the system is the addition of KT, then more I- ions are adsorbed at the interface, pe becomes more strongIy negative, and
A G = ~~~ IHKI-
(,uKI)o
(j)tKI)o
+
+ksttk
-
(~k)o(%c)o
VW
collects the contributions of the uncharged comwhere &.&?1rk - (&o(9z& ponents k which arise as a consequence of the Gibbs-Duhem rule. In the case of dilute solutions these contributions are small. It is not necessary here to consider the I<+ ions and the I- ions as separate components. SimilarIy, for a double layer consisting of a fatty acid HA one has:
AG
= pHAf?bIi;5 -
@HA>CN
(nC~h)*
+
+km)k
-
(pk)&kb
(61)
Addition of HA means an increase of the adsorption of both HA molecules and Aions, but this aspect does not appear in eq. (61). Thirdly, consider the adsorption of Oe on a solid surface where it becomes partly ionized and gives adsorbed Os- ions, the electrons necessary for this ionization being extracted from the rurderlying solid, a semiconductor_ Then:
AG
= PO, mos&-
bOz)o
(ntOp)o
+
&&mk
-
(pk)o(mk)o
(69
Again the role of the charged component (the Os- ions) does not explicitly enter in the expression of AG. However, the charge of the dotible layer dots appear explicitly if expressions A&m.
Cdoid
Interfucc
Sci.,
1 (1967)
277-333
for A0 {the change of the Gibbs function pertaining to the interfacial phase) are written down. In order to gix*e such expressions, it is necessary to reconsider the chemicaf potentiaf of charged components, i.e. of ions, e1ectrous and holes. For charged components, adsorbed of non-adsorbed. it is cu?ztomar\r to split up the chemical potential into a “~xifel~- chemicat” gxu+tam1 a “purely electrostatic” part. This separation has met with objections -1*ahut, at least to ILtirst approximation@, it is justifiable. The chemicni potential of a charged component j is called etectrochemical potential and denoted by t/J. IFof charged Ixwticles in a space-charge region where the pntentiai is; y, ant? k~s:
space-charge-free region, in the bulk of phase I, sny, the electrochemical potential reduces to the chemical potential. Since grarly = 0, the potential is a constant. This constant will be taken zero. Then: fn
a
*7i =
P3
W
A third espression for *]j is important, namely if the component j is pfesent at the interface and participates in the formation of the interfacial charge cT. Assuming that the adsorption has taken place at n plane where the potentla: is y*o -+- g we
hZX\Ve: “ s.” inrlieates tkkt the plfdy chemiczll paft Of #)J iS a CflafaCwhere the superscript teristic of the interface (surface). Often the z-potential isaassunled to he independent of the charged state of the double tayer. Then, for t+ = 0, we write: (?]Jf o $iow
=
(pJ’)o
in A@ X&J x&J’
them
t
we distinguish VI&f -
three
contributions.
Firstly:
(qr)o (~~~jShJ=
them figJs
-
@Js)o
This is the contribution provide
=Jt%
the interfacial
ehem
(‘jzJ”)o
+
airy0
ii]
=
h (;J"
due to charge carriers adsorbed
charge
at the interface:
(67) they
cr.
The second contribution is provided by charge carriers which must necessatily be present to compensate the interfacial charge (T. Their total charge nmst be rrd_ Labelling these charge carriers with the subscript *‘a*’ one has for the second contribution:
Advati. Colfoid Ink&ace Sci.. 1 (1967) 277-333
X. J.
312
SPARNAAY,
SEMICONDUCTOR
The third contribution is provided by neutral components. by the subscript “b” one has: ~&bmtf
-
(@&&W)d
=
il
SURFACES
Labelling
Gbs
them
w9
AGE one has A0 = AGf + AGk + AG# Concerning Gj *. this can be written as AGja = AGjs ctwm t AGes For
(71) (72)
where AGl’
ehem
=
C&f
cbem
fjtiB -
@f)o
ehem
(73)
(f@)oI
and AGe= =
b?#!Po A
(74)
Concerning the sum AG,a + AC b8, it has been shown that the ionic components can always be arranged so that the charge -u is provided by only one charged component, the rearrangement being such that further only nexhxd (salt or otherwise) components remain. This rearrangement leads to AGas +
hGb* =
,&~=a8 +
T;‘&b~=b’
-
(75)
(,llb)&b’)oJ.
where the prime on the summation sign indicates that the summation is not identical with that in eq. (69). The term yrrp~t a* can be written as: aA (+e)-rp8. Eq. (71) can now be rewritten as AG* =
CTA(PO i-
(&e)-’
pa) 3-
x’@b
?*tbs-
(cLb)&bs)ol
$_
&GJ*
ebem
(76)
Thus AGs contains a term which is characteristic of the double layer. For the A@--aq_-soIn_ interface the interfacial charge is: 0 =
e(I-&+
-
I‘I-)
(771
where we have used the notation 1” = A --ItztS.If the double layer is formed through the addition of AgNOs, then I’ dg+ > 0 and rr-- < 0. The compensating charge M is partly, not completely (see ref. 27~1, given by nitrate ions, and in eq. (76) nitrate ions can be chosen as component “a”. For the oil-HA-aq.-soln. system we have: d =
er.&-
(755)
Non-ionized molecules HA will also be adsorbed. H+ ions wiil contribute to the compensating charge --o-. and the H+ ions can be chosen as component “a”. For the semiconductor-& system we have: a=-Zers
a-
(79)
Non-ionized 0% molecules will a&o be adsorbed_ The compensating charge is here given by holes, and as p, we use here minus the Fermi energy. We now want to find au expression for AF. This can be done in a number of ways. We prefer here to discuss first expressions of AhyA (or of Ay, because the interfacial area A can often be kept constant when a double layer is formed). A&a*.
Colloid I&++zc
Sei.,f (1067) 277-333
ADSORPTIOZ:
313
EQUATIOSS
The reason is that, according to eq. (56) or eq. (57), the main difference between the Helmholtl: free energy and the Gibbs function is provided by a term containing the interfacial tension. This is an experimentally accessible quantity. Jloreover. the interfacial tension can be used to discuss equations of state of adsorbed particles;. For the interfacial tension we have the Gibbs adsorption equation, which is written here in the following form dy =
-
a (dly, -j- e,-1 d&
--
27’rb clp~b
(60)
using the adapted notation of eq. (76). The Gibbs adsorption equation can easily be cl-.-rived if it is remembered that the differential dGs can be written in two ways. \!‘riting GS = X piilfi’, where the subscript “i” now denotes ionized or non-ionized components, we have dCS = and also dCs =
Cpi dtJi*’ +
SSdT Combination leads to dy=-X1‘idpr
Zrllis dpi Ady
+
SpidntiS
’
where it is assumed that the temperature is constant (d?‘ = 0). Eq. {SO) is a specific form of the Gibbs adborption equation, which is derived with the aid of eq. (76). In eq. (SO) the differential dp, can often be taken a5 zero. Thus, in 2he Aglaq.-soln. case the addition of A@&, necessary for building up a positix*e interfacial charge, often leads to only a small increase of the nitrate ion concentration. Similarly, if I<1 is added. a considerable negative interfacial charge may be formed but the relative increase of the potassium ion concentration may be small Concerning the oil-HA-aq.-soln. system, the fatty acid HA can often be added wrthout appreciably altering the pH value. In the semiconductor-02 case, adsorption of O+ and the subsequent ionization usually occurs at a constant Fermi energy. fntegration of eq. (SO) and combination with eq. (58) gives
AF =
AG -
API’ -
A (J,“~ adyo +
,fradpa
+
\3’j&d&
(61)
where AC is given by eq. (59). Thus the Helmholtz free energy of the whole s_vsle#jc contains the contribution ---A J? adtyu. Apart from the integral j&dpa. which, as we have seen, can often be neglected, this is the only contribution explicitly pertaining to the double layer. It is called the free energy of the double layer27 and will be denoted AAF,.
Advan.
CoLIoid
Interface
Sci.. 1 (1967) 277-333
M. J. SPARNAAY, SEMICOXDUCTORSURPACES
314 5.3 COMPARISON
WITHTHE“FIRST
METHOD”•PVERWEY
ANDOVERBEER
In their “first method” to obtain an expression for AF Verwey and Overbeek (V-0.) imagined the interface brought into contact with an infinitely large amount of the solution, ionic equilibrium at the interface not yet being established*. Thus, considering the example of the AgI-aq.-soln. interface (this example was also used by V.O.) it was assumed that initially the I--ion concentration was “too high”; the interface should be negatively charged, but this charge had not yet “arrived”. Owing to the high I--ion concentration this arrival set in spontaneously and resulted in a free-energy gain per unit area of --crye. Since during this process the (negatix-e) kterfacial charge WZLGgraduall_v built up. electric work WZLSneeded to bring more I--ions toward the interface and it was conclucled that this amount of work was + jzytedo. This quantrty was called the electrical part of the free energy (per unit area) of the double layer. The total free energy per unit area of the doubIe layer was then found by addition. It is readily seen that the expression --ova has the character of a Gibbs function (see eq. (7-l)). The integral -O~we adyo is part of the Helmholtz free energy per unit area of the whole system and the integral +,fuo lyedo is part of the Helmhoitz free energy Fs of the interfacial phase. To see this, eq. (56) can be used: FB =
0
-
y-4
(56)
The role of the compensating charge (-0-A) is a pas&-e one in the “lirbt method”. This charge would only be subject to a “secondary rearrangement” and would “not contribute to the free energy of the double layer”, its role only being _ implicit in the value of AFe. However, the role of the compensating charge is explicitly given by the integral IX’* dpu,. As pointed out in subsection 5.2, its value is often very small. It is even exactly zero in the “first method”, because no salt containing ions “a’* was assumed to be added. I?renkel’s method*9 is quite similar to the “first method”, including the passivity of the role of the compensating charge. 5.1
EXPRESSIONS
FOR
,~%rd~‘,_
LTS RELATIVE
JIdCSITUDE
A useful expression of the integral el weadve can be obtained if the model of Fig. Sa is assumed. We put dps = 0 and identify this integral with AF,. There is a charge G located on a plane where the potential is lye_ At a distance 6 from this plane, a charge ~1 is located and the potential is lyb. Enally, a space charge rrscis present. We have the condition: * This is a disadvantage equilibrmm throughout
of the first method. the
whole
The treatment
in 5.2 presupposes
thermodynamic
process.
Advan.
CMoid
Interface
Sci.,
1 (1967)
277-333
ADSORPTIOS
u +-
a1
This model charge
-f-
USC
applies
ions;
tor-oxide
(W
to a semiconductor at the outer
to Irhe metal-electrolyte a space
charge
charge
to
may also be mentioned_
are
ade,
b leading present.
extreme
6) where the
slow states
interfaces
to a charge
ions of small
It also
where a layer of
~1, and at Gouy
Surfactants.
partly
ste,
layer,
or completely by the surfac-
and use by the GOUJ
cases can be distinguishtxl:
7. ‘fi~ere is w
space cfaarp
fusr =
0). Therefme
a-j-al-0 The
whole
double
(W charge
layer 4E‘, 2.
that
compensates
is a flat condenser
=
-
~0-l
ase =
(I iies in a plane
with 7yd =
(W
3.1 (we assume
be given
that yto =
for the three approsimattnns
in these approximations. and FrenkeP.
Mott-Schottky
a~jroximatiorr.
using eq_ (3G)
This is in accordance
\Ve consicler a strongly
for the space
dealt
ylOf. Jtrst as in eq. (SJ), AFe turns out
has been said by V.O.“?C
values,
chmge asc. Then (83
to be a negatix*e quantity
By
the
0
with in subsection
2a.
5. Then
0. Then
cow~pzmatcd b,~ cr s&m
The value of AZ;e can (in case 2) easily
conductor.
at a distance
2xbas
Tlrc cltau~r: o is mfird_s
a +
of
or
at the semiconcluc-
the semiconductor.
Here the surface charge a iriprovided
the charge CTIby specificail~-adsorbed Two
fast states
inside
or the AgI-electrolyte
ionized. layer.
of charged
consists
by charged
ions at a distance
leading taut,
and
and asc? is the space
specifically-ads;orbed
oxide layer (thickness
with an
surface
the charge aJtis provided
interface,
applies
0
=
G is situated
adsorbed
313
EQuAT~oss
charge
doped
and assuming
M ith what r:-type
large
semi-
negative
z-
one has: AF,.
=
(3E _hrlr,e)-’
,3b. Low
@et&X,
&a’
(0 negative)*
Ln.v imptwity
roatetzt
(W fttt,
13~ using
smnltj.
eq_ (37)
one
obtains: AFe! = Zc.
-
~-1 27t L a2
Large absolrite
that the charge
value
a is formed
(67) of the Poterttial
by charge
carriers
with a charge
\Ve assume
cohmt.
yro, tow im@rify
-e
each.
After
using
eq. (39) one has: AFe The Computer. and
=
2kT
e-l
curves
of Fig.
=
-
13 cover
be derived
x-1
(a negative)*
+Stb)b
For an electrolyte
(88). can both AFe
(1 +
qkT(cosh
(W
the cases 2a. 2b and 2c and were obtained
solution from f-z -
(rib =
0) bordering
the well-known
a charge4
wall,
by a
eqs. (87)
expression:
1)
(89)
* For u positive. a minus sign shoutd be added. Advan.
Co&&i
Interface Sci.. I (1867)
277-333
Xl. J.
316
SEIIICO?r’DUCTOR SURFACES
SPARNAAY,
This expression can be obtained by putting iCb = 0 in eq. (35), then using eq. (85) and integrating_
Fig. 13. The integral -
values of ub = &ky‘.
0j “0 bdy,, in units I”lim-lkT (or Snf_.kT), pbtted against rpofor various PYOC. IIId I&enr. Cotrgr. Surface Acftvit~. Cologne. From Sparnaay.
7960, X’ol. II. p_ 232. Only for ub = 0 (intrinsic case) are the curves symmetrical around the vertical line yJo= 0. This can be inferred from Fig. 7. A sign reversal of both ub and 9’0 provides the scl?sc safue of the integral.
The actual situation in a double layer system will usuall_v lie in between the extreme two cases with 01 = 0 and a,, = 0. Verwey and Overbeek ha\-e given estimates of the ratio of a1 to crSefor various values of a specific adsorption
energy, the free-ion concentration,
and values of
po and yb (Verwey and Overbeek, p. 41). These will not be discussed here. In order to consider the behaviour of the adsorbed particles. the chemical parts of the Gibbs adsorption equations must also be discussed. \Ve shall assume in what follows that these chemical parts follow “ideal” behaviour, i.e. we assume that we can write dpp = kT d fn rf. Also, adsorbed neutrai particles are assumed to exhibit ideal behaviour. Finally we assume dps = 0 in all cases to be considered. Then Ar can be written as AY =
-
kT I;[r,
-
(rj)o]
+- Ape -
kT c’
If--& -
(f’&f
(90)
If the adsorbed species is completely ionized, AF, usually dominates the other contributions. This is notably the case for the [email protected]. system. For incomplete ionization the situation is complicated. The correlations between eq. (90) zmd the pertinent adsorption isotherms will be discussed in subsection 55 5.5
EQUATIOrJS
An equation
OF STATE.
ADSORPTSON
ISOTWERMS
of state relates the surface pressure or the interfacial pressure
(both of which we denote by n) to the amount adsorbed per unit area of a certain component under consideration. This component may be ionized or non-ionized. d dVUM_ co&&
rntcrfatG sci..
1 (1967)
2774333
Equations
of state often provide a useful ph_vsicaI insight into the beha\*iour of the
adsorbed component_ In this paper we are naturally interested in the way which the charge of adsorbed particles is reflected in the equation of state. The introduction of z arises from a certain interpretation of the Gibbs adsorption equation; the decrease of y, which is the result of the adsorption of a considered component, is equivalent to an increase of -7. The classical example of the spreading pressure measured in a Langmuir trough immediately* provides evidence for this interpretation_ Thus dy =
-
(+31)
An adsorption isotherm relates the adsorbed amount of a certain (charged or neutral) component to the equilibrium pressure of the same component. In manJ relevant cases the chemical potential of such a component is proportronal to In c or In p (c = concentration, fi = pressure). In equihbrium the chemical potential of the adsorbed particles should be equal to the chemical potential of these particles in a bulk phase: pLB(1‘) =
ph (c or p)
(92)
where r is the adsorbed amount per unit area. I~nowleclge of the adsorption isotherm leads to an expression for pS(r)_ This expression can be inserted in the Gibbs adsorption equation, and finafly eq. (91) provides (after integration) an equation of state. This sequence can also be followed in relersecl order. Then a proposed equation of state predicts a certain adaorptiorr isotherm. Unfortunately this procedure can rarely be carried out without ambiguities. Thus, one adsorption isotherm can often be correlated to more than one equation of stateSo. Furthermore, specifying the adsorption of ionized components. the complication arises that we can add only neutral components to the system (XgSOa or I<1 in the XgI-aq.-soln. case; the fatty acid HA in the oil-HA-aq.-soln. case; 02 in the semiconductor-02 case). Therefore eq. (92) is an oversimptification: the relation between adsorption isotherm and equation of state requires knowledge of such quantities as the degree of ionization and its dependence upon the \-alue of ~0. For these reasons we only deal with some extreme cases. \Ve first write down expressions of four well-known equations of state for the case of a one-component adsorption, disregarding for the moment the influence of the charge. First we have: n=
rkT
(93)
This is the “ideal”
equation of state, r particles being adsorbed per unit awe. Here
the chemical potential PS (r) p” (r)
=
*u*e (P)
is
-j- kT In P/P
(94
where ~c”~(I’e) is a standard value of the chemical potential at a given temperature T and a given adsorption P. izing parameter, namely P,
There should be no confusion between this standard-
and the adsorption at yo = 0, which will be denoted Rdvun.
Colloid
Interface
Ssi.,
1 (1967) 277-333
318
hf. J. SPARNAAY,
(In our examples we will usually take (r), the chemical potential can often be written as:
by (I’),.
pb(c) =
ale
+
SEMICOXDUCTOR
SURFACES
= 0). The bulk expression
kT In c[cO
of (95)
which is analogous to eq. (91) except for the use of bulk quantities (c and co can be replaced by p and PO as the case may be). After applying eq. (92) it is seen that ris a linear function of c. Thus the adsorption isotherm is a straight line. This is called a Henry isotherm. The slope of the straight line is determined by A$’
=
r”e( P)
-
PbO(~“)
(96)
which is the (free) energy of the adsorption. The second equation of state is the Van der Itraals equation (,Tt+
(1 -
aP)
bI’) =
I-XT
(97)
where a is a constant accounting for the intermolecuIar interaction between pairs of adsorbed particles, (u > 0 indicates attraction, Q ( 0 indicates repulsion) and b is a constant accounting for the excluded area due to the size of the particles. Eq. (97), rewritten in virial form up to the second virial coefficient, gives n =
I-‘kT +
(b -
n/kT)
PkT
(99)
The second virial coefficient (b - a/kT) gives only a correction for the adsorption of non-charged particles. Thus, since the diameter of the adsorbed particles, and the range of the intermolecular interaction are of the order of Angstrom units, the magnitude of (b - a/k-T) is at most of the order of IO-11 cm?. \Ve shall see below, when the charge of the adsorbed particles is taken into account, that a much larger second virial coefficient is found. From eq. (98) and eq. (92) we derive: P
=
pSa(P)
+
(the Gibbs adsorption eq. (95) immediately The Freundlich x =
kT In I’f I‘0 +2kT(b---/kT) equation
(I’---“)
is here simply dy =
-
rdp)_
(99) Combination
with
leads to the pertinent adsorption equation. equation of state is our third case
7rAcT
( LOO)
where 9~is a number, not necessarily an integer_ The Freundlich equation of state is
easily derived r=
from the more-commonly
constant
known Freundlich
where 8 =
-
equation
x Nn
(101)
Finally (fourth case) we write down the Langmuir z=
isotherm
APkT In (1 -
equation of state:
0)
(102)
P/Ne is the coverage and NB is the number of available adsorption sites
per unit area of the interface. This number is, as a general rule, not identical to b-1 of eq. (97), the Langmuir adsorption being of a different character than the Van der Waals adsorption. In the Langmuir adsorption mechanism localized sites are assumed, whereas the adsorbed particles according to Van der Waals behave as a Advrm.
Cdlooid
Interfetce
Sci..
1 (1967)
277-333
ADSORPTiON
non-ideal
319
EQ?UATIOSS
two-dimensional
gas. The chemical
potential
in the Langmuir
case is (103)
Combination of eqs. (95) and P b = CL=,immediately leads to Returning to the influence that the Van der \\‘a& equation
(103). together with the equilibrium conclition the well-known Langmuir adsorption equation. of the charge of the adsorbed components, be bee of state and the Freundlirh equation of state are
of particular interest. This will be demonstrated by some examples_ First complete ionization is considered and expressions referring to the AgI-aq.-soln., the oil-HAaq.-soln. and the semiconductor-02 interfaces are given. Then incomplete ionization is considered and it is briefly indicated what effects can be expected for the oil-HA-aq.-soin., tilt: semiconductor-02 interfaces, and for the “adsorption” of the electrons in surface acceptor states of a semiconductor. (u)
&I-aq.-sok
i,hykct:
Under experimental circumstances~ 7.59 the surface charge consists of about 1Orr elementary charges per cm= at the most. This mean5 that the adsorption of Agf ions or I- ions hardly exceeds this number. Confinin, cJour5elves to the adtlitic>n of KI, the condition for the adsorption isotherm can be written as: or prSUeh*m + The low-potential
zreyo _t k-1 In l;/IYru approximation,
=
prrJe +
kl‘ In cr/~re
ccl. (371, gi\*es yu = -kcx-‘0.
(105)
where it can duly be
assumed that a = zr el’r. If these expressions are inserted in eq. (LOS), this equation becomes formally identical to ccl. (99)J the second virial coefficrent 13 now being given by B =
3 z (:I e)e (~kl’)-~
*
( 106)
but its value is now much larger than the orrgmal Van der \Vaals tale. Thus, with x-1 = IO-6 cm, the value is about 3 x IO-r1 cmz. This means that the etectrostatic term predominates in eq. (105) and a plot of In cr (or of In cas) against 0 should show linear parts. Fig. 14 gives such plots, but it should immediately he added that it is precisely the deviations from linearity that have led to the present detailed insight into the structure of the double layer! (b)
Oil-HA-aq.-soln. z’rztevface An equation similar to eq. (106) often applies to this case. Only z~ should be
replaced by z~. It is here relevant to note that Vrij51 considered a discreteness-ofcharge effect of the ions A-. This effect leads to a B-value which is about 10 y/o smaller than the “smeared-out model” which is used throughout this paper. The reason is that the introduction of “discrete charges” also leads to theintroduction of a redistribution of the A- ions. This redistribution leads to a lower value of the surface pressure. One important aspect of Vrij’s treatment is that the theoretical disddwm.
Colloid
lnterfuce
Sci..
1 (1967)
277-333
31. J _ SPARNAAY.
320
tinction
made
adsorbed
here between
particles,
the second
SEMICONDUCTOR
virial coefficients
can no longer be made,
i.e. they
of charged
are no longer
SURFACES
and of neutral
to be considered
as additive.
Fig. I-1. Adsorption of I-ions plotted
drfferent KSOs
Also the Iri@ potential because
the adsorption
circumstancesss. AF+$ =
agamst
p1 ( or kT/e
In cl). From ref. 52 (Lyklema).
here can exceed
If po = -
approximation
is of importance
at the interface.
This
in this connection,
1012 molecules per cm3- under experimental then this approximation gives
0 at zero adsorption.
2kTra
(107)
We assume complete ionization and an ideal behaviour n =
Five
concentrations.
of the chemical
part of l;lA
Then
3l”AkT
isotherm equation with IZ = 3. Experimental evidence for this equation has been reported 53. The electrostatic contribution here still dominates although to a lesser extent than in the low-potential approximation. (c)
corresponds
(108) to a Freundlich
Stmziconductor-02
interface
If the adsorbed O?; is completely ionized, a situation similar to those described under a) or b) exists. The only difference is that the compensating charge is provided by holes instead of ions. Another difference may be found in the chemical part of the electrochemical potential of the adsorbed 02. Its contributions to the surface pressure will be given by eq. (102) rather than by r,,r- kT, because the adsorption
may
The second B =
have virial
27~ (zoz-e)2
taken
place
coefficient
through
an adsorption-site
mechanism.
B is now
L(kT)-1
where L is often of the order of lO+cm,
(10%
i-e_ much larger than the Debye-Hiickel Advaa.
Cothid
Interface ScL. 1 (1967) _
277-333
ADSORPTION
321
EQUATIOXS
length in most ionic solutions. For roe we take in our example the value -2. If the high-potential approximation applies and if the value of 1&bis small, no new aspects arise. the case being analogous to eq. (lOSj_ Tf the semiconductor is strongly doped and of “-type conductivity, then; 4n!=
-
X&T
In (I -
1y,V8) +
(3 irX&-r
32 z es p
(110)
but there seem to be no experimental indications for this equation_ Evidence for adsorption isotherms which can be explained by means of eq. (109) will be given below. First incomplete ionization will be dealt with. \Ve shall consider three cases: (s) the adsorption of surfactant HA at an oil-water interface, (b) the adsorption of oxygen on a semiconductor surface, (c) the contribution of electrons in acceptor-surface states to the interracial tension. (a J The chemical equilibrium is HA + H + + A-. The adsorbed components are HA and X-. If the degree of ionizatmn is ~1s at the interface, then to a first approximation: pWA*
interltwe
q 2~”interface
SHAKO
= =
p A”O i-
In (1 -
+
kT
kT
In
crsF
-t
us) r (111)
z .keyo
where r is the sum of the adsorbed amounts of HX and _-\- per ems. Then: AJ.J=
-
AZ =
-
fil‘
+
AFe
(112)
where we have chosen again the initial stage to be at zero ad&orption. Furthermore it is assumed thet lye = lyO_ The interracial charge a is equal to zaerus. The integral of AFe is the same as discussed before but, as indicated by Payen 3BSit is a tedious problem to obtain us and r separately. For us one can obtain, with the aid of eqs. ( 110) and (111) and of pn4
=
3a -t- ?#Qr
(113)
(where Y~I = FH = PH b* + kT In cn/caO) the expression
a8=
c
1+-
=K
exp
CFIO
-
A prro + kT
zAev)o
-’
>
where A~~o=~~~~o-~~~--~nbo is the free energy of the adsorption of H+ ions. For small a* one has a Boltzmann-type relationship: oa =
a08 exp
WO kT
(115)
where a06 is the degree of ionization at v. = 0. One way to write the condition for the adsorption isotherm is: PF?AO
bulk +
kT
In CRA/CHA~
=
~CHA~O
+
kTIn
(1 -
a*) ~/~HAO
(116)
For small 08 a Henry-type adsorption law is found: the concentration cx~ is proportional to the amount adsorbed. For not too small aa, a quite involved adsorption law is found (see Payens). .4dvan.
Co&id
Interface Sci.. 1 (1967) 277433
322
M. J_ SPARNAAY,
(b) For oxygen adsorption equilibria will be assumed:
on a semiconductor
02($w
+G
Ot(ads)
PO,
Oz(ads)
f
2 O(ads)
PO, _ ads
0 (ads) +
O+
(ads) +
SEMICOSDUCTOR
2h+
PO
gns =
PO2
=
ads =
SURFACES
surface the following
(117)
ads
ZpO
ads
910 ads
three
(118)
-2EEF
(119)
Writing PO2 gas =
PO2 go +
kT ln
/JO2 ads =
~02=”
kT
+
~o,I~o~~
ads =
PO s* i_
kl‘ In ro/roO
‘lo
ada =
PO so +
kT
Ay 5
-
An =
-
(121)
In I’o,lro,”
PO
the following expression
(120)
In
(122)
r02-/ro2-0
-
(123)
2eyo
for A? is obtained: (1’0,
+- I’,
+
I’oz-)
kT +- AFe
(124)
where again it is assumed that at y~y~ = 0 the adsorption is zero. The degree of dissociation of the equilibrium eq. (I 18) is denoted as CLIand the degree of ionization in eq. (119) is denoted as ~tr. Then: =
roz
(1 -
al)rl;
1-o =
2~(11’1 =
(1 -
~22)r*;
ran-
For weak ionization and strong dissociation, a Freundlich-type
=
a%Tz
isotherm is obtained
with tt = 2. For strong ionization the formation of a double layer can be hampered if the dissociation dissociation ~0,~’ =
is weak and a Henry-type adsorption cau then be found. For strong and strong ionization the following equation is relevant: i-
2$-f
0 +
In Po,/PoQ
=
2 kT In rogf-_lPo~--O -
4evo -
4Ep
(125)
Taking ~0 = (-2er&) (capacitance)- 1, the same situation can exist as was met in the AgT-aq_-soln. case and as we have reproduced in eq_ (109). As % example 54, the adsorption at room temperature of oxygen on oxidized Si may be mentioned. Fig. 15 gives curves pertinent to this case. The adsorption is not measured directly here. In fact, the quantity plotted against In PO2 (the width of an electron resonance line) requires some explanation. If a Si-crystal is crushed in high vacuum (10 -s-10-10 torr), no electron resonance lines are found. This points to the absence of “dangling bonds” at the surface, because free electrons at the surface (the ‘*dangling bonds”)
may be expected
to show resonance in an applied
magnetic field. Absorption lines in a magnetic field are found, however, if the Sisample is heated in a poor vacuum. In particular if the sample, after being crushed in high vacuum,
is heated between 450°C and 500°C at oxygen pressures between Aduan. Colloid Interface Sci.. 1 (1967) 277-333
ADSORpTION
323
EQUATIONS
I i-
i-
.
Fig. 15. Line broadening AN (which is roughly proportional to the adsorbed amount of oxygen, on the Si-sample. AH = 1 Gauss corresponding to about 1011 adsorbed oxygen molecules per
cm:) plotted against In pan. Room temperature. From ref. 53.
10-s and 10-5 torr, a resonance line (line “A” in Fig. 15) appears with a g-factor of 2.0029 and a width of about 3 Gauss, Line “B” in Fig. 15 appears, when the Sisample is crushed and sealed in a quartz tube under moderate vacuum. These lines are attributable to paramagnetic centres formed by a chemical reaction between oxygen and silicon. Their density is about 1012 cm-e.
If at room temperature
(or
temperatures roughly within 50’ from room temperature) the oxygen pressure is increased, the width of this absorption line is increased. This is a reversible phenomenon. A&an.
Colioid Interface Sci.. 1 (1967) 277-333
324
M. J. SPARNAAP, The line broadening is understandable
SEMICONDU~OR
if it is assumed that
SURFACES
(paramagnetic)
Dxygen molecules are adsorbed and distort the local magnetic fields, thus providing
for a wider scale of absorption energies. It should he noted that diamagnetic gases have no effect. Theoretica calculations make it probable that the line broadening is proportional to the amount adsorbed, the proportionality factor being of the order of 1011--1012oxygen molecules (atoms) per gauss line broadening. If so, the slope of the curve indicates a “virial coefficient” which is of the same order of magnitude as that found in the &I-aq.-soln. case and 3 to 4 orders of magnitude larger than predicted on the basis of the usual intermolecu1a.r forces. -4nother case where such absorption lines are found is that of crushed GaAs. Here a heat treatment at 200°C in IO-4 torr of oxygen results in the formation of paramagnetic centres. A reversible line broadening takes place if the oxygen pressure is varied at room temperature. However, in this case the line broadening is proportional
to the pressure
itself
and not
to the logarithm
of the pressure.
This
points to a Henry type of adsorption, i.e. to the absence of strong ionization_ A’ number of analogous cases may be investigated in the future. In fact electrostatic effects, mostly connected with the adsorption of osygen, ha\-e been widely discussed in the literature of the past 15 yearsJ5, and even earlier_ We only mention here the case of oxygen adsorption on CdS. According to Afark56, light is needed here for the adsorption, and, once the oxygen is adsorbed, light is again needed for desorption. This behaviour may well be explained if the electronegative properties of oxygen are combined with the well-known photoelectric properties of CdS. Thus, light shining on a CdS crystal creates electron-hole pairs. Electrons are needed to saturate the demand of adcorbing oxygen. Again, if desorp tion is to take place (a decrease of the osygen pressure) the influence of light is very helpful: created holes are now net essary to neutralize the osygen before desorption can take place. Finally we note that the degree of ionization Q?:can be written in a way analogous to that of the equilibrium HA -L H+ + A- namely: 0%
[I
=
where Ama
=
+
(_%)_-I
f(tro2-so
-
z * m” ;-go+-evt,]
esp
pose}
Ei and where tz now enters as a parameter in-
-
stead of CH and A&O instead of Ap~o. fc) E8edrotrs irz acceptor swface states. Eq. Ep =
EAO + kTh
m
zr, _
r
e
(18) can be written
{free) energy at flat-band conditions.
AZ=
FAkTln
(
1 -
as:
-eye
where we have written the “adsorption energy “EA as EaO -
4 =-
(126)
For
Ay
x)”
one obtains: +
AFe
pyre, EAT being the
APPLICATIOSS
For smatf f,
OF
(F,
COLLOID
4
CHESIISTRY
325
31ETHODS
PA) this equation reduces to the “ideal”
be noted that the Langmuir model of gas adsorption
equation.
It should
(or also the Stern model of
ionic adsorption) leads to equations similar to those we ha\-@ seen here; eqs. (127) and (128). In these cases I -A is the number per unit area of adsorption sites and re is the number per unit area which is occupied by a gas atom or an ion.
(6)
Methods used in colloid chemistry and applicable
to semiconductor
probkms
tVe mention two such methods: (n) The study of electrokinetic phenomena, whit h leads to knowledge concerning [-potentials. {b) The study of the contact angle of a droplet placed on a semiconductor surface, as a function of a potential difference applied between the droplet and the semiconductor. Once more germanium is the semiconductor applications 6.1
tvhich is most frequently used in
of these methods.
ELECTROKISETIC
PHESOJflZ:SA
EIectrokinetic phenomena are connected with the tangential movement of two adjoining phases with respect to each other. One of these phases is a liquid. The c-potential is the potential difference between the slipping plane of these txvo phases and the bufk of the liquid phase. It is not always clear where to locate the slipping plane and what the exact physical nature of such a slipping plane is. In the case of an aqueous solution bordering a solid. a layer of kvater molecules of. say, 10 w thickness will be immobilized and attached to the surface of the solid. This marks the Idcation of the slipping plane. AltholAgh electrokinetic phenomena have been studied at a very early date 57” the interpretation of the results is still not unambiguous 5s. %Ve confine ourselves to the most simple interpretation, but this wiil be sufficient for our purpose because we are chiefly interested in the semiconductor aspects of the experiments, An important achievement of the theory was the proof, given by De Groot, Xazur and Overbeekss, that for a given s-stem all the different electrokinetic
phenomena prox$de information concerning the same electrokinetic coefficient, which can usually be identified with the c-potential. Thus, Ge-particles (both of *c-type and of P-type conductivity) immersed in dilute aqueous and moving under the infhxence of an solutions (10 --S .N acids and I-I-electrolytes) electric field (electrophoresis). have a ( potential which is between 0 and -40 mV (depending on the pH, see below) and this is in accordance with the results of streaming-potential measurements 60. The streaming potential is the potential produced by a liquid flow which is pressed through a porous plug of, in this case, germanium particles and measured by two electrodes placed at either side of the Advan.
C&oid
1mferfat-c
Sci.,
1 (1967)
277-333
326
bf.
J. SPARKAAI,
SEM ICOHDUCTOR
SURFACES
plug. In Eriksen’s measurements of streaming potentials no specification was made as to the conductivity type of the germanium. More detailed measurements were carried out by Sparnaayer, who applied the method based on electro-osmosis. In this method an electric current is passed through the liquid contained in the plug and the liquid transport (x*olume transported per unit time u) is measured, Then:
where C is the zeta potential, i! the specific conductivit>* of the liquid, ye the viscosity* and e the dielectric constant of the solution. The pH of the solution was varied at constant ionic strength (provided by mixtures of HXOe and KXOa solutions) between 2 and 4-i. Electrolytes with a concentration greater than lO-2 .N coulcl hardly be used because it was then difficult to measure c’ with any precision. Solutions in which the pH was higher than 4.5 were avoided because germanium &ides were beginning to show a tendency to dissolve, All the experiments were carried out in thoroughly-steamed glassxvare. For 10-z Q. N-Q&? germ,anium ft linear relationship bekveen c-potential and pH x-as found between pH 2f and pH 4, the slope being about 30 mV per pH unit. The f-potential at pH 4 is about -40 mV. For P-type germankrm this was not the cse. .a11 calculated C-potentials being found around -10 mV. This discrepancy must be explained as follows: eq. (129) can only be used if electricity transport across the semiconductor particles is absent. However, the specific resistance of the semiconductor samples was lo- f Q cm, whereas the specific resistance of the solution was about 104 Q cm. If the electric current was wholly transported across the particles, no liquid volume transport would have taken place and application of eq. (129) would have led to a zero value of the c-potential. 1Ve conclude therefore, that the “real” (;-potential for both P-type and n-type Ge is the same (as is also suggested by the electrophoresis experiments) but that there is more electricity transport across p-type material than across St-type material \t’e return to this point below, The measurements (now only n-type Ge is discussed) indicate that H* ions are potential-determining. The slope of the t-pH curve is the same as that found for glass-aq.-soln. interfaces, although the values of the c-potentials are much less negative. In both cases the pH-dependence points to the presence of acidicgroups at the interface. Calculations based on Gouy’s theory have indicated a density of charged acidic groups of about 1012 cm-%, which decreases upon a decrease of the pH. Acidic groups can be formed through an oxidation of the Ge atoms at the surface followed by a reaction with water molecules. The density of Ge atoms at the surface is about 7 x 1014 cm-s_ If the same density of acidic groups is present, * There should be no confusion between the symbol q, used here for viscosity, and its use elsewhere for electrochemical potential_ Advan. ColCoid In&f&c
SC.., 1 (1967) 277-333
APPLICATIOS
OF
COLLOID
CHE.\IISTHI-
327
3IETHODS
after the osidation and reaction with water has taken place, rhe degree of the ionization is usuaIly somewhat less than lWz. This points to an ionization con&ant which ~11 be significantly higher than that of free tfissol~~ed H&eOs. Such a phenomenon, a variation in the ionization constant of an acid, placed tirst in the bulk of a solution and then at an interface, has been relMwtet1in the literature. for example by Scheraga6” and by Payens 38. They attributed the difference to dielectric influencel;_ If oxygen had been absent, the t-potentials might ha\w been different. Ilrwever. during the electro-osmosis experiments electrochemical reactionr at the Ge particles are taking place. Each particle has an anodlt Ale (lvhere there is a tenclency to oxide formation) and a cathodic side (N here hydrogen formation may take place). The total interface in the plug will thercfcxe be highly heterogeneous;. The influence of the pH can not only be noted in c-potential measurements. but nlst, in the measurement of the capacitance and of the surface conductance of (freshly anodized) Ge-electrodes which were discussed in section 1. The celts used for these measurements contained three electrodes: the Ge-electrode, a platinizeri Pt electrode whit-h has a large surface area. and (in the ver&n of IM:uenkamp and Engell) a Ag-A&l electrode_ \Vhen. through application of a potential difference between the Ge-electrode and the ptatinizecl R-electrode, the flat-band condition was maintained, the potential difference mea-Jured during the first minutes after anodization between the Ge-electrode and the _Ag- A&I ekctrocte proved to be ZL function of the pH. the slope of the potential-pH cm-\-e being claw to 60 mV per pH unit. In a ceI1 containing only a Ge eIectrode (U hi& has not been pre-.modized) and a reverdble electrode. the relation between the mensuretl (floating) potential and the pH is qualitatively the same, but it is cliRicult to obtai!: constant potential values, fluctuations in time of -10 m\: being possible. This points to an imperfect reversibility of the Ge-electrode in combination with mixed potentials. Although the H+ ions will be potential-determining, the finding of a perfect Xemst law is obscured here by secondary effects. However, as discovered by Brouwer63, a perfect Nemst law is found if Hg,Oe is added to the electrolyte_ This may be interpreted as a removal of unwanted side effects. In this respect it is useful to note that Hz02 is a constituent of many etchants and an increased ciissolution rate of Ge has been reported. Thus the reversibility of the Ge-electrode will be improved by the presence of H20~. According to Brouwer 6-L. Hz02 stabilizes the surface because it directly provides sufficient oxygen. Oxide layers are then formed, the formation being especially rapid (formation time of the order of minutes) if K+ isns are present in the solution_ One of the side effects is caused by the presenee of Cu”-+ ions in the electrolyte. As mentioned in section 4. Cu’+ ions, after contacting the Ge surface, can form surface states. These can be measured by means of radioactive tracer methoc?s at concentrations on the pH,
as low as IO-7 mole44b. The density of these states depends strongly
especially
at pH
>
5. The density Advan.
increases with increasing
Coltoid
Xnferfucc
SC&, 1 (1967)
pli.
177-333
328
M. J. SPARNAAY,
i
SEBIICOXDUCTOR
SURFACES
Measurable desorption of Cut+ ions takes place only in very acidic solutions. the reaction at pH K l+ being as follow+: Ce + x(H~0 -f- Cu”-f) % GeOz -+ x (Cu + 2 H*) (130) In passing we note that this reaction can be interpreted as a competition for electrons between the reaction Cu”+ + 2 e- + Cu. and Ge -f- x Hz0 +S GeOx + 2x H+ + 2~ e-. Since the standard potential of the copper reaction is known (0.34 V) the standard potential of the second reaction can be estimated to be somewhat lower and does not contradict the value given by Lovrecek arxd Bock&66 (0.11 V), These
authors
potential system.
also
1-s. plf curves It is evident
electrochemical a paper
have
greatly
that
depend
the
of the reaction
and Faktor6;.
slope
and
upon the state
that one of the most
aspects
b>r Carasso
shown
dangerous
between
the reproducibility of purification
impurities
Ge anti Hz0
of the
of the whole
is Cu”-+ ions.
The
have been reviewed
in
See also Latimers*.
-terpretation of ;L slope different from that predicted by the Nernst law might be given in terms of the -discreteness-of-charge concept of the charges at the interface_ It was assumed hitherto that the charges at the interface could be considered as smeared-out and consequently that the potential y5 had the same value everywhere on the interfacz Owing to the discreteness of the charges, the “adsorption energy” EA of the electrons in the acceptor surface behaves as if it were a function of the density of the charges present at the interface and their distribution over that interface. This function has been discussed extensively in the literature of the Hg-aq.-soln. and the AgI-xl_-soin. interfaces~~.69. For hexagonal and tetragonai array-s of the charges and densities of about 1013 cm-2 or less, the energy EA has been found, by numerical computations to decrease at increasing coverage, implying that at a higher coverage the adsorption is “easier” than at a lower coverage. However, so far no experimental results seem to be available in the physical chemistry of semiconductor surfaces which need. as their unavoidable interpretation, the concept of discrete charges. Therefore this concept will no longer be discussed here. Returning to the marked difference which was found in the electro-osmosis expetiments of n-type and of j-type germanium samples, this difference is probably connected with the mechanism of dissolution during anodization. As mentioned in the Introduction, section 1, a hole mechanism operates, because there is a limiting anodic-current density in n-type Ge, which is absent in P-type Ge. As far as the electro-osmosis experiments are concerned this means that the current can more easily find its way across P-type samples than across s-type samples, in accordance with the experimental data. Finally we notethat although surface states can be removed both by anodic An aiternati -
and by cathodic treatments076,such a “state-free” situation is by no means a stable one. States may rapidly form again. their rate of formation probably being connected with the composition of the solution. The purifiation of chcnzicats ad apparatus shouiX be carried out extremely
thoroughly. Advan. Colloid Interfact
Sci., 1 (1967) 277-333
APPLfCATfCW
6.2 rOWTACT
OF COLLOID
CHEDIISTRY
xETHOf%
329
ANGLBS
A droplet of an aqueous eiectrotyte placed on the surface of a semicwductor, has a contact angle (denoted as a), whose value depends on the history of the senliconductor surface, Par instance, if the semiconductor is a germanium crystal, etching with “CP-4” folirnved by rinsing with water provides a “h_vcfrrjI,fIt,t>ic”,i.e. a non-wettable surface, the contact angle being about 180”. This is not a stable situation. In the course of 15 minutes or so the surface hecomes more ‘“h) clrqd~iiic” or wettable and the contact angle becomes about SO degrees7”-~*. If a thin Pt-wire is introduced in the dropfet and used as ;m electrode then, after application of a potential difference between the li’t-electrode and the semiconductor, the contact angle decreases further and mxy dtirnately become zero. The surface has then been completely wetted 7r*7p.The potential difference (V) may be positive or negative. The rate of decrease of rt is about prr>pwtionaI to the absolute value 1VI. For 1V] < S S, this rate is trsualiy very smrrll. To mention a typical example: if in the case of a fre&ly-etched Ge-surface a potential difference of 10 V is used, complete wetting occurs in about 5 min. A current pnsses across the interface between semiconductor and solution if a potential difference is applied, the current density being about 10 ,w& cm-* at \I’! = 10 \‘, If some time (a day or so) had elapsed after the et&ml: procedure had taken place. a higher value of \I/‘! was needed for the same rate of decrease of u and of the same current density. These results should be interpreted in terms of a varying liquid--solid inter facial tension?e. \j’hat is measured here is. in fact, an electrocaptltary curve, which is well-known in the study of Hg-aq.-soln. interfaces. However. in the underlying case we cannot hope to obtain more than qualitative information. The most important reason for this is that the surface is “dirty”. It contains an oxide Iaj*er, obtained by etching in our case, and this oxide layer cannot be removed under the conditions suitable for the underlying experiments. In fact it is impossible to remove any oxide layer in the course of experiments concerning contact angles between the semiconductor (germanium) surface and an aclueous 4ectroIyte. Some information as to numerical values may he obtained as follows: YS =
ysl4 +
3% eos a
(131)
This is Young’s equation. In this equation, ys, ys~ and ye are respective& the surface tension of the solid. the solid-Siquid interfacial tension, and the surface tension of the liquid. It is seen that the observed decrease of u. as the result of the application of an electrostatic potential difference, is due to a decrease of yst. We explain this decrease by the”adsorption“ of electrons in slow surface states (for positive V) and by the “adsorption“ of holes (for negative V). Now for fast surface states it was reasonable to assume Fermi statistics, as expressed in eq. (18). The same will be assumed for slow states, although the concept of slow surfaces states is still rather vague-The A&an.
Co&id
Xnterfice SW., I (1967)
277-333
M.
330
J. SPARSAAY,
SEMICOSDUCTOR
SURFACES
two parameters which are usually considered to be characteristic of slow states are their relaxation time and their density, rss. For etched Ge-surfaces the relaxation cm-“. If, time rgs. is of the order of minutes, and IIBs is of the order of 10’~-101* upon applying the positive potential,
V. electrons are filling the great majority
of
the slow states, then with the aid of eq. (IS), in which the exponential term is now considered to be small compared with unity, the electrostatic contribution to ys~ is A~SL
=
-
~TBe(a)
(132)
leyol
where the added subscript (a) indicates slog acceptor type states. If fey,,! = 5 . lo-13 erg, i.e. if it is about IO kT, and if the density of states is about 1014 cm-z, then Ays~ equals about 50 erg cm -2_ \Vith this value, which is of the same order of magnitude &asthat of yr, (for xvater, ye = SO erg cm-?), the observed decrease of u times, can be explained. This explanation, together with the observed relaxation provides evidence for the conclusion that slow states are involved in the phenomena described. Ellis70 ascribed the increase of the wettability during the first minutes after etching (i-e_ without the appIication of an electrostatic potential difference) to a replacement of fluoride groups, formed by the etchant at the outermost surface, by hydroxyl. In view of what has been argued here, this increase of the nettability may also be connected with the charging up of slow states. Alternatively, the interpretation of the observed decrease of czafter application of the potential difference, should leave room for the possibility that slow chemical reactions are involved. The nadure of rhe slow states is far from clear and the best definition that one may give is that slow states are non-stable ionic, atomic and molecular configurations, with slowly-varying donor and acceptor character. Slow states have also been found at the etched surfaces of other semiconductors and of less noble metals72. Therefore it may well be that they occur more generally than hitherto assumed, in other words, they may build up in many instances where new interfaces are formed.
ACKNOWLEDGEMENT
The author
is deeply
indebted
to Professor
Overbeek
for his constructive
criticism_
REFERENCES Numerous references can be found in the Proceedings of Semiconductor Surface Conferences: a R. H. KINGSTON. (ea.). Semiconductor Surface Physics. University of Pennsylvania Press. - Philadelphia. 1957. b /_ PlryJ. Chem. Solids. 14 (1960). c H. G. GATOS. (ed.), The Surface Ckemisfry of &IetaZs and Semiconductors, Wiley, 1960. d Ann. N-Y_ Acad, Sci., 101 (1963)_
e Sut$xcrrSci., 2 (1964).
Advsn.
ColIoid Inlerface
Sci.. 1 (1967) 277-333
REFERENCES See also surface sections Phpico. 20 (1951).
Semrcond,
331
1
Phosphor
in the Proceedintis of Solid State Conferences: a. z: Proc.
Iwern.
Cotloq
. Garnlisck-Purtorkirckeu
19Jb.
Welker.
l3runs-
wick. 19%. Proc. Illd lut. Conf. Semicond. Rochester 1938. Pkys. Chem. Solrds. 8 (1959) Ptac. Itzttrn. CoHf. Srmlcomf. pkyrr., Pmgue 1960, Publ. House. Czech. Xkad. Sci.. Prague. 1961. 7962. Inst. Physics and The Physical Sot.. Proc. Intern. Con/. Phys. ,I’emico#Jd., Exeter. London. 1962. The
following book isof special inter-z&: MANY, Y. GOLDSTEW ASD S. U. GROVER. Semrconductor
A.
Surfaces,
Amsterdam,
1965.
Neview articles G. L. PS~ARSON ASD \V. 2-L.BRATTAI~. History of Semiconductor Research, Proc. I.R.E.. 13 (1955) 1794. b R. H. K~XGSTON. Review of Germanium Surface Phenomena. J. Appt. r.5ys.. 27 (1956) 101. Bl. GREEN, Electrochemistry of the Semrconductor-Electrolyte Interface. MO& Asptict~ c Eiectrochem.. Z (1959) 343. _-1dv. Electrorheru. Electrochem. L’ng.. II. GERISCHER, Semiconductor Electrode Reactions, d 1 (1961) 139. e T. 13. War~rss, The Electrical Properties of Semiconductor Surfaces, in Progress an Semiconductors. Vol. 5. Hevwood and Co.. London, 1960. p. 1. f G. HEILASD, Herstellung und Eigenschaften reiner Halbleiteroberflachen. I;ortschr. Pkysik. 9 (1961) 393 11. FLIETNER, Eigenschaften gewohnhcher Halbleiteroberflrichen. Pkyslka Stutus Soltdr, g 2 (1962) 221. h P. J, BODDY. Structure of the semiconductor-electrolyte interface. /. Electvoanal. Chew.. 10 (1965) 199. i See also review articles in Electrockewc. Semicor-duct.. t (1962).
a
Reinhold 1959 (a compilation of revlexv papers). N. B. HANMAY (ed.). Semicoprdrcctors. See for example: a) \V, SHOCKLEY. Electrons aHd Holes ZH Semico?zdrtctors, Xran Sostrand. Prmceton, 1950; b) E. SPENKE, ElectroGsche Halblemr, Springer, Berhn. 1955; c) .A. F. Josre. Firika Potypravodnikov. JXoscow. 1957: d) C. H. \\‘ASNIER. SolrJ state tkeory, Cambridge University Press, London, 1960. See also ref 1. 3 J. R. IfacDoNarD ASD C. A. BARLOW JR.,J. Chem. Phys.. X (1962) 306%. 4 J. LYKLEMA. Thesis, Utrecht. 1957. 5 A. N. FRUMKKIN. Russ. J_ of Phys. Chem. (EqCish Transl.). 35 (1961) 1064. 6 See for example ref. 1 and: 31. S BROOKS .CLSD J. K. KESS~D~ (eds.). Semiconductor MateGats, JLacXillan. London, 196% Researches in Electricity, I. 1839. in ,!Zuev)murr’s Library 1922. 7 M. FARADAY, Experimentul p. 44. 8 G. L. PEARSOX AND W. H. BRATTAIH. PYOG. Z.R.E., 43 (1955) 179J. 9 L. GALVANI. De virrbus etectricitafrs in tnotu nrusculari comme~rtuvirrs, Prague. 1791, (Handwarterbuch der _B7atrrrwisserlchaftelr. IV. 462. Jena. 1913). 10 F. MASSARDI, his pile L-opera di Atessandro Volta, Jfrlano, 1927. p. 155 (He constructed in 1799). H. HELMHOLTZ. Pogg. Ann., LXRXIX. (1853) 211. :: I. TAMX, PhysiR. Z. Soujctunion. 1 (19%) 733. 13 W. SHOCKLILY. Phys. Rev.. 56 (1939) 317. 1-l V. HEINE, Surface Sci.. 2 (1964) 1: J_ KOUTECKY. J. Whys. Ghem. Solids, 14 (1960) 233; T. B. GRIMLEY. J. Phys. Cirem. Solids, 1-I (1960) 227. and private communication; 51. TOM ASEK, Surface Sci., 2 (1964) 8. 15a P_ HANDLER. in Semicond. Surface Fhys. Con/.. Pheladelphia. 7956. Univ. of Pennsylvania Press, Philadelphia, 1957; R. MISSJ~AX AWD P. HANDLER. J. Phys. Chews. Solids. 8 (1958) 109: G. HEILAND AXD P. HANDLER, J. Appl. Phys.. 30 (1959) 446. 15b A. H. BOO~ZSTRA. J. VAN RULER AND M. J. SPARNAAY, Proc. Konbkl. Ned. Rkad. Wetenschap., B 66 (1963) 70: ?Lf_J_ SPARNAAY, A. H. B~ONSTQA AND J_ VAN RULER. Surface Sci.. . 2 ( 1964) 56. 16 G. HEZLAND AND H. LAMATSCH. Surfncs Sci.. 2 (196.l) 18. 17 J. BARDEEN, Phys. Rev.. 71 (1947) 717. 18 J- H. BRAI-TAIN 15~~ J_ BARDZEN. Bell System TecRn. J.. 32 (1953) 1. :
Advan.
Colloid Interf’e
Sci..
1 (1967)
277-333
MI. J. SPARNAAY,
332
SEMICONDUCTOR
SURFACES
19 I<. ~jOIiNEKK.%MP AXD fi. J. &W.XLL, %. i?~CC~rOCilUH., 61 (1957) 1184. 19a H. GERISCHER. in P. I)ELAHAY led.), Rdv. in Elcclruchewislry, Vol. 1. lntersclence. Kew York. 1961. ch. 4. 20 f). TURXER. J. IZCectrochon. Sot.. IO3 (1956) 251. 21 W. H. BRATTAIZU AND C. c. B. GARRETT. Bell Sysleru Tedwc. J_, 34 (1955) 129. 22 C. HERRISG AND M. 14. S~cnom, A’eu. !Uod. Phys., 21 (l949f 185. 23 IZ. El. FOWLER AHD I,.W. NORDHEIM, Pmt. Roy SW. London. A119 (1926) 173. 24 1;. G. _h.te.~ AND G. \V. Goserr. Ph_l~. Rev., 127 (1962) 190; .-f~n. S.Y. dcod. Sci.. 101 (1963) 647; Proc. IM. Co~f. Phys. Senricond. Exctcr, 1962. p 815. 25 J. VAN Laa~ I% J. J_ SCHEER, PYOC. Intern. Cobtf. Phys. Srmtcotrd. Exeter. 1962. p. 827; J. SCHEER ASD J_ v_+s La~=, Pkys. Lcffers, 3 (1963) 31. ‘2% 25b
C. A. 3fr_~in, Solrd-skde Eleclrorrrc,p, 3. VAN L.&AR ASD J_ SCXEER, to be
9 (1966)
1023.
pubhshed in -~urfuce
.%iewe.
26 G_ GOUY. .-fn~. C/rim. Physique, (8) 8 (1908) 391; a review in .+IM~. Pkvs. (Paris), (9) 9 (1917) 129. 26a D. C. CET=.PMA~, Phil. Jfag . (6) 25 (1913) 175. 27a J. T. G. OITRBEEK, in H. R. KRUYT (en.). Colfoid Science. Vol. T, Elsevier, Amsterdam,
lk%?;
28 29 30 31
b) T).
v_ DERVACXS
ASD
L.
LASDA; . .-icta
Physicockim.
fYR.SS,
14 (I 94 1) 633;
J. E.r@.
‘I’Aerw. (Russ ), 11 (1941) 803; c) iI. J, \V. VERWEY MD J_ T. G. O~E%BEE#, Theory offhe 1948; d) B. J_ \c’. X'ERW'EY, Ckcm. Stabtlily of I_sopkabic c-olloids, Elsevirr. .imsterclam. Il’eekllad, 39 (19-E) 563. I-. ONSAGER, Chews, Recks.. 13 (1933) 73: J. G, ~CIRKW~OD. /_ Ckem. Pkys., f (1933) 767 \V. I?. \I'ILLI~MS.Proc. Roy. Sot. /Londo>:), A 66 (1953) 372, see also F. P. DVFB AXD I?.H. STILLXSGER, J. Ckcm.I’kys ., 39 (1963) 1911; H. L FRIEDMAN. Ionic Suiufio~ Theor> Based o)t Cjusier Brpanslou Jrethods. Interscience. Xew York, 1962. 12. Sl?IWilZ AND >f. GREEN. /. rtppr. Pkys.. 29 (1958) 103-1. q7-.Y. ST. J. MURPHY. Surface Ser.. 2 (1964) I;: DZWALD. Ann. .V. Y. .-fcad. Sci.. 101 ( f963) 87,. 86. 51. J. SPARNAAY,
Rev. ‘Tract. Chim.. 77 (1958) 872. 1’. S. KRYLOV. Dokl. .-lkad. .Vauk SSSR, I44 (1962) 155. E. H. KINGSTON ~SD S. F. SEUSTADTER. /_ Appf. Phys., 26 (1955) 718: GARRETT AND 1%‘. H. DRATTAIS. Pkys, Rev.. 99 (1955) 376. 35 h’. I;. MOTT, Proc. Roy. Sot. ILotrdon J. A 171 (1939) 27. 2. Pkysik, I13 (1939) 367; 118 (1943) 539; IV. SCHOTTKY 36 W. SCIIOTTKY,
see also G. G. B.
AXD
R. SPESKE.
lYiss.Ycrorffcrrt~. Siemctts,18 (1939) 3.
J. T. 39 a) in 40
41 42 43
44 45 45 46 47 47 48 48 49 50
51 52
F. DEWAX_D. SelC System Tcchm. J.. 39 (1960) 615. A. J_ P~QESS, Pirrlips Res. Repts., 10 (1955) 425. D. C. GRAHAVE. Chem. Revs., 4 1 (1947) 44 1: b) R. PARSOXS m P. DEL~HAY (ed.). A duanccs EZectrochcmisfry. Vol. 1. Interscience. Xew York, 1961. R. DE LEVIE, Efecbothim. Acta. 10 (1965) 1136. E. A. EPIMOV AXD I. G. ERUS~LIMCHIK, Zh. Fiz. Klriw~.33 (1959) 4-11: and book. ~~ecfroclewgistry o/ Gerwutriur>~ a>td Stlicorr, Sigma Press, 1963. 109 (1962) 571: 110 (1963) 570. a) W. H. BRATTAM AND P. J. BODDY.~. Electrochcm.Soc.. b) P_ J_ BODDY AND \V_ I-l. BRATTAIS. Surfuce Sci., 3 (1965) 348. b1. D. KROTOVA AND Yu. 1'. PLESKOV. Phvsika Status Solidr. 2 (1962) 111. a) P-J. BODDY AND W. H. BRATTMN. ].~‘lcctrocherrr.Soc.. 109(1962)812:b) R.MEMMING. Surface Sci.. 2 (1964) 436. a) Yv. V. PLYZSICOV,Dokl Akad. Nauk SSSR, 126 (1959) 111. b) IV. IV. HARVIZY. J. Phys. Chew Solids, 15 (1960) 82: W. 1’. HARVEY AND H. C. GATOS. /. AppI. Phys., 29 (1958) 1267. P. J. BODDY, /_ ElectroaHal. Ckem_. 10 (1965) 199. H. GEREXHER.M.HOPFMAN-PEREZ AND \V. ,MINDT, Ber.BunsengeseClsckaft. 64 (1965) 136. a) R. MHHMING AND G. SEUMANN. Phys. Lefters. 24X (1967) 19. J. W. GISBS. The Collected Works. Vol. 1. Yale Univ. Press. 1948. a) E. A. GUGGENHEIM. /. Phys. Chum., 33 (1929) 842: Thermodynamics, North Holland Publ. Co., 1959. 4th ed. J, FRENKEL. Ki#dic Theory o/Ltqrrids, Oxford Univ. Press. London, 1955 (first Eagllsh ed.1 ch. 6 (especially p. 361). D. H. EVERETT. PYOC. Chcm. Sot.. (1957) 38. A. VRU, GrensCaagvcrsc~rjnseZor, Coffoq. Koninki. Vlaamse Acad. Wetenschap.. Dru=els. 1965, p. 13 (in English). J. ~YKI.ehfA, Tmns. Faraday SOG.. 59 (1963) 418. Advan.
Colloid Interface
SC&. 1 (1967) 277-333
333
REFEREXCES
58 59 60 61 62 63 63 6-l 65 66 67
a) El. GERKCHER ASD \V. MINDT, Surfrrce Sci.. -l6 (1966) 440. G;. BROUWER. to be prtbhshed. 31. 1. SPARNAAY. Sur/uce Ser., L (1964) 102. u.I_ 31. FAK COR. in The Electrocbemistvy o~Sonico,ldnct~rs. hcatlemic Sew \*ork. 196’). p_ X34. Ilridatio,t Pute~rtruls. Prentice 1fall, Sew York. 1952. 68 IS. 31. I_ATI~~R. Yb. I-I=. Kbtm .. 20 (19J6) 679. U. C. GRAHAVE. Z. illektroclem.. 6’1; (1958) 69 11. V. Basn~~x. 26-l; \*. G. I.ewc~, \‘. A. ~
Advan.
Colloid
Inter-ace
Sci..
1 (1967)
277-333
Semiconductor
PktCips Rrsearck
List
(1)
of
symbols
Surfaces
Laboratones,
and the Electrical
S. V. Pkzlrps’
Gloeitampe,lfubriekeH.
319
Introduction ............ Outline ........ (1.9) The bulk of a semlcoaductor._&t (I 3) Physical reasons for the existence
....
....
notions. k;m&cai of a double layer.
.
\:alud Surface
Potential distributions Potentials . . _ .................. (3-l) The chenucal potential of the electrons. The band scheme (2.2) \Vork function and photoelectrrc threshold. Experrments
(3) The (3.1) (3.3) (3.3)
....
Capacitan e equatwns &d&s Experiments ........................
of ma&&de
:
:
:
. :
:
(3.5)
Equations
of state. Xdsorptron
References
. .
: : 1 1 : .....
.
: :
:
:
.... 1 :
: %bl . 286 291 291 . 293
......... 1 :
. !i%Jz
:
1 :
. 397 1 * . 397 298
-e.l _.
..... : :
: : :
:
301 : .. 301 305
Isotherms
308 308
309 ...
314 311
....
316
.................
Methods used in colloid chemtstry and appltcable (6.1) Electrokinetw phenomena ........................ (6-Z) Contact angles ............
Acknowledgement
.
states
(5) The integral j3_ nd*y,. Adsorptron equations ................. (5.1) Adsorption and double-layer parameters . ................ ..................... (3.2) Thermodynamw functions (5.3) Comparison with the “first method” of XVerwey and Overbeek (5.4) Expressions for Jz arly~, . Its relative magnitude ............. (6)
. .
.........
space charge. . _ Gnuy layer or space ch&e_la;_e; : : 1 : : 1 : 1 : 1 : Simplafying assumptions. . 1 .................... The Por~on-lloltzmann equation. Three approrlmatlons
(4) The capacltazce (4.1) (4.3)
The :VcthcrKands
Eindhoum.
..............................
(1.1)
(3)
Double Layer
to semiconductor ................
problems
...
325 335 399 330
..............................
..................................
330
Advan.
Colloid Interface
Sci., 1 (1967) 277-333
279 LIST OF SYMBOLS
The right hand column gives the place of first appearance in the text according to page number and, where appropriate. number of equation {in brackets) and Figure number. A A
a l-3 b
c CH, CG
c
%C
C *Y
Cion C Lb-p.
co D E=A Ee &----EY
ED -EF E cal
Eh
El? e
eF FS 271,PI AFtZ! AF
acceptor atom interfacial area, surface area Van der \%raaIsparameter accounting for intermolecular attraction (ct > 0) or repullsion {U < 0) second virial coeflitient Van der 1Vaals parameter accounting for excluded volumes of molecules differen tiai capacitance of double layer differential capacitances of ltiehnholtz and Gouy layer differential capacitance of space charge layer differential capacitance of a layer of surface states differential capacitance of a layer of ionic groups adhering to a solid concentration in dilute soiutions contact potential difference standard value of ~~~n~entrati~~n
285 309 318 319
donor atom energy level of electrons in acceptor states bottom of conduction band forbidden band gap (considered temperature-indepeudent in this paper; in fact, Ec--Ev = 0.67 V at room temp. and 0.35 V at a~s~~ut~ zero, see alany’s hook p. 37) energy level of electrons in donor states Fermi energy, or efectrochemical potential of electrons potential difference in galvanic cell containing a calomel electrode potential difference in gakanic cell containing a hydrogen electrode top of valence band charge of proton (- 4.80 x 10-l@ e.s.u.) electron Helmholtz free energy (often if in American literature) Heimholtz free energy of interfacial phase Helmholtz free energy of bulk phases I, If (Helmholtz) free energy of double layer per unit area =F - Fo; difference of two Helmholtz free energies, Fatyo#OandFoaty~===O Advan.
Co&id
Intnvfacs
Sci..
318
302 304 302 303, 302 317 296 31s 285 292 28-4
2&1 292 29 I 290 305 284 285 285 309 309 313 310 1 (1967)277-333
I.IS-l-
280
G I;6 G*, G”
AC b” N
h h+ i
k I, fJfi Wi)o 7JLe*, 9JzJJ*
7JCe NA xc
7J 7t ni
‘Jj P
P PO --Q T
Gibbs function (often F in American literature) tiibbs function of interfacial phase Gibbs function of bulk phases I, II = GG,: difference of two Grbbs functions. yo # 0 and G, at ye = 0 factor appearing in Fermi-Dirac statistics magnetic field strength
309
Coiloid
SYXIROLS
(53) (53) (53)
G at
Planck’s constant (=: 6.62 x 10-a’ ergsec) hole current Holtzmann’s constant (zz 1.3S x IO-l8 erg.deg-l) Debye-Hiickel length (semiconductor notation) number of particles of component i number of particles of component i at ye = 0 effective masses of electrons and (heavy) holes respectively. For “heavy” and “light” holes, see textbooks. electron mass (z 9.11 x 10-2a g) concentration of acceptor atoms = 2~1~~~. number of states per cma available to conduction electrons concentration of donor atoms number per unit area of adsorption sites (Langmuir theory) concentration of conduction electrons coefficient in Freundlich isotherm equation intrinsic concentration of conduction electrons concentration of ions j concentration of holes pressure upon system standard value of pressure charge of an electron (solid state notation) absolute temperature El)/kT in the bulk of a semiconductor (EF(EF-- El)/kT at the surface of a semiconductor volumes of bulk phases I and IT applied potential difference potential difference across oxide layer liquid volume transported per unit of time work function of a metal and of a semiconductor distance from interface or surface exponentials related to the degree of occupation of acceptor (A) or donor (D) surface states at potential tyo Advan.
OF
310 291 323 291 285 326 285 285 310 310
(59) (12) Fig.
15
(9) (2) ( 129)
(61 (6) (59) (59)
291 291 295
(26)
291
(1-I)
295
(25)
318
(102)
2S-l 318 285 285
(101) (5)
2S-4 309 318 293 2S5 299 299
(1) (7) (1) (55) Fig. 4 (6)
(55) 329 295 326 296 293 303
Fig. 6c ( 129) Fig. 7 Fig. 4 (49)
Inferface SC<.. I (1967) 277-333
LIST
OF SYJlbOLS
osygen y
281
to germanium
ratio in c&de con~pouncl GeO,
bw-~
299
partitifjn function ijQ(ei]krr) vnlency of cliarged c0mp0nent
301,328 292 299 285
j
contact angle degree of ionization in the adsorbed state degree of ioni.*ation in the adsorbed state at vt, = 0 degree of dissoc;&on of 02 molecules at interface degree of ionization of 0 atoms at interfrtce factor accounting for electron spin number of acceptor (A) or donor (0) surface states per cmJ_ number of occupied srcceptor surfxe states per cm?
329 321 32L 322 3!!2 291
number of empty donor surface states per cm2 number of excess mo,lecules (ions) 0,f component i per unit area at the interface or at the surface standard value of the number per unit area nf esc’e+s molecules (ions) at the interface or at the surface number of excess molecules or ions per unit area at the interface or at the surface if ~0 = 0 interfacial tension, surface tension distance from pfane x = 0 where a surface charge cq is located (sometimes identified with thickness of Helmhcftz Iayer) dielectric constant dielectric constant in medium of Helmholtz layer dielectric constant in medium between planes x = 0 and x = S zeta-potential (in eiectrokinetic phenomena) viscosity
292
electrochemical potential of component j electrochemical potential of component j at yo = 0 degree of coverage (in Langmuir”s theory) Debye-Hlickel lehgth (ionic solution notation) specific conductivity chemical potential of component i chemical potential of component i at fyo = 0 chemicaf part of ekctrochemicai potential of component j
A&an.
Coltoid
Zderjacc
Sci.,
292 292
312 317 31s 309
314 265 304 315 323 326 311 311 318 283 326 310 310 311
1 (1967)
277-333
LIST OF SYMBOLS
chemical part of electrochemical potential of compnent j in the adsorbed state chemical part of electrochemical potential of component j in the adsorbed state at ~JQ= 0 chemical potential of adsorbed particles, considered as a function of the amount adsorbed chemical potential of considered component in a bulk phase as a function of concentration or pressure standard value of chemical potential of adsorbed particles at given T and given amount adsorbed standard x*alue of chemical potential of considered component in the bulk at given T and gix*en concentration ehxtrochemical potential of electrons or Fermi energy electrochemical potentintof holes or minus the Fermi energy interfacral pressure. surface pressure charge density conductance (of cylindrical Ge single crystal) interfaciat charge, surface charge; often expressed in this paper in number of electronic charges per cmz value of the space charge per unit area value of the charge in surface states, per unit area value of the charge in ionized layer, per unit area relaxation time (51ow states) energy parameter, (Ep - Et} in the bulk energy parameter, (EF - El) at the surface photoelectric threshold of a semiconductor chi- or Lange-potential; potential difference originating from a layer of polarised atoms or molecules electrostatic potential in space charge or Gouy layer with respect to the bulk of the phase (J = oo) electrostatic potential at surface or interface (.,, = 0) with respect to the bulk of the semiconductor phase (x ?= w) potential drop across Helmholtz + Gouy layer (or Stem + Gouy layer) total potential drop across double layer potential difference between the plane x = 6 and the plane x = 00 frequency, cycles per set AhaH.
Colloid
TnlerfucslSci.,
311 311 317 317 317
318 291 291 317 298 288 286 299 301 301 330 293 293
296 295 293
293
302 302 30-L 305 1 (1967) 277-333
~~~i~onduc~or Surfaces
and the Eleetricaf
Double Layer
(1) Xntroduction
Electric:al double layers are found at interfaces and surface5;. Speciaf attention will be given fo semiconductor surfaces, and to interfaces between semiconduetnrs and an a~pcous electrolyte. The most extensively studied surfaces and interfaces are those in ~rhich germanium participates, but properties of systems with other semiconductors (CciS, ZnQ, 5%) wilt be reported, EIectritrtl properties pertinent to semiconductor systems wi!t he comprecl with “cl;~s4~;11” esamptes of double layer systems, notably those at the f-fg-acl. sofn. interface, the AgI-aq. soln. interface, the doubfe layer-~ formed by ionized surfaetant molecules at a water surface or at a water-oil interface, and also, with properties of ionic solutions, In this respect it is useful to note that it is possible to distinguish two groups of workers in the field of double-layer systems_ The first group is mainly concerned with the “classical” systems mentioned here, and the second li;roup is concerned with systems in which semiconductors play a preponderant role. This paper is an attempt towards a more or less unified view of aii the systems, although it is primarily written ft)r students of the “classicaY’ systems, An electrical double iayer is formed by the unequal distribution of charge carriers, which reflects the physical and chemical discontinuities at the interface neutral system. The charge carriers with which we (surface). It is an electrically are concerned here should be mobile, They are the conduction electrons and holesl~~ in a semiconductor, the conduction electrons in a metat, and the ions in 3~1 aqueous solution. Apart from contribution due ta mobile-charge carriers, the molecules of the main constituents at the interface can take part in the doublelayer system. They form a moIecular condenser. AR important evampfe is that of water molecules present at the water surface or at the water-AgI or the: water-Hg interfaces, Hero the (polar} water molecules are oriented with respect to the plane of the interface, the orientation being different for thedifferent cases. For thewaterHg interface, the water molecules may have turned their positive side to the mercurys, in the ease of the water-AgI interface, the negative side may be4 turned Advan.
Co&id
Inter/ace
Ssi.,
1 (1967)
217-333
284
X.
J. SPARNAAT,
SEJIICONDUCTOR
SURl=ACES
to the AgI. Concerning the water surface, it is as yet uncertain which orientation is preferred by the water molecules5. 1.2
THE
BULK
OF
A
SEXICOXDUCTOR,
FIRST
NUXERICAL
XOTIOSS,
VALUES
Before proceeding with double-iayer systems some numerical values of pertinent quantities will be mentioned and some definitions will be given. \Vhere semiconductors are inx’olved, we refer to germanium. Thusin apure-Gesinglecrystal at room temperature (3OO”K), about 2.-l x 1013 Ge atoms per cm3 are “ionized”, i.e. a valence electron of such a Ge-atom is brought into a state of higher energy and is, to a certain extent, free to move. This valence electron thus has become a conduction electron. The minimum difference of the energy levels of the conduction electrons and the valence electrons is usually denoted as E, - E,., where E, is the (minimum) energy of a conduction electron and E, is the (maximum) energy E, is about 0.67 V. Quantum of a valence electron. For Ge the value of Ee theory requires that the electrons cannot occupy an energy level between Ec and Ev. The energy distance Ee - E, is therefore referred to as the forbidden energy gap. At the ionized Ge-atoms, there is an electron vacancy. This is called a hole. A hole is mobile because a valence electron of a neighbouring atom can occupy the vacancy. According to views which are now standardle”, a hole can be considered as a mobile positive charge with a positive effective mass. For Ge single crystals at 3OO”K, the drift mobility* ,D)&of the conductivity electrons is 3800 cm”/Vsec, and that of holes py, is 1800 cm’f\‘sec. lt is interesting to note that these values are about 107 times higher than those of ionic mobihties in aqueous solutions. The case just described is that of intrinsic conductivity. The concentrations of conduction electrons and of holes (denoted respecti\-ely by n and by p) are equal: n=p
(11
If the crystal contains impurities then as a general rule n # ~5.Thus if pentavalent impurity atoms (P, As) are present at lattice sites in the Ge-crystal, then n > 9 (*r-type conductivity). Four electrons of the pentavalent atoms are required for chemical satnration, the fifth has become a conduction electron. The pentavalent atom is called a donor atom, and has obtained a positive charge, lt is fixed to the lattice position. If trivalent impurity atoms (B, Al) are present on lattice sites, then TV< p (P-type conductivity), and there are immobCized negative charges. The technology trivalent impurity atoms are acceptor atoms. In semiconductor standard techniques are now available6 (for example diffusion at elevated tempera* The drift mobility is measured by conductance measurements. This may be distinguished from the Hall mobdlty. If a magnetic field is applied normal to the direction of the electric field. the deflection of the movement of the charge carriers leads to a potential difference. This Ha11 effect, as it is called, is often used to determine the concentration and the mobility of the charge carriers. The Hall mobihty differs somewhat from the drift mobdity’*s. Advaa.
Colloid
Interface
Sci.,
1 (1967)
277433
ISTRODUCTIOS
265
tures) for the preparation of crystals with almost any desired \*alue of rr-type or p-type conducti\-ity. The required impurity content is u~uall_v \-cry small. A Gecrystal with a conductivit_v (tr-type or P-type) of 0.1 R cm contains about 3 X lOI impurityatomspercm3oralmost 1 ppm. (A pure crystal at 3OO’K has a conductivity of about 46 Q cm and contains 2.4 x 1013 impurity atoms per ems). A result of solid-state theory is that the ionization reactions of Ge and of the impurity atoms can be considered as chemical equilibria, which can be treatecl in the same way as those occurring in ionic solutions for e-uample. Thus:
and
Ge + Gef + eD 7 D’ + eA % A-
+
or 0 s
h+ +
e-
12) (3)
h+
(1) where e- denotes an electron, h+ a hole. D a donor atom and A an acceptor atom. \Ve consider in this paper fully-ionized donor and acceptor atoms. Furthermore: n x p =
constant
=
~12
(5)
where ni is the intrinsic concentration of conduction electrons and holes;. (Thus eq. (1) could be written as PI = p = 9Li) Eq. (5) is only correct if the temperature and the influence of light are held constant. The value of ILLcan be increased by thermal anti by photoactivntecl stimulation of the ionization of the Ce-atoms in the crystal. Then new electron-hole pairs are created. Conversely, a lowering of the temperature and/or a decrease of the mtensity of Illumination leads to recombination of holes and electrons_ The recombination process is a complicated one nncl requires specific recombination centres- impurity atoms ancl imperfet tions in tiw bulk serbweas such and so too does the surface of the cry&al. For our purposes it is important to note the latter. The conductance of a semiconductor sample increases with increasing temperature (although the mobitity of the charge carriers decreases). This was first noted by Faratlay?m* on A&. The conduction electrons and the holes are in thermal movement just as are the ions in a solution, and just as in ionic solutions, it is possible to define a DebyeHEeke length in semiconductors_ This is often written as L in semiconductor literature:
Where E is the static dielectric constant (for Ge. E = 16). k is the Boltzmann constant, T is the absolute temperature, and e is the charge of an electron (e = 4.8 x lo-10 e.s.u.). At room temperature kT = 4.1 x 10-l” erg and L becomes about 2 micron (2 x 10-a cm). The definition of the Debye-Hiickel length for ionic solutions (usually denoted by z-1) is somewhat different (7) Advatr. Colbid Interfucc Sci.. 1 (1967) 277-333
Jf. J. SPARSAAY,
2236
SE?4ICOSDUCTOR
SURFACES
where the summation is over all the ionic species present in the solution, zj being the vakncy for ion j, and f&J its concentration. For a 10-3 tV l-1 electrolyte the value of the summation is X2 x 1017 cm-3 and. at 3oO”IC. x-1 becomes 10-S cm (for water, E = SO). On the whole, even in dilute aqueous solutions, the DebyeHUckel length is several orders of magnitude smaller than that in semiconductors. 1.3
PNYSICAE.
SURFACE
REASONS
FOR
THE,
RSISTESCE
OF
A
DOUBLE
LAYER.
STATES
There are two physical reasons far the existence of double layers. The first is the preference that mobile charge carriers may ~zhowfor one of the two substances bordering the interfczce, Three examples will be given: (a) 2% %kte@zce be&z-em IWO tiifirmtt metds. Here the electmns have different afCnities for the two different metals. The obsertqation and the interpretations of the phenomena concerning metal-metal interfaces have a Lang history, starting with the work of Gaivanig and of YoltatQ, IVe shall only mention here that it was probably Welmholtzll who introduced the name “electrical double layer” in 1853, assuming that at the metal-metal interface a very thin ~~osit~vely-eharge~~ layer was facing a everythin negatively-charged layer. As will be pointed out below. most double-layer systems have not as simple a structure as suggested by this view and the name “efectrical double Xayer” must now be considered as somewhat misleading. (6) A ~~?~~~~u~~~i&~i~~ s~+t5 cvystaG wkid.. catt be &vi&d into two rlrgims. One region has #-type conductivity, the other region has P-type conductivity. II’here the two regions touch, a double layer is formed. Electrons of the pc-type region diffuse somewhat into the j-type region (and partly recombine), and holes move to the n-type region. The dictible layer is quite diffuse, not only because the conduction electrons and the electrons are in thermal movement, but also because the transition from It-type to P-type conductivity regions, as determined by the donor and acceptor concentrations, is not so abrupt as for example the transition from one metal to the other in example (u). (c) A membrcrnesejkxratlng ho solutions f an3 II of th same salt CA, tdtidt we asstrme
compZei%Jy ionized,
ad
wlticR have
&+erent
ctmce~ttrations
cz and
en
The membrane is assumed to be permeable only to ions C+, These diffuse across the membrane from T to IL, Then f contains an excess negative charge formed by A-, and 11 contairzs an excess positive charge. Thus a double layer is again formed which again is of a diffuse character. The latter two examples are closely related to the Donnan equiXibrium_ fcx >
c&.
The second reason for the existence of a double layer is the adsorption of mobile charge carriers at the interface or surface, which then assumes an interfaz;irtlcharge (surface charge). This is denoted by CFand is expressed per unit area. R&an.
Co&M
Inlerfucc §cci., 1 (1967) 277-333
287
IXTRODUCTION
A charge -u should be present nearby, and should make the double-layer system electrically neutral. The adsorption of neutral species can aho lead to the formation of a double-layer system. Owing to the adsorptive forces, the adsorbed molecules or atoms can be ionized, thus leading to a charged surface, with the nentraliaing charges nearby. There are numerous eramp!es of double-layer systems ia which the arlsorption of ions is preponderant, notably the “classical” sybtems and the surftictant systems mentioned in the first paragraph. Examples of neutral molecules which are ionized after adsorption are those of O-, adsorbed on Ge and Si, on CdS, aA on ZnO. A number of polar gaseous ambients probably hehave in the same way on one or more of these adsorbents. A special example i5 that of the adsorption of electrons (or holes) at a semiconductor surface. The atoms at the surface have, as a general rule, a different affinity for electrons than those in the bulk. This was first pointed out by Tammr~ in 1932. These atoms, sites for charge storage, are the surface states. Tamm considered in his theory an ideal crystal bordering vacuum. States found under t’xperimental circumstances which approach this ideal case, are called ‘famm states, Another approach to the theory of this ideal ca+e was given by Shockleyl~ in 1939. The reasons why the electrons can show different affinities have been the subject of much theoretical work 14, but up till now the problems here ha\penot’” been solved. Surface states are often conceived of as adsorption sites for electrons or for holes. If electrons can he adsorbed. the states are acceptor states. if holes can be adsorbed (i.e. if valence electrons can be repelled) the states are donor states. It is not impossible that one single state can act both as a donor state and as an acceptor state. Acceptor states have a negative charge when occupied. Donor states have a positive charge when empty. The occupancy of these “adsorption sites,” by electrons and holes is calculated by means of Fermi statistics and the same equations are obtained as for the occupaccy of adsorption sites in Langmuir’s well-known theory (1917). This theory is in turn similar to that of Stern (192-f) for ionic adsorption (see ref. 27a). The occupancy of a surface can be varied esperimentally by placing a metal electrode parallel to the surface of a (grounded) semiconductor crystalandapplying an electrical potentiaf difference between the metal electrode and the semiconductor, Thus, if the metal electrode is made positive, electrons are transported to the surface of the semiconductor. Some of them are then in (acceptor) surface states. The others either have recombined with holes or are in thermal movement and &x-m a diffuse layer bordering the plane of the surface states. These electrons participate in the electrical conductance of the sample. Consequently a change of the electrical conductance of the semiconductor can be measured as the result of the application of the transverse field. This change is called the field effect. If it is small, a relatively large fraction of the electrons will be,in the surface states. The semiconductor behaves like a metal. If it is large, there WI! be few electrons in these Advaaa. Colloid fnterface
Sci.. 1 (1967) 277-333
M. J_ SI’hRNAAY.
Fig_ 1 Equipment crystallsb.
fcr merisuring conducrance
SE3%IICOE;I3WCTOR
SURFACES
and field effect of a thin cy%ndricaS Ge singte
states. Fig,. 1 showz the experimentaf arrangement, and Fig. 2 gives some results_
From the fidd effect the denftit~- of states can be estimated. Tf~us, cleaned Gel5 and 5316s-m-faces in vac--tum have about 101s states per cm%_This is the same density as
suggested bj- TZrmm’Y theory in \GGch every atom at the surface (there are ahout 101s atoms per crnl at the s&face) mav behat-e as a state. There are both donor and acceptor states, but acceptor states art: (weakly,‘1st~dominant. Handlerls* was the first to carry out this type of experimen t. He found a stroq dominance of acceptor states at the Ge surface.
i !
Now
r
gaseous
impurities,
such LS CO, even in quite small
i L-J-
a>
I
Fig. 2. Fieid effect; influence ~5 Fxygen gas a) high vsr*.um (c 10-S Torr): h\ n-_iddIeregion. oxygen pressure - 104 Tow; c) sigh presswe, about ii.01 Torr. The vertical scale shows the relative change of the conducta.nce_ Tfre horizontat scale shows the salues of the electrostatic potentialdifference between Pt ekctrode and Ge wire, AXso given is a tfme scis$e,showing that in region c) (high oxygen presstre) slow states are formed. A&an.
Colltid Interface Sci., 1 (1967) 277-333
289
ISTRODUCTIOX
quantities,
git-e the states an enhanced
difference
may
at least partly
acceptor
be found
character.
in the methods
Thus,
the origin of
of purifying
the
the surface.
Just as in Langmuir’s theory an “adsorption energy” can be defined. Estimatesbf such adsorption energies for the cases considered here will not be given, since these estimates
are little more than guess work. The states are “fa&”
that the time they need to adapt themselves
to changed
states. This means
eIectrica1 circumlstances
small. less than IO-~ sec. If s.uy*gen is allowed to react with the germanium at room temperature,
oridcs
are formeri. The density
of the tast states
is
vr*rfnce
derreaass,
finally to about NW per cm=, depending on the final oxygen pressure, tird “slow &ate5 ” are formed with relaxation times of the order of minutes. The o&latior~, as carried out in the manner described, w&l-defined
relationship
gas pressure.
between
is an irrever=ible process. There is therefore no the
In the experiments,
degrtte
osidation
of
of the surface
the gas pressures were in the range
xnd the
lc)-s-10-e
torr, and at the latter pressure there are about 101s fast states per ems. If the ouidation has taken place through an etchant, for example ‘“CY--1” (a mixture of 25 parts 6Ou/, HSOa, Bre),
15 parts
then the density
100 y/u CIIsCOUH, of fast states
states per cm” with relaxation
12 parts 3Es7& HF.
and 0.2 parts lOQy$
is still lower and there are about
times extending
1Ota slow
to hours, the whole situation
being
leas well-defined from a molecuhtr puint of view than in the case of careful gaseous oxidation of II cleaned swface. Htzxvex~er,ON ing to the much greatez- experimental accessibility
of etched
surfaces.
thctse were the first to be investigatecl.
theoretical concepts were used in 1947 hy anomalously low field effctcts found at etch4 suggested
states
layer of, say,
IO0 A (10-g
the chemical
nature of the semiconductr_r,
surface.
finrdzenr’ Ior the interpretation of Si surfct,-es. Barcleen and 13rattainia
in 1953 the moclel of an oxiciidecl semiconductor
layer of fajt
at the interfwe
between
cm) thick delxnding
In the case of etched Ge surface,
ems and about tionedabove,
101s slow states per ~4,
about lO0A.
Aspects
Tamm’s
surface.
semiconductor on the twchod
consisting
and oxide,
of a
zn oxide
of oxidation
and on
and a layer of slow states on the aside to lOra--IOll fast states per
this amounts the oside
layer
thickness
being,
as men-
of the potentials used in this model will be dis-
cussed in the next section, Very small densities nium stirface is brought iced, The anodic
of (fast) sn_f, _ states hove been iound when a germar +-e
into contact
reaction
with aqueous
leads to the formation
reaction with water, Is dissolved.
The remaining
electrolytes
andisslightlyanod-
of a surface aside,
G=-II@
interface,
which,
after
which naturally
concdns excess oxygen atoms, is almost “ideal” (or “perfe
Ge electrode,
because it is easy to increase the anodic current for
whereas there is a limiting current density (of about ImA/
cma) for n-type Ge-efectrodes (see Fig_ 3). Thus, holes are transported from the bulk of the germanium
to the surface, or rather, valence Advan.
Cdioid
efectrons are transported
iaterfate
Ssi..
1 (1967)
277-333
?r!. J_ SI’ARNAAY,
290 from the surface initiates
the
Garrettzl,
to the bulk,
dissolution
have
given by the American
Fig. 3a. Xnod~ aq.-soln. (pLI.
=
of
investigated
leaving
a broken
oxidized this
workers
chemLa1
compounds.
case.
SEMICOSDUCTOR
that the anodic
bond at the surface.
Tumer%o,
Gerischer
et aL.l9a
reaction
density vs. applied potential (ref. electrode, I)-calomel electrode (from ref. 20).
SURFACES
and
and
Brattain
inferred
the data
from
characteristics
calomet
This
eh-ctrode)
should
be
in the cell
Ce-
Pig. 3b. Model of the anodlc dissolution reaction: a) formatlon of oxidized compound intxface; b) further oxidation foIloww1 by dissolution of a germanic mxd**~.‘Q. According to Gerischer13=. A, + le = 1.1 1.8.
affected
by the addition
the solution. transfer
of electrons
mechanism
Other
Most
example,
compounds
(for example
mechanism
one state
to another.
can be expected
Cesf-Ce4fsulphate)
in a redox This
to affect
system
mechanism,
each other
requires
to the
and the hole
at the interface.
to be the case.
experiments, and surface
attention
because
from
just described
This appeared
potentials
of redox
The reduction-oxidation
at the
will
notably
be paid in what
these are more closely questions
those
recombination.
concerning
related surface
concerning
follows
eIectron
emission,
contact
the general picture outlined here.
confirm
to questions
to the “classical”
concerning dcuble
layers
potentials, than,
for
recombinationAdvan.
Colloid Interface
Sci.,
1 (lb’i7)
277-333
POTESTIAL
DISTRIBUTIOSS.
POTESTIALS
291
(2) Potential distributions. Potentials 2.1
THE
CHEMICAL
POTESTIAL
OF
THE
ELEGTROXS.
THE
SASD
SCHEJIE
The conduction electrons will be con&lered as analogous to ideal gas atoms. Two differences compared with ideal gas atoms ~111 be taken into account. First, there is the influence of the crystal lattice. As discussed in textbookslJ on solid state physics, this can be satisfactorily accountecl for by the introduction of an effective mass, MC,*, instead of the real mass 2~~. For germanium, we* z $ w,. Secondly, electrons have two spin states. Considering these two differences, the chemical potential, pe, of the electrons can be written ast: w = E, + kT ln f~rA,~ (8) where E, is the (minimum) energy of the conduction electrons and where fi, = h (2% wc’,kT)- i (9) Here h is Plan&s constant. The two spin states of the electrons are accounted for in eq. (S) by the factor _Sin the logarithmic term. The equatron of the chemical potential of the holes, ph. is the symmetrical counterpart of that of the electrons* CL,,= -E, _t kT In &&A,,~ (10) where E,, is the (maximum) energy of the valence elecrrons and where Ah is simrlar to rle except that wte* is replaced by the effective mass p)ch*of the holes. For germanium, ML~* z + me. It is a parameter with a positive numerical value. The equilibrium situation, given by eq. (2). requires: re tCLh=O (11) The chemical potential, pc,, is usually calLed the Fermi energy and denoted as Eg. The reason is that the electrons obey Fermi-DIrac statistics, which is concerned with the number of way 3 in which M particles can occupy LV quantum states with the same energy. This number, say g, is: IV! g= (12) ts! (N 9L)! B” where /2 accounts for the electron spins. If ILUre are no paired electrons at T = O°K at the considered energy levels, then #4 = f. If all electrons are paired. then /3 = 2. For n $ N this becomes: g = NA (n!)-’ fin (13) This corresponds to Boltzmann statistics. The condition n g N is often fulfilled: The p,;spression ZAe3. denoted as N, in solid state physics, can be identified with the number of states per cm3 available for tc tonduction electrons (with p = 2) : 211,~9 = NC z @.8 x lOID cm-3 whereas +zis of the order of lo*7 cm+ at the most. t
There
should
be no confusion
between
the notation Advan.
p here
(14)
and iCs use elsewhere
Colloid
Ittterfuce
Sri..
for mobility.
I (1967)
277-833
%I, J_ SPARNA.&S,
292
SEMXCOSDUCTOR
SURFACES
With the aid of eq. (131, it is not difficult to derive eq. (8) by means of the methods of statistical thermodynamics. The partition function Z of it eiectrons, ah having the energy E,, becomes:
Here the assumption The Helmholtz
F
*c
-
exp
Nen (n!)-1
Z=L:
Ee
ICT
is made that E, does not depend on the temperature.
free energy F is
=-kTln2
IIf3
and the chemical potential is
fi~Eps=
117)
After tvriting In tz! = rtln 7t - n (Stirhng’s approximation), eq. (S) is easily obtained. Similar reasoning can be applied to the holes. For surface states the situation is different because it is improbable that the Boltzmann approximation can be used. If there are I’r acceptor surface states per cm” and re are occupied by an electron, then by using eq. (12) instead of eq. (I?);:, unefinz&Iyobtains:
where E& is the “adsorption {free} energy”. Each neutral (i.e. empty) state is assumed to have one unpaired electron. Each charged state is assumed to haxe two paired electrons. Thus /3 = 2. For I’D donor states per cm9- and rr, holes, one has in a similar way [here the energy Ea is now replaced by ED. \Ve assume that the neutraf (filled) state has one electron which can occupy two spin states: /$ = +] rh
rrJ
=
1 + 2 exp -
(19)
ED--F
(
kT
>
The same chemical potential EF appears in eq_(lQ) and in eq.(18), because it is assumed that the system is in eqtrilibri~_;m*_ If* returning now to the conduction electrons in a semiconductor, EC, EY and EF are plotted against distance from the surface, the band scheme is obtained (Fig. 4). Disregarding the individuality of the atoms of the crystal lattice, this consists, in the bulk of the crystal, of three paralfel horizontal lines. Of course there are more energy levels in the crystal. These would be represented by hofizontal lines in the picture higher than EC and lower than E, (the name “band scheme” is then elucidated), Moreover, in impure crystals, energy levels arising from the presence of * “Quasi Fermi” This is the case
energies
st the surface are sometimes
when li~;ntis shining on the surface.
introdueed
if there is no equilibrium.
A&JCZSL Colloid Inlevfaca SC&, 1 (1967) 277-333
POTEXTIAL
DISTRIBUTIOSS.
293
POTESTIALS
donor and acceptor impurity atoms would be indicated by short horizontal lines representing levels in the forbidden energy gap. Their position with respect to distance would indicate the position of these atoms. h pentavalent -4s impurity in the Ge-lattice results in a donor level close to E,. Little energy is gained if a conduction electron “falls” into a state determined by an ionized As-atom, and it has already been assumed that ionization of such a donor was complete. Similar considerations apply to the trivalent B impurity which provides an acceptor level close to E,_ However, we shall not consider such energy leveis because we are interested
+ es ,/- _-_--____ $3 ---_--_ +r/
(n-type chttiv~ty)
EF
6
E r
~__*Ep\
W=U
____-
I
Dlstaxe
x -
X=0 Fig. 4. Band scheme.
Symbols
explained
in text.
only in the band scheme close to the surface. Just as in an aqueous solution, close to its surface or the interface with other substances, there will be a non-zero electrostatic potential, y, whose value depends on the distance x from the interface (surface). The energy EC of the conduction electrons then contains a contribution -ey, where --e is the charge of an electron, and so does the energy Ev, the energy of a valence electron. In Fig. -4 the symbol q is used for the charge of an electron. For this reason the bands near an interface or surface are, as a general rule, curved. Owing to the negative sign of the charge of an electron, if the bands are bent upwards, the electrostatic potential is bent downward and vice versa. Eqs.(S) and (10) can be written as: Ep
-
=
pe
Ep =
=
Ee
ph =
bulk
-
-
ey
+
E, bulk +
kT ln +1&s
G33)
ey +
(31)
kT h f&/1$
In the bulk of the semiconductor (x = 00) the electrostatic potential y = 0 and also d+lx = 0. It is seen that for any value of y the product n x p has the same value, given by:
Taking A, = An =c- 4 and denoting 90 = Ep - Es bulk. where El bulkis the energy vafue just helfway between Ee and E,.in the bulk, or also: where El is the Fermi energy ior intrinsic material, one obtains for >z and $:
tt = ttt erp
Qlb k2.
eyt exp kT
ni = 2/l-”
exp
-
(Ee 2kf
E.4
t-Z3 ,
Me&d
bkculor denser
‘-L=
-
Li m_m- 88
_e3
-
m
Fig. 5. Comparison between Gouy and spacecharge:a) liquid. Circles surrounding(+-)or (-) signs represent ions: b) semiconductor. Squares surrounding (+) signs represent charged donor atoms, fixed in thelattice. Open (+) signs represent holes. Open (-} signs represent conduction ektrons. The surface charge is chosen negative (bands curved downward). and this is effected by meazrs of afczptor states. The Iquid corresponds to an ‘“intrinsic’” semiconductor.
For n-type material the Fermi energy is in the upper half of the energy gap, i.e. q% > 0, and for #-type material it is in the ~owferhdf. The case qb = 0 (intrinsic conductivity) corresponds to that of l-1 ionic solutions (see Fig. 5). Since we have
POTESTIXL
DISTRIRUTIOSS.
assumed
full ionization
POTESTIALS
rf donor
ND and A*_%respectl\.el~.
these
concentrations
Flat
surfaces
2%
and
placed so as to coincide
their concentrations,
as:
being independent
and interfaces
impurities,
ilCCeptOr
can be written
c>f the electro5taric
Gil
be consldered
only
potential.
and the plane x =
with the plane of the fast surf-ace states.
0 Is
Here, the electro-
static potential is denoted b_v PO_ The energy of the electrons in the surface states thus contains a contribution --Ed+ It is assumed that the outermost layer of atoms is polarized. potential
Then,
jump.
interfaces.
just
as at the surface
the x-potential,
If there
of other,
is present.
is an oxide
layer
Similar
solid
or liquid.
potential
jumps
materials.
a
also occur at
OS in the Rardeen-~rattai11~ttaixl model,
the in-
---_ --Jr+-
T +!w
0
---XT
----$2 _-_’ ____ V&i
_ _-- -_
3i_--__ *
cj
xE
Fig. 6. Bardeen-&attain model’8. a) Semiconductor covered with an oxide layer. An electrode for measuring field effects is also shown. An electric current J II Indicated. The charge in the states is modified by the applied potential and so is the charge CT*, sf the space charge region. The modification of the value of the space charge leads to the field effect. b) Energy Itwels. e) Potentials.
fluence
of the slow
surface
should
JV,
=
state;and
~&-OX
d-
se
where xseWox and x,,~-,,~~ oxide
interface
difference
the molecular
also be considered bulk
-
condenser
eYo
+
Xox-vae -
refer to layers of polarized
and to the outer
across the oxide
at the outermost,
(see Fig. 6). The work function
oxide
surface.
IV,,
EF -i- VOX atoms
oxide,
is (27)
at the semiconductor-
respectively.
VOX is the potential
layer. -4dvaa.
Colroid
later/ace
Sci..
1 (1967)
277-333
296
M.
J. SPARNAAY,
SEMICONDUCTOR
If no oxide is present [for example because the semiconductor been thoroughly cleaned). eq.(27) reduces to: rv,, =
x + E~t,ur~-eyo-E~
SURFACES
surface has (27a)
The work function is defined as the difference between the electrostatic potential just outside the crystal and the chemical potential of the electrons inside the crystala=_ The value of IvSc will vary from face to faee of the crystal and these variations can be due to the difference of the X-potential and those of yo when going from one face to another. If two semiconductors. or a semiconductor and a metal, are electrically connected, equilibrium requires that the chemical potential of the electrons (the Fermi energy) has the same value in both materials. The difference between the two work functions of the interconnected materials is the Volta-potential or the contact potential difference, 4c.p. (see Fig. 7). This potential difference can be measured if one of the materials is set in vibration. An alternating current then flows.
Fig.
7. Potentials
plained
in text.
pertinent
to metal
surfaces
and
to semiconductor
surfaces.
Symbols
es-
A variable potential difference is usually applied to the circuit which makes this alternating current zero. The value of the variable potential difference gives the value of the 4c.p. This is the Kelvin or vibrating plate method, which is used for a wide variety of surfaces of semiconductors and metals, solids and liquids. The photoelectric threshold, vse differs from the work function in that it should be counted from the edge of the valence band (see Fig. 7j. q&xc=
W*e --
EF +
Ev surface
WI
incident on the surface penetrates in many materials Light of frequencies 2 h-+,, to a depth of several atomic diameters and leads to the excitation of valence electrons. A fraction of them leaves the crystal and is measured as the photocurrent. The photoeIectric threshold is given by 10 = h-rpse. Since the penetration depth is, in a number of cases, an order of magnitude smaller than the depth over which the bands are curved (this depth is of the order of the Debye-Htickel length in the material), the values obtained in these cases are characteristic of potential values at the surface. Advan.
Colloid Interface Sci.. 1 (1967) 277-333
THE
SPACE
CHARGE
297
In metals the difference between vae and Ws,, disappears, at least in principle. There is no energy gap and the concentration of conductivity electrons is of the order of 103z cm-a. Fowlerfa calculated the photoemission of these electrons as a function of the photon energy, the photoelectric threshold appearing as a paraThe \*alue of 2w.p. meter in this theory. “Fowler plots” provide a value of T1-urerar. between the metal and a semiconductor then gi\*es a value of Il’ser,~tCOndUCtOr. This value can be compared with the value of r-faeot thesame semiconductor surface. R-v means of eq.(ZS) a value of EF - EY lur~aee is thus obtained_ Theexperiments can be carried out on various samples of the semiconductor with different impurity concentrations. The balk x*alue of Ep - E, can be varied by varying the impurity concentration_ As pointed out. measurements of qsc and IV,, provide srlrJnce values of E,. Thorough investigations of the case of a clean Si surface have shown that EFthe srtrfuce value of E B - E, varies much less than the balk value. Thus the value of elyo varies almost as strongly as the bulk value of EF - E,. This means that the space charge in highly doped p-type material is negative, this negative space charge being compensated by excess positive charges residing in the surface states (empty donors). In highly doped tl-type material the space charge is positive and there is an excess negative charge in the surface states (occulted acceptors). of the energy distance EF Ev at the surface can Such a “stabilization” evidently take place only if the den&v of states is high. It should be large compared with the number per cmf of charge carriers in the space charge, which 1s limit of the density of states to usually less than $01” per cma. Therefore a lok%eer have the required stabilizing effect will be at least 101s per cm2. In fact, for cleaned Si surfaces in high vacuum (pressures less than 10 -9 torr), state densities (donors and acceptors) of about 101s per cmz have been reportedz’=ss. From field effect measurements even higher values have been foundl6. For metal-semiconductor interfaces Mead--5a formulated a remarkable rule. based on numerous experimental data concerning a wade x-ariety of combinations of metals and semiconductors brought into intimate contact. According to this rule the Fermi energy at the interface is stabilized in such a way that (E, - EF) =: ~(EF - EY). It should be stressed that a large density of states such as is necessary here does not imply a large density of states at the clean surface of the pertinent semiconductor. This is pointed out by Van Laar and Scheerssb for the case of (cesiated) GaAs.
(3) The space charge 3.1
GOUY
LAYER
OR
SPACE
CHARGE
LAYER
Gouy*-6 was first to give a quantitative treatment of the space charge problem. He appIied his results to the Hg-aqueous solution interface. Gouy’s theory Advaw. Colloid Znfsrfacts Sci.,
1 (1967)
277533
31. J. SPhRh’hhY,
298
SC3lICOSDt*CTOR
SURFACES
has now become standard. The space charge in ionic solutions bordering a solid, a
liquid or a vapour is now called the Gouy layer, or, less frequently,
the Gouy-
Chapman26a laver, Gouy’s theory has also been applied to the problem of the distri_ bution of ions around charged hydrophobic collo;dal particles and it has been extended to the case of two particles at a distance which is of the order of the DebyeHEckel length and less. More recently (see Fig. 5b). Gouy’s theory has been applied to space charge problems in semiconductors although it was presented there as a new theory. In both cases (semiconductors and ionic solutions) the Boltzmann eqs. (23) and (24) (and eqs. (25) and (26)) should essentially be combined wit?1 the Poisson equation: div (c grady)
=
-
42~
(29)
where 9 is the charge density at the position considered. This is written as: e =e(fi3.2
It +
SIJIPLIFYING
f\‘n -
n’,)
(39)
ASSUJIPTIOXS
1e consider the charge carriers as point charges. This is certainly a better approsimation here than for ions in an aqueous solution which are surrounded by
hydration shells_ Furthermore it may be recalled that the concentrations p. 12,f\‘D and LVX indicate average values. This makes the potential y in eqs. (23) and (24) an average potential. In contrast, the potentia? appearing in the Boltzmann equations
is the potential of the average force, grady [=(kT/lre) gradlt for electrons; = (kTf+) grad+ for holes] as pointed out long ago by Onsager”*. x+Ireidentify these potentials. In doing so, fluctuations terms are neglected, as can be shown by a statistical analysis_ The problem has been thoroughly studied for ionic solutions (ref. 28,!29). The result is that the potential calculated when neglecting thefluctuation terms approaches the zero value somewhat “too rapidly” as the distance is increased, the whole effect amounting to a few percent in the final ,numerical values of double-layer parameters. We assume the same to be true for semiconductors. Finally we recall that eqs. (23) and (24) are Boltzmann approsimations of an analysis based on Fermi statistics and that this is justified if - (EF - &) s kT and (EF - EY) 9 kT. Special numerical examples with Fermi equations instead of Boltzmann equations were worked out by Seiwatz and Greenso, who also took into account possible effects of incomplete ionization of donors and acceptors. Another
correction which may become of importance is that which arises from a possible quantization of the movements of the conductivity electrons in the space charge. This quantization may occur if the value of ~JUvaries much more than kT over an interatomic &stance, irrespective of the values of EF - Ee or Ep - E,. This interesting problem is as yet far from solvedsr. Neglecting the possible corrections enumerated here we shall use a simplified form of the Poisson equation: divgradv=-4n$
(31) Aduaa_ Colloid Interface
Sci..
1 (1967)
277-333
THE
SPACE
299
CHARGE
Thus, it is assumed that the dielectric constant E is inciependent of the field strength, This is certainly a good approximation for most, if not all, semicouductor crystals in which we are interested. For aqueous solutions, in which the influence of the field strength upon E is likely to be greater, the effect can usuaIIy be neglected=, For the flat-plate case the (Laplace) operator, div grad, becomes dZ/cW. Any plane .Y = constant, is an equipotentlal plane.. For x # 0 it can be shown that this assumption is consistant with the identification of the ax*erage potential with the potential of the average force. For x = 0 it means that the surface charge is assumed to be smeared out, i.e. that “LTiscrcteness-of-charge” effects are neglected. It is difficult to give more than a quantitative estimate of the consequences of the equipotential assumption, for one thing because it is not known how the charges on the surface are distributed. I(ryIo%~33showed that in the Stern layer the potential should decay faster to zero with increasin,= x than would be the case assuming a smeared-out sumface charge. \Ve return below to di&reteness-of-charge effects (section 6). For the present, however, we assume a smeared-out surface charge. 3.3
THE
POXSSOX-BOLTZMAXX
EftUATfOX.
THREE
APPROSI.UATIOSS
The Poisson-Boltzmann equation that fits our pcrpose is obtained by combining eq. (31) arxl eq_ (30). Eq. (30) is rewritten as follows: e=
-2
e8zi
5 ey (siair ‘Fb kT -
siuh
qb kT )
(32)
We shall use the following notation:
the Poisson-Boltzmann
equation reads:
=
sinh 2(b)
L-s(sinh(ub
+ r) -
The surface charge, compensating is given by: E dW use = 4n ( -Z-Z=O=>
(33)
the total charge per unit area of the space charge, ekT one
dy ( d.r ) xiliio
(34)
eq. (33) should be integrated_ The integraIn order to obtain an expression for a Ottr tion can be carried out by means of: _
Then the root must be taken. The positive r-f negative. Then:
is chosen if the space charge is
31. J. SPARS,4AY,
4tkTL ,
use = *
42
Kingston and Neustadter numerical function
values
[cash (zzz-,-j- Z) -
3-1 have
SURFACES
SEMICOXDUCTOR
cash zzb-
Z sihn z&j*
given ubCas a function 6f zzSand Zzband calculated
(see Fig. 8). The second
of x. will not be undertaken
integration,
here. Instead,
which
should
provide
y as a
we give three useful approuima-
Cons of eq. (35).
-16
-12
-8
-4
0
f6
Accomuiotlon layer
Fig. 8. Value of the space charge use as a function of u J and ub_ The function ZQ,bSc. can be obtained from eq. 35 and is i; = &2 [cash (ut, f =) -cash t = itstl,,
F_ proportional to z sinh ub]: hmwhere
(a) The Mot&Schottky apjmrimation 35a36. This is obtained for 1Cb 9 1 and “-type material, one 3 4 0, or for -#lb 3 1 and z 9 0. Thus, for strongly-doped has; &
bee
=
(
ND kT t [eL > 2Jz
,I -
114
For large negative z-values. cr,c is proportional to lzli. Thus we have here the case of an n-type semiconductor with the bands bent up quite strongly. It should
be noted
that
a Debye-Hiickel
length
can be defined
here by using ND
instead of ~1. (b) Both tib mrd E me smatC, positive OYnegalive. Then: Qac =
e 4nt
(37)
w0
and zzbhas altogether disappeared i.e. only a term containing #b’ would have been Advaw
Colioid I~~terfuce Sci.. I (1967) 277-333
THE
301
ChPACITASCE
present_ Eq. (37) is the same as that obtained ersf?a .C_
For jq >
for small potentials in Gouy lay-
1, this simplifies to:
where ose > 0 for z > 0 and czc ( 0 for J -=z 0. These three approximations fill prove to be ver>* useful in the following sections, where the differential capacity, 3a/3ye, and the free energy of the double layer, - Jce adye. are considered. *
(4) The capacitance 1.1
C.4PACITASCI:
EQL’rlTIONS.
ORDEIPS
OF
JIAGSZTUIJE
Capacitance equations xvi11 be given here for a semiconductor-aq -soln. interface rather than for a semiconductor surface, capacitance measurements m the latter being relatively mnccurate. Knowledge concerning dr_v surfaces has mainly been obtained by other methods (see previous sections). Lvhereas knowledge of semiconductor-aq.-soin. intertaces is to a fairly (see, however, ref. 63) large extent due to capacitance measurements. The capacitance of such an interface can usually (prox4ding electrical leakclgc across the interface is not preponderant) be measured if it is made part of a cell in which, apart from the semiconductor electrode, the other electrode has ;L large surface area, thus making the capacitance of this electrode large. A simple model of the interface will be considered, in which it is assumed that the charge at the semiconductor side of the interface can be written as 0 =
orlc +
CSs +
oion
(-10)
where otscis the contribution of the space charge, asa that of the surface states and where oren is the contribution of an ionized layer which arises from the chemical interaction between water and the semiconductor. For the (anodized) Ge-aq.-sohl. interface this layer is composed of ionized HeGeO,+l groups. Here _1cis a number which is 2 if the germanic acid is an independent molecule. However. we consider a germanic acid group which is still chemically bound to the Ge crystal and it is likely that x is smaller than 2. It is possible that this “ionized la_ver” can be identified with a layer of “slow states”, but, since little is known of the physical nature of slow states. we prefer in our simple model to use the concept of an ionized layer. ~dvan.
Colloid
lntcrface
Sci., 1 (1967) 277-333
bf. J. SPARNAAY,
302
SEMICONDUCTOR
SURX’ACES
At the sofution side of the interface the charge o is compensated by the charge of a Gouy layer and a charge in the Stern layer, which is formed by specifically adsorbed ions. The potential drop Vtot across the interface will be written as: ytot =
yo +
g +
yso1n
(41)
where T,UO has the same meaning as before, x is the potential drop across the molecular condenser at the interface and vaoln is the potential drop across the solution (Stem layer and Gouy layer)_ Following the same method as that indicated by Dewald37, the capacitance C is written as (42) where, assuming that 2 remains constant if ytot is varied, WY0
=
I--
‘;)ytot
ayso1n
(43)
bytot
The foltowing differential capacitances are defined:
and
Csoln =
aa dqJso1 II
Tt is not difficult to obtain the following expression for the capacitance: c-1
=
C&n-- 1 +
(CM2 5
C&1
(1 -
CionCsoln-1)
Orders of magnitude are now given. From capacitance measurements of the Hg-aq.-soIn. and the AgI-aq.-soln. interfaeess’a, it has been found that C Boln is at least of the order of 10 FF cm-‘, depending on the electrolyte concentration_ For C=+ theoretical curves have been given by Bohnenkamp and Engell 19. These are reproduced in Fig. 9. The curves show minima just as do the capacitance-potential curves of the Gouy layer. The numerical value of the capacitance C sc in Fig. 9 is of the order of 0-l FF cm-l. It shouId be emphasized that this is a much lower value than that of Csoln. Indeed in many cases one has: CfJc% Csoln
(45)
Concerning C Bc, consider the three approximations given for ose at the end of the previous section. a) The Molt-Sclwttky zzpproximation. Differentiation of eq. (36) with respect to ‘190and taking z large negative, gives:
Advan.
ColIoid Interface
Sci., 1 (1967)
277333
THE CAPACITANCE
’
Fig. 9 Theoretical capacitance-potentid curve for Ge-soln. sidered. From ref 38 (Bohnenkamp and Engelt).
b) c
Low 2x?=
irx@rit_y & 4xL
cowetrtration
interfaces.
arrd low ~ote-nfial.
303
So surface states con-
One snnpl~ has
which is the same equation as that obtained for the low-potential approximation of the Gouy capacity. Only the Debye-Htickel length x-1 is replaced by its equivalent L for semiconductors. Eq. (47) in particular illustrates why the inequality (43) is -not surprising: in many instances L S z-1. c) Low iwq5zfrity coxcedration and high ~0tehkI. From eq. (39). one has:
c
SC
=
-
e Qsc
2kT
Concerning surface states, let us consider one set of donor states and one set of acceptor states. From eqs. (18) and (19) and differentiation with respect to ~0, one obtains:
c
ES
=
- tT
where .%A
=
2 exp.
zD
=
2-l
t
e
r*.xa
(1 + .ZAJ2
(491
1
exp
kT and EA and ED are the energies of the state at yo = 0, Plots of Css against ~0 should show maxima at eye = EA Advas
Culioid
EF -j- kT
ln 2 _
Interface Sci.. 1 (1967) 277-333
51. J. SPARSBAY,
SEMICO?r’DUCTOR
SURFACES
andateyo= ED-EE~kT In 2. For very small x and very large x, Boltzmann approximations are valid. On the whole, CSSis of the order of 0.1 PI? cm-l, if there are about 1010 states/cm”. Its value is of the same order of magnitude as Csoln if the surface-state density is about lOl%m-2’. Concerning the contribution of the ionized layer, Iet us consider the case of weak ionization. As derived by Payens 3% for the case of weakly-ionized monolayers at the wnter*il or water-air interface. the surface charge oion is proportional to esp eyttioln/kT. Then:
c
e iOR
=
k?’
uion
As we shall see in section 5, the case of adsorbed gases, which are weakly ionized, is similar to the case considered by Payens;. In fact. the cases of weakly-ionized monolayers, weakly-ionized groups chemically bound at the surface, weakly-ionized gases on the surface and charged surface states have a common theoretical b&as&. In view ot eq_ (14). it is useful to compare values of Cion with those of Csoln. If crion is providecl by IO’“- charged groups/cm”, the same order of magnitude is obtained for Cion and Csoln_ IVe conclude that the measured capacitance C is often determined by the value of CS, or of (Cse + C,,). Finally we note that the contribution Csoln may be analysed in a way analogous to that followed by Dewald. Thus, if it is assumed that the charge u of a double-layer system is determined by the charge in the Gouy layer only. and if it is furthermore assumed that the thermal movements can bring the ions to a distance from the charged wall, which is not shorter than a certain value 6. then C-1 can be written a@: cion-1
=
3WbOln
=7i
30
3 (Y8oln aa
3w
W)
=
co-1
+
en-r
(51)
where yb is the potential difference between that of the plane -2:= 6 and that of the bulk of the solution (x = co). Co is the Gouy capacity and CH is the Helmholtz capacity. This is often written as .&&~a where EII is the dielectric constant in the region between the planes n = 0 and x = 6. Eq. (51) corresponds to eq. (44) in the absence of surface states and ignoring the contribution Clan_ If now. apart from the Gouy layer, specifically adsorbed ions (charge da,-&per cm=) are present. i.e. if a Stem layer is present, eq. (51) should be modified. The charge in the Gouy layer is now no longer determined by Q, but by (0 -
dads)-
Then instead of Co-l
Wd = 3u
one has:
3 (a -
3yd 3 (0 -
uads)
-
'hds)
3a
-
cc-1
(52)
It is assumed here that the specific adsorption has taken place at the plane x = 8. This is a simplification. Often the adsorption takes place between x sz -&a and x z b (see ref. 39a, and b). Advan. Colloid InferfaceSci..I (1967) 277-333
303
THE CAPACITASCE 4.2
ESPERIMESTS
3feasurements
Bohnenkamp current
of the capacitance
and Engell IQ. Their
density
being
of Ge electrodes were first carried out by
electrodes
10 p_?(Lcm-* - or
were
alightly
anodized,
the
anodic
more_ They obtained curves which to a large
extent could be explained by space charge theory aione (Fig. 10). Thus, the curse
7
600
Fig. f0. Experimental capacitance-poteatiatl curves fur Ge-soln. interface. Ge-electrode anodized_ Solution, 1 X KOH. Eh potential referring to hydrogen electrode. Frequency range indicated. From ref. 19 (Bohnenkamp and Engell).
capacitance versus applied potential was found to have a minimum. The capacitance minimum was of the right order of magnitude to be expected from space charge theory. This indicates that under these circumstances surface states were absent. They also found a frequency effect at frequencies ranging from 10y cycles/ set to lo6 cycles/set. This may be ascribed to a decreased participation of the ionized layer in the charge transfer at increased frequency. This would lead to an the time effects apparent decrease of Cr,, at increased frequencies. Alternatively may be ascribed to time effects of the X-potential. It does not seem probable that surface roughness plays a role here- Frequency effects due to surface roughness can be expected to occur at frequencies between zero and 10s cyclesjsec as pointed out by De Levie40. If, prior to the measurements,
the Ge electrode is cathodized, Adva~~ Celloid Interface
rather different
Sci., 1 (1967) 277-333
306
3%. J.
SPAKNAAY,
SEbIICOKDUCTOR
SURFACES
capacitance values are found. Thus, Efimov and ErusaIimchik4r, applying a cathodic current density of 3 mA cm-e, found capacitances (in acid solutions) of about 10~ F cm-s, the capacitance-potential curve showing a shallow minimum accompanied by “shouIdcrs” (see Fig. 11). Here surface states will be present, their
60-
&I--
20 -
Fig. Il. Experimental capacitance-potential curves for Ge-soln. interf;sces. and Erusalimchtk). dized. Freq. 200 cycIes/sec From ref. 4 1 (Efimov
densitybeingof
the order of
Ge-eiectrode
catho-
The frequency used was low (200 cycles/set). Therefore surface roughness and electrical leakage might h;we played a role here. (A method for obtaining “state-free” interfaces by means of a cathodic treatment, was recentl+a reported). At the same pH value, the position of the capacitance _ minimum in the “cathodized” case is about 500 rnv more negative than in the “anodized” case. These two cases may be seen as two extremes~r. This is in agreement 44th the findings of &attain and Roddyao, who measured capacitances as a function-of the time elapsed after switching off the anodic current, applied during 1011 cm-z.
the measurements. It appeared that the potential of the capacitance minimum shifted about 200 mV in the negative direction. This may be interpreted as a shift of the X-potential combined with a change of the charge of the ionized layer and/or a change in the charge of a Stern layer. These effects are probably also related to the crystallographic orientation at the interface, because different results were found for different crystal planes of the Ge electrodes. The low-frequency objection was raised by Krotova and Pleskov4s against the measurements of Boddy and Brattain. However, the latter authors found little differencedzb between data obtained bv d-c. and by low-frequency a.c. measurements, at least for neutral 10-r M &SO4 solutions. Krotova and Pleskov, working with 10-Z M NaOH solutions, reported the existence of 1012 surface states per cma. Therefore the disagreement must probably be attributed to different densities of surface states. Advun.
Colioid
Intevface Sci., I (1967) 2774333
THE
307
CAPACITAXCE
At extreme pH values, fast states are rapidly formed and, as reported by Bohnenkamp and Engell, a more or less state-free interface can be maintained only for a few minutes after switching off the anodic current. If, however, Hz02 is added to the solution, the interface is stabilized. This was found by Brouweres. A full interpretation of Brouwer’s results, which is at variance with the interpretation in refs. 19, 42 and 46, will be published. The surface states formed upon cathodic polarization may be due to germanium hydride complexes, or to adsorbed cations. Thus, BoddJ* and Brattain”” concluded from the increased capacitance x-alues after Cu’+ addition to the solution that surfac= states due to coPper are formed. State formation at low pH values can probably not be explained along these lines, because at low pH, eopper is desorbed. Ions of less-noble metals are desorbed even at higher pH values. Other methods, such as the measurement of surface recombination velocity (Pieskov rt 11.458, Ham-ey45b) are considered in reviews by Cerischerl9 and. most recently, by BoddylG. The role of the pH, which seems to be more related to that played in colloid chemistry, mil! be discussed in section 6. For the ZnO-aq.-soln. (pH = 8.5) interface. Dewald found from capacitance
Fig. 12. (Capacrtance)-2 plotted against potential for %&borate buffer soln. (pi-f, 8 5). Straight lines indicate validity of Moot+Schottky approach in this region. From ref. 37 (Dewald), Rdvna.
Colloid Inter$zrc Sci..
I (1967)
277-333
308
M. j. SPARNAA’S-,
measurements
that
in carefully
purified
surface states were almost completely
borate-buffered
SEMICONDUCTOR
1 X
KC1
SURFACES
solutions
fast
lacking, their density being less than ID* per
cm*. He used ZnO crystals with impurity concentrations leading to bulk electron densities up to 101* per cm 3. At this density an unexplained anomaly occurred. Apart from this anomaly, ZnO is particularly suitable for these measurements, because the impurity content can readily be varied and, since the energy gap is about 3 V, a wifie range of potential variations can be covered by the experiments. approxiThus, Dewafd found that over a range of nearly 3 V the Mott-Schottky mation applied (see Fig. 12) _ Measurements were also carriedout in the region where the Poisson-Bultzmann erjuation should be replaced by a Poisson-Fermi Here also gcod agreement with space charge theory was obtained.
(5)
The integral J**~dy+
3.1
ADSORPTICIS
AKD
equation.
Adsorption equations DOUBLE-LAYER
PARAMETERS
The adsorption of neutral substances may affect the value of the parameters
of the double layers at the interfaces
or surfaces where the adsorption
has taken
place. In principle it is immaterial whether these substances +a.reionized zx nonionized prior to the adsorption. 1Ve illustrate this statement by two esamples which can be considered as estremes and by a third example \vhich is intermediate. Firstly, in the case of a colloidal solution of Agl particles, the addition of a completely ionized sotution of Agh’Os results in the adsorption of Ag+ ions, which makes the charge at the AgI-aq,-soln. interface more positive and the surrounding solution more negative. Secondly, the addition of the completely non-ionized system often leads to charge transfer, the a+ oxygen z;“= to a semiconductor-gas sorbed oxygen being negative and the semiconductor becoming positive. A third example may be added: the addition of a partly ionized long-chain fatty acid HA to an oil-aqueous
solution interface leads to the adsorption
of non-ionized
mole-
cules HA together with ions A- at the interface. The degree of ionization at the interface usually differs from that of the non-adsorbed acid. This example may therefore ‘be considered as intermediate between the other two. These three systems: (u) the AgI-aq.-soln. system, (b) the oil-HA-aq.-soln. system, (c) will often adsorbing interface,
the semiconductor-& gas system, serve to illustrate the considerations presented in this section. -Other substances may only indirectly affect the charge distribution at the for example because they have a special dipole orientation. These will not
be considered.
Knowledge of the adsorption isotherm of ionizing agents should yield inA&an,
Gdt5hi
fnlerfacc Sci..
1 (1967)
277-333
ADSORPTIOS
309
EQUATIOSS
formation concerning the charge redistribution at the interface. Such isotherms. are considered here from a thermotogether with appropriate equations of state, dynamic point of view. 5.2
THERMODYSXMLC
FUSCTIOSS
An electrically-neutral system will be considered which consists of two bulk phases and an interfacial phase. each phase being electrically neutral (see, however, subsection 5.3). The reason for the introctuction of a separate interfacial phase is our use of the interfacial tension as an independent variable. The int rfacial tension is a characteristic of the interracial phase. Our treatment bears a close resemblance to the “first method” of \‘erweyp7d and Overbeek (V.O.)“e.d and to the method of Frenkel 4Qfor the determinatirx~ of the free energy of the double layer. This 141 be discussed in section 5 3. Charge transfer from one phase to another is assumed to be reversible. Thus we preclucle here a system containing an ideally polnrizahle interface such as the Hg-aq.-soln. system. \Ve follo\v Gibb5a8, who detined the interfacial phn+e by a subtraction procedure. First the two bulk phasea were taken to be homogeneous right up to a hypothetical dividing surface. Gibbs showed that the choice of the location of this surface is allowed to be arbitrary, at least for planar interface-. Second, the phys~cat situation, Le. the heterogeneity of the interfac e, was considered. The thermodynamic quantities were then defined by subtraction. The cluantlties pertaining to the interfacial phase are then excess quantities. The locatiw of the Gibbs dividing surface is usually chosen such :LS to make the excess mash of one ot the main constituents a bulk phase) zero. The Gibbs function of the interface is: G” =
G -
(G1 +
where the superscripts
Gl[)
(53)
s, I and II pertain
to the interfacial
The quantity without phases, respectively. whole. Similarly one has for the Helmholtz Fs =
F -
(P
+
(of
superscript
phase and the two bulk
pertains
to the system as a
fret: energy:
PI)
(34
For bulk phases the relation between the f_;ibbs function and the Helmholtz free energy is given by: Jx =
Cl _
plYI;
FIX =
GII _
&l/II
(55)
where p is the pressure upon the system, and Vi and I/lx are the volumes. Only if the interface is curved are the pressures in phases I and II different. Forinterfacial phases one has: I;” =cP$_yA
(56)
where y is the interfacial tension and A the interfacial area. Thus the Helmholtz free energy of the whole system can be uiritten as: Advan.
Colloid
Xntcrfaca
Sci.,
1 (1967)
277433
.M. J.
310 F =
G
SPARNAAY,
SEBIICOSDUCTOR
SURFACES
--_pV3-~A
(57)
where V = Vr + VII_ The interfacial phase in Gibbs’ sense has no volume of its own. Assume now that at a certain composition of the system the double layer potential is zero, and that the reversible addition of a neutral substance leads to a non-zero value ps. For each of these two compositions of the s_vstem an equation similar to eq. (57) can be written down. Subtracting
AF=
these two equations
gives:
AyA
AG-AapV+
(5%
where AF is the difference between two Helmholtz free energies, to be denoted as F - Fo, and similarly for the other quantities. In pnrticular, the difference AC can be written as: 4 G =
G -
where ,ur and @I),,
G, =
C plttrl -
r: (&&z~)~
are the chemical
potentials
of the neutral components
(59) i, pai
and (PM&, being the number of molecules of these components in the system. Thermodynamic equilibrium requires a uniform value of pi in each phase and, if reversible transport is possible between two phases, ~1 should have the same value in these two phases. Characteristics of the double layer do not appear in the sum X ~_LS $>rrand C @t)O(ll~t)O. Thus, considering by way of example the double layer at the AgI-aq.soln. interface, we can vary the potential ye by the addition of the neutral salt AgNOs or of KT. Tf the only change taking place in the system is the addition of KT, then more I- ions are adsorbed at the interface, pe becomes more strongIy negative, and
A G = ~~~ IHKI-
(,uKI)o
(j)tKI)o
+
+ksttk
-
(~k)o(%c)o
VW
collects the contributions of the uncharged comwhere &.&?1rk - (&o(9z& ponents k which arise as a consequence of the Gibbs-Duhem rule. In the case of dilute solutions these contributions are small. It is not necessary here to consider the I<+ ions and the I- ions as separate components. SimilarIy, for a double layer consisting of a fatty acid HA one has:
AG
= pHAf?bIi;5 -
@HA>CN
(nC~h)*
+
+km)k
-
(pk)&kb
(61)
Addition of HA means an increase of the adsorption of both HA molecules and Aions, but this aspect does not appear in eq. (61). Thirdly, consider the adsorption of Oe on a solid surface where it becomes partly ionized and gives adsorbed Os- ions, the electrons necessary for this ionization being extracted from the rurderlying solid, a semiconductor_ Then:
AG
= PO, mos&-
bOz)o
(ntOp)o
+
&&mk
-
(pk)o(mk)o
(69
Again the role of the charged component (the Os- ions) does not explicitly enter in the expression of AG. However, the charge of the dotible layer dots appear explicitly if expressions A&m.
Cdoid
Interfucc
Sci.,
1 (1967)
277-333
for A0 {the change of the Gibbs function pertaining to the interfacial phase) are written down. In order to gix*e such expressions, it is necessary to reconsider the chemicaf potentiaf of charged components, i.e. of ions, e1ectrous and holes. For charged components, adsorbed of non-adsorbed. it is cu?ztomar\r to split up the chemical potential into a “~xifel~- chemicat” gxu+tam1 a “purely electrostatic” part. This separation has met with objections -1*ahut, at least to ILtirst approximation@, it is justifiable. The chemicni potential of a charged component j is called etectrochemical potential and denoted by t/J. IFof charged Ixwticles in a space-charge region where the pntentiai is; y, ant? k~s:
space-charge-free region, in the bulk of phase I, sny, the electrochemical potential reduces to the chemical potential. Since grarly = 0, the potential is a constant. This constant will be taken zero. Then: fn
a
*7i =
P3
W
A third espression for *]j is important, namely if the component j is pfesent at the interface and participates in the formation of the interfacial charge cT. Assuming that the adsorption has taken place at n plane where the potentla: is y*o -+- g we
hZX\Ve: “ s.” inrlieates tkkt the plfdy chemiczll paft Of #)J iS a CflafaCwhere the superscript teristic of the interface (surface). Often the z-potential isaassunled to he independent of the charged state of the double tayer. Then, for t+ = 0, we write: (?]Jf o $iow
=
(pJ’)o
in A@ X&J x&J’
them
t
we distinguish VI&f -
three
contributions.
Firstly:
(qr)o (~~~jShJ=
them figJs
-
@Js)o
This is the contribution provide
=Jt%
the interfacial
ehem
(‘jzJ”)o
+
airy0
ii]
=
h (;J"
due to charge carriers adsorbed
charge
at the interface:
(67) they
cr.
The second contribution is provided by charge carriers which must necessatily be present to compensate the interfacial charge (T. Their total charge nmst be rrd_ Labelling these charge carriers with the subscript *‘a*’ one has for the second contribution:
Advati. Colfoid Ink&ace Sci.. 1 (1967) 277-333
X. J.
312
SPARNAAY,
SEMICONDUCTOR
The third contribution is provided by neutral components. by the subscript “b” one has: ~&bmtf
-
(@&&W)d
=
il
SURFACES
Labelling
Gbs
them
w9
AGE one has A0 = AGf + AGk + AG# Concerning Gj *. this can be written as AGja = AGjs ctwm t AGes For
(71) (72)
where AGl’
ehem
=
C&f
cbem
fjtiB -
@f)o
ehem
(73)
(f@)oI
and AGe= =
b?#!Po A
(74)
Concerning the sum AG,a + AC b8, it has been shown that the ionic components can always be arranged so that the charge -u is provided by only one charged component, the rearrangement being such that further only nexhxd (salt or otherwise) components remain. This rearrangement leads to AGas +
hGb* =
,&~=a8 +
T;‘&b~=b’
-
(75)
(,llb)&b’)oJ.
where the prime on the summation sign indicates that the summation is not identical with that in eq. (69). The term yrrp~t a* can be written as: aA (+e)-rp8. Eq. (71) can now be rewritten as AG* =
CTA(PO i-
(&e)-’
pa) 3-
x’@b
?*tbs-
(cLb)&bs)ol
$_
&GJ*
ebem
(76)
Thus AGs contains a term which is characteristic of the double layer. For the A@--aq_-soIn_ interface the interfacial charge is: 0 =
e(I-&+
-
I‘I-)
(771
where we have used the notation 1” = A --ItztS.If the double layer is formed through the addition of AgNOs, then I’ dg+ > 0 and rr-- < 0. The compensating charge M is partly, not completely (see ref. 27~1, given by nitrate ions, and in eq. (76) nitrate ions can be chosen as component “a”. For the oil-HA-aq.-soln. system we have: d =
er.&-
(755)
Non-ionized molecules HA will also be adsorbed. H+ ions wiil contribute to the compensating charge --o-. and the H+ ions can be chosen as component “a”. For the semiconductor-& system we have: a=-Zers
a-
(79)
Non-ionized 0% molecules will a&o be adsorbed_ The compensating charge is here given by holes, and as p, we use here minus the Fermi energy. We now want to find au expression for AF. This can be done in a number of ways. We prefer here to discuss first expressions of AhyA (or of Ay, because the interfacial area A can often be kept constant when a double layer is formed). A&a*.
Colloid I&++zc
Sei.,f (1067) 277-333
ADSORPTIOZ:
313
EQUATIOSS
The reason is that, according to eq. (56) or eq. (57), the main difference between the Helmholtl: free energy and the Gibbs function is provided by a term containing the interfacial tension. This is an experimentally accessible quantity. Jloreover. the interfacial tension can be used to discuss equations of state of adsorbed particles;. For the interfacial tension we have the Gibbs adsorption equation, which is written here in the following form dy =
-
a (dly, -j- e,-1 d&
--
27’rb clp~b
(60)
using the adapted notation of eq. (76). The Gibbs adsorption equation can easily be cl-.-rived if it is remembered that the differential dGs can be written in two ways. \!‘riting GS = X piilfi’, where the subscript “i” now denotes ionized or non-ionized components, we have dCS = and also dCs =
Cpi dtJi*’ +
SSdT Combination leads to dy=-X1‘idpr
Zrllis dpi Ady
+
SpidntiS
’
where it is assumed that the temperature is constant (d?‘ = 0). Eq. {SO) is a specific form of the Gibbs adborption equation, which is derived with the aid of eq. (76). In eq. (SO) the differential dp, can often be taken a5 zero. Thus, in 2he Aglaq.-soln. case the addition of A@&, necessary for building up a positix*e interfacial charge, often leads to only a small increase of the nitrate ion concentration. Similarly, if I<1 is added. a considerable negative interfacial charge may be formed but the relative increase of the potassium ion concentration may be small Concerning the oil-HA-aq.-soln. system, the fatty acid HA can often be added wrthout appreciably altering the pH value. In the semiconductor-02 case, adsorption of O+ and the subsequent ionization usually occurs at a constant Fermi energy. fntegration of eq. (SO) and combination with eq. (58) gives
AF =
AG -
API’ -
A (J,“~ adyo +
,fradpa
+
\3’j&d&
(61)
where AC is given by eq. (59). Thus the Helmholtz free energy of the whole s_vsle#jc contains the contribution ---A J? adtyu. Apart from the integral j&dpa. which, as we have seen, can often be neglected, this is the only contribution explicitly pertaining to the double layer. It is called the free energy of the double layer27 and will be denoted AAF,.
Advan.
CoLIoid
Interface
Sci.. 1 (1967) 277-333
M. J. SPARNAAY, SEMICOXDUCTORSURPACES
314 5.3 COMPARISON
WITHTHE“FIRST
METHOD”•PVERWEY
ANDOVERBEER
In their “first method” to obtain an expression for AF Verwey and Overbeek (V-0.) imagined the interface brought into contact with an infinitely large amount of the solution, ionic equilibrium at the interface not yet being established*. Thus, considering the example of the AgI-aq.-soln. interface (this example was also used by V.O.) it was assumed that initially the I--ion concentration was “too high”; the interface should be negatively charged, but this charge had not yet “arrived”. Owing to the high I--ion concentration this arrival set in spontaneously and resulted in a free-energy gain per unit area of --crye. Since during this process the (negatix-e) kterfacial charge WZLGgraduall_v built up. electric work WZLSneeded to bring more I--ions toward the interface and it was conclucled that this amount of work was + jzytedo. This quantrty was called the electrical part of the free energy (per unit area) of the double layer. The total free energy per unit area of the doubIe layer was then found by addition. It is readily seen that the expression --ova has the character of a Gibbs function (see eq. (7-l)). The integral -O~we adyo is part of the Helmholtz free energy per unit area of the whole system and the integral +,fuo lyedo is part of the Helmhoitz free energy Fs of the interfacial phase. To see this, eq. (56) can be used: FB =
0
-
y-4
(56)
The role of the compensating charge (-0-A) is a pas&-e one in the “lirbt method”. This charge would only be subject to a “secondary rearrangement” and would “not contribute to the free energy of the double layer”, its role only being _ implicit in the value of AFe. However, the role of the compensating charge is explicitly given by the integral IX’* dpu,. As pointed out in subsection 5.2, its value is often very small. It is even exactly zero in the “first method”, because no salt containing ions “a’* was assumed to be added. I?renkel’s method*9 is quite similar to the “first method”, including the passivity of the role of the compensating charge. 5.1
EXPRESSIONS
FOR
,~%rd~‘,_
LTS RELATIVE
JIdCSITUDE
A useful expression of the integral el weadve can be obtained if the model of Fig. Sa is assumed. We put dps = 0 and identify this integral with AF,. There is a charge G located on a plane where the potential is lye_ At a distance 6 from this plane, a charge ~1 is located and the potential is lyb. Enally, a space charge rrscis present. We have the condition: * This is a disadvantage equilibrmm throughout
of the first method. the
whole
The treatment
in 5.2 presupposes
thermodynamic
process.
Advan.
CMoid
Interface
Sci.,
1 (1967)
277-333
ADSORPTIOS
u +-
a1
This model charge
-f-
USC
applies
ions;
tor-oxide
(W
to a semiconductor at the outer
to Irhe metal-electrolyte a space
charge
charge
to
may also be mentioned_
are
ade,
b leading present.
extreme
6) where the
slow states
interfaces
to a charge
ions of small
It also
where a layer of
~1, and at Gouy
Surfactants.
partly
ste,
layer,
or completely by the surfac-
and use by the GOUJ
cases can be distinguishtxl:
7. ‘fi~ere is w
space cfaarp
fusr =
0). Therefme
a-j-al-0 The
whole
double
(W charge
layer 4E‘, 2.
that
compensates
is a flat condenser
=
-
~0-l
ase =
(I iies in a plane
with 7yd =
(W
3.1 (we assume
be given
that yto =
for the three approsimattnns
in these approximations. and FrenkeP.
Mott-Schottky
a~jroximatiorr.
using eq_ (3G)
This is in accordance
\Ve consicler a strongly
for the space
dealt
ylOf. Jtrst as in eq. (SJ), AFe turns out
has been said by V.O.“?C
values,
chmge asc. Then (83
to be a negatix*e quantity
By
the
0
with in subsection
2a.
5. Then
0. Then
cow~pzmatcd b,~ cr s&m
The value of AZ;e can (in case 2) easily
conductor.
at a distance
2xbas
Tlrc cltau~r: o is mfird_s
a +
of
or
at the semiconcluc-
the semiconductor.
Here the surface charge a iriprovided
the charge CTIby specificail~-adsorbed Two
fast states
inside
or the AgI-electrolyte
ionized. layer.
of charged
consists
by charged
ions at a distance
leading taut,
and
and asc? is the space
specifically-ads;orbed
oxide layer (thickness
with an
surface
the charge aJtis provided
interface,
applies
0
=
G is situated
adsorbed
313
EQuAT~oss
charge
doped
and assuming
M ith what r:-type
large
semi-
negative
z-
one has: AF,.
=
(3E _hrlr,e)-’
,3b. Low
@et&X,
&a’
(0 negative)*
Ln.v imptwity
roatetzt
(W fttt,
13~ using
smnltj.
eq_ (37)
one
obtains: AFe! = Zc.
-
~-1 27t L a2
Large absolrite
that the charge
value
a is formed
(67) of the Poterttial
by charge
carriers
with a charge
\Ve assume
cohmt.
yro, tow im@rify
-e
each.
After
using
eq. (39) one has: AFe The Computer. and
=
2kT
e-l
curves
of Fig.
=
-
13 cover
be derived
x-1
(a negative)*
+Stb)b
For an electrolyte
(88). can both AFe
(1 +
qkT(cosh
(W
the cases 2a. 2b and 2c and were obtained
solution from f-z -
(rib =
0) bordering
the well-known
a charge4
wall,
by a
eqs. (87)
expression:
1)
(89)
* For u positive. a minus sign shoutd be added. Advan.
Co&&i
Interface Sci.. I (1867)
277-333
Xl. J.
316
SEIIICO?r’DUCTOR SURFACES
SPARNAAY,
This expression can be obtained by putting iCb = 0 in eq. (35), then using eq. (85) and integrating_
Fig. 13. The integral -
values of ub = &ky‘.
0j “0 bdy,, in units I”lim-lkT (or Snf_.kT), pbtted against rpofor various PYOC. IIId I&enr. Cotrgr. Surface Acftvit~. Cologne. From Sparnaay.
7960, X’ol. II. p_ 232. Only for ub = 0 (intrinsic case) are the curves symmetrical around the vertical line yJo= 0. This can be inferred from Fig. 7. A sign reversal of both ub and 9’0 provides the scl?sc safue of the integral.
The actual situation in a double layer system will usuall_v lie in between the extreme two cases with 01 = 0 and a,, = 0. Verwey and Overbeek ha\-e given estimates of the ratio of a1 to crSefor various values of a specific adsorption
energy, the free-ion concentration,
and values of
po and yb (Verwey and Overbeek, p. 41). These will not be discussed here. In order to consider the behaviour of the adsorbed particles. the chemical parts of the Gibbs adsorption equations must also be discussed. \Ve shall assume in what follows that these chemical parts follow “ideal” behaviour, i.e. we assume that we can write dpp = kT d fn rf. Also, adsorbed neutrai particles are assumed to exhibit ideal behaviour. Finally we assume dps = 0 in all cases to be considered. Then Ar can be written as AY =
-
kT I;[r,
-
(rj)o]
+- Ape -
kT c’
If--& -
(f’&f
(90)
If the adsorbed species is completely ionized, AF, usually dominates the other contributions. This is notably the case for the [email protected]. system. For incomplete ionization the situation is complicated. The correlations between eq. (90) zmd the pertinent adsorption isotherms will be discussed in subsection 55 5.5
EQUATIOrJS
An equation
OF STATE.
ADSORPTSON
ISOTWERMS
of state relates the surface pressure or the interfacial pressure
(both of which we denote by n) to the amount adsorbed per unit area of a certain component under consideration. This component may be ionized or non-ionized. d dVUM_ co&&
rntcrfatG sci..
1 (1967)
2774333
Equations
of state often provide a useful ph_vsicaI insight into the beha\*iour of the
adsorbed component_ In this paper we are naturally interested in the way which the charge of adsorbed particles is reflected in the equation of state. The introduction of z arises from a certain interpretation of the Gibbs adsorption equation; the decrease of y, which is the result of the adsorption of a considered component, is equivalent to an increase of -7. The classical example of the spreading pressure measured in a Langmuir trough immediately* provides evidence for this interpretation_ Thus dy =
-
(+31)
An adsorption isotherm relates the adsorbed amount of a certain (charged or neutral) component to the equilibrium pressure of the same component. In manJ relevant cases the chemical potential of such a component is proportronal to In c or In p (c = concentration, fi = pressure). In equihbrium the chemical potential of the adsorbed particles should be equal to the chemical potential of these particles in a bulk phase: pLB(1‘) =
ph (c or p)
(92)
where r is the adsorbed amount per unit area. I~nowleclge of the adsorption isotherm leads to an expression for pS(r)_ This expression can be inserted in the Gibbs adsorption equation, and finafly eq. (91) provides (after integration) an equation of state. This sequence can also be followed in relersecl order. Then a proposed equation of state predicts a certain adaorptiorr isotherm. Unfortunately this procedure can rarely be carried out without ambiguities. Thus, one adsorption isotherm can often be correlated to more than one equation of stateSo. Furthermore, specifying the adsorption of ionized components. the complication arises that we can add only neutral components to the system (XgSOa or I<1 in the XgI-aq.-soln. case; the fatty acid HA in the oil-HA-aq.-soln. case; 02 in the semiconductor-02 case). Therefore eq. (92) is an oversimptification: the relation between adsorption isotherm and equation of state requires knowledge of such quantities as the degree of ionization and its dependence upon the \-alue of ~0. For these reasons we only deal with some extreme cases. \Ve first write down expressions of four well-known equations of state for the case of a one-component adsorption, disregarding for the moment the influence of the charge. First we have: n=
rkT
(93)
This is the “ideal”
equation of state, r particles being adsorbed per unit awe. Here
the chemical potential PS (r) p” (r)
=
*u*e (P)
is
-j- kT In P/P
(94
where ~c”~(I’e) is a standard value of the chemical potential at a given temperature T and a given adsorption P. izing parameter, namely P,
There should be no confusion between this standard-
and the adsorption at yo = 0, which will be denoted Rdvun.
Colloid
Interface
Ssi.,
1 (1967) 277-333
318
hf. J. SPARNAAY,
(In our examples we will usually take (r), the chemical potential can often be written as:
by (I’),.
pb(c) =
ale
+
SEMICOXDUCTOR
SURFACES
= 0). The bulk expression
kT In c[cO
of (95)
which is analogous to eq. (91) except for the use of bulk quantities (c and co can be replaced by p and PO as the case may be). After applying eq. (92) it is seen that ris a linear function of c. Thus the adsorption isotherm is a straight line. This is called a Henry isotherm. The slope of the straight line is determined by A$’
=
r”e( P)
-
PbO(~“)
(96)
which is the (free) energy of the adsorption. The second equation of state is the Van der Itraals equation (,Tt+
(1 -
aP)
bI’) =
I-XT
(97)
where a is a constant accounting for the intermolecuIar interaction between pairs of adsorbed particles, (u > 0 indicates attraction, Q ( 0 indicates repulsion) and b is a constant accounting for the excluded area due to the size of the particles. Eq. (97), rewritten in virial form up to the second virial coefficient, gives n =
I-‘kT +
(b -
n/kT)
PkT
(99)
The second virial coefficient (b - a/kT) gives only a correction for the adsorption of non-charged particles. Thus, since the diameter of the adsorbed particles, and the range of the intermolecular interaction are of the order of Angstrom units, the magnitude of (b - a/k-T) is at most of the order of IO-11 cm?. \Ve shall see below, when the charge of the adsorbed particles is taken into account, that a much larger second virial coefficient is found. From eq. (98) and eq. (92) we derive: P
=
pSa(P)
+
(the Gibbs adsorption eq. (95) immediately The Freundlich x =
kT In I’f I‘0 +2kT(b---/kT) equation
(I’---“)
is here simply dy =
-
rdp)_
(99) Combination
with
leads to the pertinent adsorption equation. equation of state is our third case
7rAcT
( LOO)
where 9~is a number, not necessarily an integer_ The Freundlich equation of state is
easily derived r=
from the more-commonly
constant
known Freundlich
where 8 =
-
equation
x Nn
(101)
Finally (fourth case) we write down the Langmuir z=
isotherm
APkT In (1 -
equation of state:
0)
(102)
P/Ne is the coverage and NB is the number of available adsorption sites
per unit area of the interface. This number is, as a general rule, not identical to b-1 of eq. (97), the Langmuir adsorption being of a different character than the Van der Waals adsorption. In the Langmuir adsorption mechanism localized sites are assumed, whereas the adsorbed particles according to Van der Waals behave as a Advrm.
Cdlooid
Interfetce
Sci..
1 (1967)
277-333
ADSORPTiON
non-ideal
319
EQ?UATIOSS
two-dimensional
gas. The chemical
potential
in the Langmuir
case is (103)
Combination of eqs. (95) and P b = CL=,immediately leads to Returning to the influence that the Van der \\‘a& equation
(103). together with the equilibrium conclition the well-known Langmuir adsorption equation. of the charge of the adsorbed components, be bee of state and the Freundlirh equation of state are
of particular interest. This will be demonstrated by some examples_ First complete ionization is considered and expressions referring to the AgI-aq.-soln., the oil-HAaq.-soln. and the semiconductor-02 interfaces are given. Then incomplete ionization is considered and it is briefly indicated what effects can be expected for the oil-HA-aq.-soin., tilt: semiconductor-02 interfaces, and for the “adsorption” of the electrons in surface acceptor states of a semiconductor. (u)
&I-aq.-sok
i,hykct:
Under experimental circumstances~ 7.59 the surface charge consists of about 1Orr elementary charges per cm= at the most. This mean5 that the adsorption of Agf ions or I- ions hardly exceeds this number. Confinin, cJour5elves to the adtlitic>n of KI, the condition for the adsorption isotherm can be written as: or prSUeh*m + The low-potential
zreyo _t k-1 In l;/IYru approximation,
=
prrJe +
kl‘ In cr/~re
ccl. (371, gi\*es yu = -kcx-‘0.
(105)
where it can duly be
assumed that a = zr el’r. If these expressions are inserted in eq. (LOS), this equation becomes formally identical to ccl. (99)J the second virial coefficrent 13 now being given by B =
3 z (:I e)e (~kl’)-~
*
( 106)
but its value is now much larger than the orrgmal Van der \Vaals tale. Thus, with x-1 = IO-6 cm, the value is about 3 x IO-r1 cmz. This means that the etectrostatic term predominates in eq. (105) and a plot of In cr (or of In cas) against 0 should show linear parts. Fig. 14 gives such plots, but it should immediately he added that it is precisely the deviations from linearity that have led to the present detailed insight into the structure of the double layer! (b)
Oil-HA-aq.-soln. z’rztevface An equation similar to eq. (106) often applies to this case. Only z~ should be
replaced by z~. It is here relevant to note that Vrij51 considered a discreteness-ofcharge effect of the ions A-. This effect leads to a B-value which is about 10 y/o smaller than the “smeared-out model” which is used throughout this paper. The reason is that the introduction of “discrete charges” also leads to theintroduction of a redistribution of the A- ions. This redistribution leads to a lower value of the surface pressure. One important aspect of Vrij’s treatment is that the theoretical disddwm.
Colloid
lnterfuce
Sci..
1 (1967)
277-333
31. J _ SPARNAAY.
320
tinction
made
adsorbed
here between
particles,
the second
SEMICONDUCTOR
virial coefficients
can no longer be made,
i.e. they
of charged
are no longer
SURFACES
and of neutral
to be considered
as additive.
Fig. I-1. Adsorption of I-ions plotted
drfferent KSOs
Also the Iri@ potential because
the adsorption
circumstancesss. AF+$ =
agamst
p1 ( or kT/e
In cl). From ref. 52 (Lyklema).
here can exceed
If po = -
approximation
is of importance
at the interface.
This
in this connection,
1012 molecules per cm3- under experimental then this approximation gives
0 at zero adsorption.
2kTra
(107)
We assume complete ionization and an ideal behaviour n =
Five
concentrations.
of the chemical
part of l;lA
Then
3l”AkT
isotherm equation with IZ = 3. Experimental evidence for this equation has been reported 53. The electrostatic contribution here still dominates although to a lesser extent than in the low-potential approximation. (c)
corresponds
(108) to a Freundlich
Stmziconductor-02
interface
If the adsorbed O?; is completely ionized, a situation similar to those described under a) or b) exists. The only difference is that the compensating charge is provided by holes instead of ions. Another difference may be found in the chemical part of the electrochemical potential of the adsorbed 02. Its contributions to the surface pressure will be given by eq. (102) rather than by r,,r- kT, because the adsorption
may
The second B =
have virial
27~ (zoz-e)2
taken
place
coefficient
through
an adsorption-site
mechanism.
B is now
L(kT)-1
where L is often of the order of lO+cm,
(10%
i-e_ much larger than the Debye-Hiickel Advaa.
Cothid
Interface ScL. 1 (1967) _
277-333
ADSORPTION
321
EQUATIOXS
length in most ionic solutions. For roe we take in our example the value -2. If the high-potential approximation applies and if the value of 1&bis small, no new aspects arise. the case being analogous to eq. (lOSj_ Tf the semiconductor is strongly doped and of “-type conductivity, then; 4n!=
-
X&T
In (I -
1y,V8) +
(3 irX&-r
32 z es p
(110)
but there seem to be no experimental indications for this equation_ Evidence for adsorption isotherms which can be explained by means of eq. (109) will be given below. First incomplete ionization will be dealt with. \Ve shall consider three cases: (s) the adsorption of surfactant HA at an oil-water interface, (b) the adsorption of oxygen on a semiconductor surface, (c) the contribution of electrons in acceptor-surface states to the interracial tension. (a J The chemical equilibrium is HA + H + + A-. The adsorbed components are HA and X-. If the degree of ionizatmn is ~1s at the interface, then to a first approximation: pWA*
interltwe
q 2~”interface
SHAKO
= =
p A”O i-
In (1 -
+
kT
kT
In
crsF
-t
us) r (111)
z .keyo
where r is the sum of the adsorbed amounts of HX and _-\- per ems. Then: AJ.J=
-
AZ =
-
fil‘
+
AFe
(112)
where we have chosen again the initial stage to be at zero ad&orption. Furthermore it is assumed thet lye = lyO_ The interracial charge a is equal to zaerus. The integral of AFe is the same as discussed before but, as indicated by Payen 3BSit is a tedious problem to obtain us and r separately. For us one can obtain, with the aid of eqs. ( 110) and (111) and of pn4
=
3a -t- ?#Qr
(113)
(where Y~I = FH = PH b* + kT In cn/caO) the expression
a8=
c
1+-
=K
exp
CFIO
-
A prro + kT
zAev)o
-’
>
where A~~o=~~~~o-~~~--~nbo is the free energy of the adsorption of H+ ions. For small a* one has a Boltzmann-type relationship: oa =
a08 exp
WO kT
(115)
where a06 is the degree of ionization at v. = 0. One way to write the condition for the adsorption isotherm is: PF?AO
bulk +
kT
In CRA/CHA~
=
~CHA~O
+
kTIn
(1 -
a*) ~/~HAO
(116)
For small 08 a Henry-type adsorption law is found: the concentration cx~ is proportional to the amount adsorbed. For not too small aa, a quite involved adsorption law is found (see Payens). .4dvan.
Co&id
Interface Sci.. 1 (1967) 277433
322
M. J_ SPARNAAY,
(b) For oxygen adsorption equilibria will be assumed:
on a semiconductor
02($w
+G
Ot(ads)
PO,
Oz(ads)
f
2 O(ads)
PO, _ ads
0 (ads) +
O+
(ads) +
SEMICOSDUCTOR
2h+
PO
gns =
PO2
=
ads =
SURFACES
surface the following
(117)
ads
ZpO
ads
910 ads
three
(118)
-2EEF
(119)
Writing PO2 gas =
PO2 go +
kT ln
/JO2 ads =
~02=”
kT
+
~o,I~o~~
ads =
PO s* i_
kl‘ In ro/roO
‘lo
ada =
PO so +
kT
Ay 5
-
An =
-
(121)
In I’o,lro,”
PO
the following expression
(120)
In
(122)
r02-/ro2-0
-
(123)
2eyo
for A? is obtained: (1’0,
+- I’,
+
I’oz-)
kT +- AFe
(124)
where again it is assumed that at y~y~ = 0 the adsorption is zero. The degree of dissociation of the equilibrium eq. (I 18) is denoted as CLIand the degree of ionization in eq. (119) is denoted as ~tr. Then: =
roz
(1 -
al)rl;
1-o =
2~(11’1 =
(1 -
~22)r*;
ran-
For weak ionization and strong dissociation, a Freundlich-type
=
a%Tz
isotherm is obtained
with tt = 2. For strong ionization the formation of a double layer can be hampered if the dissociation dissociation ~0,~’ =
is weak and a Henry-type adsorption cau then be found. For strong and strong ionization the following equation is relevant: i-
2$-f
0 +
In Po,/PoQ
=
2 kT In rogf-_lPo~--O -
4evo -
4Ep
(125)
Taking ~0 = (-2er&) (capacitance)- 1, the same situation can exist as was met in the AgT-aq_-soln. case and as we have reproduced in eq_ (109). As % example 54, the adsorption at room temperature of oxygen on oxidized Si may be mentioned. Fig. 15 gives curves pertinent to this case. The adsorption is not measured directly here. In fact, the quantity plotted against In PO2 (the width of an electron resonance line) requires some explanation. If a Si-crystal is crushed in high vacuum (10 -s-10-10 torr), no electron resonance lines are found. This points to the absence of “dangling bonds” at the surface, because free electrons at the surface (the ‘*dangling bonds”)
may be expected
to show resonance in an applied
magnetic field. Absorption lines in a magnetic field are found, however, if the Sisample is heated in a poor vacuum. In particular if the sample, after being crushed in high vacuum,
is heated between 450°C and 500°C at oxygen pressures between Aduan. Colloid Interface Sci.. 1 (1967) 277-333
ADSORpTION
323
EQUATIONS
I i-
i-
.
Fig. 15. Line broadening AN (which is roughly proportional to the adsorbed amount of oxygen, on the Si-sample. AH = 1 Gauss corresponding to about 1011 adsorbed oxygen molecules per
cm:) plotted against In pan. Room temperature. From ref. 53.
10-s and 10-5 torr, a resonance line (line “A” in Fig. 15) appears with a g-factor of 2.0029 and a width of about 3 Gauss, Line “B” in Fig. 15 appears, when the Sisample is crushed and sealed in a quartz tube under moderate vacuum. These lines are attributable to paramagnetic centres formed by a chemical reaction between oxygen and silicon. Their density is about 1012 cm-e.
If at room temperature
(or
temperatures roughly within 50’ from room temperature) the oxygen pressure is increased, the width of this absorption line is increased. This is a reversible phenomenon. A&an.
Colioid Interface Sci.. 1 (1967) 277-333
324
M. J. SPARNAAP, The line broadening is understandable
SEMICONDU~OR
if it is assumed that
SURFACES
(paramagnetic)
Dxygen molecules are adsorbed and distort the local magnetic fields, thus providing
for a wider scale of absorption energies. It should he noted that diamagnetic gases have no effect. Theoretica calculations make it probable that the line broadening is proportional to the amount adsorbed, the proportionality factor being of the order of 1011--1012oxygen molecules (atoms) per gauss line broadening. If so, the slope of the curve indicates a “virial coefficient” which is of the same order of magnitude as that found in the &I-aq.-soln. case and 3 to 4 orders of magnitude larger than predicted on the basis of the usual intermolecu1a.r forces. -4nother case where such absorption lines are found is that of crushed GaAs. Here a heat treatment at 200°C in IO-4 torr of oxygen results in the formation of paramagnetic centres. A reversible line broadening takes place if the oxygen pressure is varied at room temperature. However, in this case the line broadening is proportional
to the pressure
itself
and not
to the logarithm
of the pressure.
This
points to a Henry type of adsorption, i.e. to the absence of strong ionization_ A’ number of analogous cases may be investigated in the future. In fact electrostatic effects, mostly connected with the adsorption of osygen, ha\-e been widely discussed in the literature of the past 15 yearsJ5, and even earlier_ We only mention here the case of oxygen adsorption on CdS. According to Afark56, light is needed here for the adsorption, and, once the oxygen is adsorbed, light is again needed for desorption. This behaviour may well be explained if the electronegative properties of oxygen are combined with the well-known photoelectric properties of CdS. Thus, light shining on a CdS crystal creates electron-hole pairs. Electrons are needed to saturate the demand of adcorbing oxygen. Again, if desorp tion is to take place (a decrease of the osygen pressure) the influence of light is very helpful: created holes are now net essary to neutralize the osygen before desorption can take place. Finally we note that the degree of ionization Q?:can be written in a way analogous to that of the equilibrium HA -L H+ + A- namely: 0%
[I
=
where Ama
=
+
(_%)_-I
f(tro2-so
-
z * m” ;-go+-evt,]
esp
pose}
Ei and where tz now enters as a parameter in-
-
stead of CH and A&O instead of Ap~o. fc) E8edrotrs irz acceptor swface states. Eq. Ep =
EAO + kTh
m
zr, _
r
e
(18) can be written
{free) energy at flat-band conditions.
AZ=
FAkTln
(
1 -
as:
-eye
where we have written the “adsorption energy “EA as EaO -
4 =-
(126)
For
Ay
x)”
one obtains: +
AFe
pyre, EAT being the
APPLICATIOSS
For smatf f,
OF
(F,
COLLOID
4
CHESIISTRY
325
31ETHODS
PA) this equation reduces to the “ideal”
be noted that the Langmuir model of gas adsorption
equation.
It should
(or also the Stern model of
ionic adsorption) leads to equations similar to those we ha\-@ seen here; eqs. (127) and (128). In these cases I -A is the number per unit area of adsorption sites and re is the number per unit area which is occupied by a gas atom or an ion.
(6)
Methods used in colloid chemistry and applicable
to semiconductor
probkms
tVe mention two such methods: (n) The study of electrokinetic phenomena, whit h leads to knowledge concerning [-potentials. {b) The study of the contact angle of a droplet placed on a semiconductor surface, as a function of a potential difference applied between the droplet and the semiconductor. Once more germanium is the semiconductor applications 6.1
tvhich is most frequently used in
of these methods.
ELECTROKISETIC
PHESOJflZ:SA
EIectrokinetic phenomena are connected with the tangential movement of two adjoining phases with respect to each other. One of these phases is a liquid. The c-potential is the potential difference between the slipping plane of these txvo phases and the bufk of the liquid phase. It is not always clear where to locate the slipping plane and what the exact physical nature of such a slipping plane is. In the case of an aqueous solution bordering a solid. a layer of kvater molecules of. say, 10 w thickness will be immobilized and attached to the surface of the solid. This marks the Idcation of the slipping plane. AltholAgh electrokinetic phenomena have been studied at a very early date 57” the interpretation of the results is still not unambiguous 5s. %Ve confine ourselves to the most simple interpretation, but this wiil be sufficient for our purpose because we are chiefly interested in the semiconductor aspects of the experiments, An important achievement of the theory was the proof, given by De Groot, Xazur and Overbeekss, that for a given s-stem all the different electrokinetic
phenomena prox$de information concerning the same electrokinetic coefficient, which can usually be identified with the c-potential. Thus, Ge-particles (both of *c-type and of P-type conductivity) immersed in dilute aqueous and moving under the infhxence of an solutions (10 --S .N acids and I-I-electrolytes) electric field (electrophoresis). have a ( potential which is between 0 and -40 mV (depending on the pH, see below) and this is in accordance with the results of streaming-potential measurements 60. The streaming potential is the potential produced by a liquid flow which is pressed through a porous plug of, in this case, germanium particles and measured by two electrodes placed at either side of the Advan.
C&oid
1mferfat-c
Sci.,
1 (1967)
277-333
326
bf.
J. SPARKAAI,
SEM ICOHDUCTOR
SURFACES
plug. In Eriksen’s measurements of streaming potentials no specification was made as to the conductivity type of the germanium. More detailed measurements were carried out by Sparnaayer, who applied the method based on electro-osmosis. In this method an electric current is passed through the liquid contained in the plug and the liquid transport (x*olume transported per unit time u) is measured, Then:
where C is the zeta potential, i! the specific conductivit>* of the liquid, ye the viscosity* and e the dielectric constant of the solution. The pH of the solution was varied at constant ionic strength (provided by mixtures of HXOe and KXOa solutions) between 2 and 4-i. Electrolytes with a concentration greater than lO-2 .N coulcl hardly be used because it was then difficult to measure c’ with any precision. Solutions in which the pH was higher than 4.5 were avoided because germanium &ides were beginning to show a tendency to dissolve, All the experiments were carried out in thoroughly-steamed glassxvare. For 10-z Q. N-Q&? germ,anium ft linear relationship bekveen c-potential and pH x-as found between pH 2f and pH 4, the slope being about 30 mV per pH unit. The f-potential at pH 4 is about -40 mV. For P-type germankrm this was not the cse. .a11 calculated C-potentials being found around -10 mV. This discrepancy must be explained as follows: eq. (129) can only be used if electricity transport across the semiconductor particles is absent. However, the specific resistance of the semiconductor samples was lo- f Q cm, whereas the specific resistance of the solution was about 104 Q cm. If the electric current was wholly transported across the particles, no liquid volume transport would have taken place and application of eq. (129) would have led to a zero value of the c-potential. 1Ve conclude therefore, that the “real” (;-potential for both P-type and n-type Ge is the same (as is also suggested by the electrophoresis experiments) but that there is more electricity transport across p-type material than across St-type material \t’e return to this point below, The measurements (now only n-type Ge is discussed) indicate that H* ions are potential-determining. The slope of the t-pH curve is the same as that found for glass-aq.-soln. interfaces, although the values of the c-potentials are much less negative. In both cases the pH-dependence points to the presence of acidicgroups at the interface. Calculations based on Gouy’s theory have indicated a density of charged acidic groups of about 1012 cm-%, which decreases upon a decrease of the pH. Acidic groups can be formed through an oxidation of the Ge atoms at the surface followed by a reaction with water molecules. The density of Ge atoms at the surface is about 7 x 1014 cm-s_ If the same density of acidic groups is present, * There should be no confusion between the symbol q, used here for viscosity, and its use elsewhere for electrochemical potential_ Advan. ColCoid In&f&c
SC.., 1 (1967) 277-333
APPLICATIOS
OF
COLLOID
CHE.\IISTHI-
327
3IETHODS
after the osidation and reaction with water has taken place, rhe degree of the ionization is usuaIly somewhat less than lWz. This points to an ionization con&ant which ~11 be significantly higher than that of free tfissol~~ed H&eOs. Such a phenomenon, a variation in the ionization constant of an acid, placed tirst in the bulk of a solution and then at an interface, has been relMwtet1in the literature. for example by Scheraga6” and by Payens 38. They attributed the difference to dielectric influencel;_ If oxygen had been absent, the t-potentials might ha\w been different. Ilrwever. during the electro-osmosis experiments electrochemical reactionr at the Ge particles are taking place. Each particle has an anodlt Ale (lvhere there is a tenclency to oxide formation) and a cathodic side (N here hydrogen formation may take place). The total interface in the plug will thercfcxe be highly heterogeneous;. The influence of the pH can not only be noted in c-potential measurements. but nlst, in the measurement of the capacitance and of the surface conductance of (freshly anodized) Ge-electrodes which were discussed in section 1. The celts used for these measurements contained three electrodes: the Ge-electrode, a platinizeri Pt electrode whit-h has a large surface area. and (in the ver&n of IM:uenkamp and Engell) a Ag-A&l electrode_ \Vhen. through application of a potential difference between the Ge-electrode and the ptatinizecl R-electrode, the flat-band condition was maintained, the potential difference mea-Jured during the first minutes after anodization between the Ge-electrode and the _Ag- A&I ekctrocte proved to be ZL function of the pH. the slope of the potential-pH cm-\-e being claw to 60 mV per pH unit. In a ceI1 containing only a Ge eIectrode (U hi& has not been pre-.modized) and a reverdble electrode. the relation between the mensuretl (floating) potential and the pH is qualitatively the same, but it is cliRicult to obtai!: constant potential values, fluctuations in time of -10 m\: being possible. This points to an imperfect reversibility of the Ge-electrode in combination with mixed potentials. Although the H+ ions will be potential-determining, the finding of a perfect Xemst law is obscured here by secondary effects. However, as discovered by Brouwer63, a perfect Nemst law is found if Hg,Oe is added to the electrolyte_ This may be interpreted as a removal of unwanted side effects. In this respect it is useful to note that Hz02 is a constituent of many etchants and an increased ciissolution rate of Ge has been reported. Thus the reversibility of the Ge-electrode will be improved by the presence of H20~. According to Brouwer 6-L. Hz02 stabilizes the surface because it directly provides sufficient oxygen. Oxide layers are then formed, the formation being especially rapid (formation time of the order of minutes) if K+ isns are present in the solution_ One of the side effects is caused by the presenee of Cu”-+ ions in the electrolyte. As mentioned in section 4. Cu’+ ions, after contacting the Ge surface, can form surface states. These can be measured by means of radioactive tracer methoc?s at concentrations on the pH,
as low as IO-7 mole44b. The density of these states depends strongly
especially
at pH
>
5. The density Advan.
increases with increasing
Coltoid
Xnferfucc
SC&, 1 (1967)
pli.
177-333
328
M. J. SPARNAAY,
i
SEBIICOXDUCTOR
SURFACES
Measurable desorption of Cut+ ions takes place only in very acidic solutions. the reaction at pH K l+ being as follow+: Ce + x(H~0 -f- Cu”-f) % GeOz -+ x (Cu + 2 H*) (130) In passing we note that this reaction can be interpreted as a competition for electrons between the reaction Cu”+ + 2 e- + Cu. and Ge -f- x Hz0 +S GeOx + 2x H+ + 2~ e-. Since the standard potential of the copper reaction is known (0.34 V) the standard potential of the second reaction can be estimated to be somewhat lower and does not contradict the value given by Lovrecek arxd Bock&66 (0.11 V), These
authors
potential system.
also
1-s. plf curves It is evident
electrochemical a paper
have
greatly
that
depend
the
of the reaction
and Faktor6;.
slope
and
upon the state
that one of the most
aspects
b>r Carasso
shown
dangerous
between
the reproducibility of purification
impurities
Ge anti Hz0
of the
of the whole
is Cu”-+ ions.
The
have been reviewed
in
See also Latimers*.
-terpretation of ;L slope different from that predicted by the Nernst law might be given in terms of the -discreteness-of-charge concept of the charges at the interface_ It was assumed hitherto that the charges at the interface could be considered as smeared-out and consequently that the potential y5 had the same value everywhere on the interfacz Owing to the discreteness of the charges, the “adsorption energy” EA of the electrons in the acceptor surface behaves as if it were a function of the density of the charges present at the interface and their distribution over that interface. This function has been discussed extensively in the literature of the Hg-aq.-soln. and the AgI-xl_-soin. interfaces~~.69. For hexagonal and tetragonai array-s of the charges and densities of about 1013 cm-2 or less, the energy EA has been found, by numerical computations to decrease at increasing coverage, implying that at a higher coverage the adsorption is “easier” than at a lower coverage. However, so far no experimental results seem to be available in the physical chemistry of semiconductor surfaces which need. as their unavoidable interpretation, the concept of discrete charges. Therefore this concept will no longer be discussed here. Returning to the marked difference which was found in the electro-osmosis expetiments of n-type and of j-type germanium samples, this difference is probably connected with the mechanism of dissolution during anodization. As mentioned in the Introduction, section 1, a hole mechanism operates, because there is a limiting anodic-current density in n-type Ge, which is absent in P-type Ge. As far as the electro-osmosis experiments are concerned this means that the current can more easily find its way across P-type samples than across s-type samples, in accordance with the experimental data. Finally we notethat although surface states can be removed both by anodic An aiternati -
and by cathodic treatments076,such a “state-free” situation is by no means a stable one. States may rapidly form again. their rate of formation probably being connected with the composition of the solution. The purifiation of chcnzicats ad apparatus shouiX be carried out extremely
thoroughly. Advan. Colloid Interfact
Sci., 1 (1967) 277-333
APPLfCATfCW
6.2 rOWTACT
OF COLLOID
CHEDIISTRY
xETHOf%
329
ANGLBS
A droplet of an aqueous eiectrotyte placed on the surface of a semicwductor, has a contact angle (denoted as a), whose value depends on the history of the senliconductor surface, Par instance, if the semiconductor is a germanium crystal, etching with “CP-4” folirnved by rinsing with water provides a “h_vcfrrjI,fIt,t>ic”,i.e. a non-wettable surface, the contact angle being about 180”. This is not a stable situation. In the course of 15 minutes or so the surface hecomes more ‘“h) clrqd~iiic” or wettable and the contact angle becomes about SO degrees7”-~*. If a thin Pt-wire is introduced in the dropfet and used as ;m electrode then, after application of a potential difference between the li’t-electrode and the semiconductor, the contact angle decreases further and mxy dtirnately become zero. The surface has then been completely wetted 7r*7p.The potential difference (V) may be positive or negative. The rate of decrease of rt is about prr>pwtionaI to the absolute value 1VI. For 1V] < S S, this rate is trsualiy very smrrll. To mention a typical example: if in the case of a fre&ly-etched Ge-surface a potential difference of 10 V is used, complete wetting occurs in about 5 min. A current pnsses across the interface between semiconductor and solution if a potential difference is applied, the current density being about 10 ,w& cm-* at \I’! = 10 \‘, If some time (a day or so) had elapsed after the et&ml: procedure had taken place. a higher value of \I/‘! was needed for the same rate of decrease of u and of the same current density. These results should be interpreted in terms of a varying liquid--solid inter facial tension?e. \j’hat is measured here is. in fact, an electrocaptltary curve, which is well-known in the study of Hg-aq.-soln. interfaces. However. in the underlying case we cannot hope to obtain more than qualitative information. The most important reason for this is that the surface is “dirty”. It contains an oxide Iaj*er, obtained by etching in our case, and this oxide layer cannot be removed under the conditions suitable for the underlying experiments. In fact it is impossible to remove any oxide layer in the course of experiments concerning contact angles between the semiconductor (germanium) surface and an aclueous 4ectroIyte. Some information as to numerical values may he obtained as follows: YS =
ysl4 +
3% eos a
(131)
This is Young’s equation. In this equation, ys, ys~ and ye are respective& the surface tension of the solid. the solid-Siquid interfacial tension, and the surface tension of the liquid. It is seen that the observed decrease of u. as the result of the application of an electrostatic potential difference, is due to a decrease of yst. We explain this decrease by the”adsorption“ of electrons in slow surface states (for positive V) and by the “adsorption“ of holes (for negative V). Now for fast surface states it was reasonable to assume Fermi statistics, as expressed in eq. (18). The same will be assumed for slow states, although the concept of slow surfaces states is still rather vague-The A&an.
Co&id
Xnterfice SW., I (1967)
277-333
M.
330
J. SPARSAAY,
SEMICOSDUCTOR
SURFACES
two parameters which are usually considered to be characteristic of slow states are their relaxation time and their density, rss. For etched Ge-surfaces the relaxation cm-“. If, time rgs. is of the order of minutes, and IIBs is of the order of 10’~-101* upon applying the positive potential,
V. electrons are filling the great majority
of
the slow states, then with the aid of eq. (IS), in which the exponential term is now considered to be small compared with unity, the electrostatic contribution to ys~ is A~SL
=
-
~TBe(a)
(132)
leyol
where the added subscript (a) indicates slog acceptor type states. If fey,,! = 5 . lo-13 erg, i.e. if it is about IO kT, and if the density of states is about 1014 cm-z, then Ays~ equals about 50 erg cm -2_ \Vith this value, which is of the same order of magnitude &asthat of yr, (for xvater, ye = SO erg cm-?), the observed decrease of u times, can be explained. This explanation, together with the observed relaxation provides evidence for the conclusion that slow states are involved in the phenomena described. Ellis70 ascribed the increase of the wettability during the first minutes after etching (i-e_ without the appIication of an electrostatic potential difference) to a replacement of fluoride groups, formed by the etchant at the outermost surface, by hydroxyl. In view of what has been argued here, this increase of the nettability may also be connected with the charging up of slow states. Alternatively, the interpretation of the observed decrease of czafter application of the potential difference, should leave room for the possibility that slow chemical reactions are involved. The nadure of rhe slow states is far from clear and the best definition that one may give is that slow states are non-stable ionic, atomic and molecular configurations, with slowly-varying donor and acceptor character. Slow states have also been found at the etched surfaces of other semiconductors and of less noble metals72. Therefore it may well be that they occur more generally than hitherto assumed, in other words, they may build up in many instances where new interfaces are formed.
ACKNOWLEDGEMENT
The author
is deeply
indebted
to Professor
Overbeek
for his constructive
criticism_
REFERENCES Numerous references can be found in the Proceedings of Semiconductor Surface Conferences: a R. H. KINGSTON. (ea.). Semiconductor Surface Physics. University of Pennsylvania Press. - Philadelphia. 1957. b /_ PlryJ. Chem. Solids. 14 (1960). c H. G. GATOS. (ed.), The Surface Ckemisfry of &IetaZs and Semiconductors, Wiley, 1960. d Ann. N-Y_ Acad, Sci., 101 (1963)_
e Sut$xcrrSci., 2 (1964).
Advsn.
ColIoid Inlerface
Sci.. 1 (1967) 277-333
REFERENCES See also surface sections Phpico. 20 (1951).
Semrcond,
331
1
Phosphor
in the Proceedintis of Solid State Conferences: a. z: Proc.
Iwern.
Cotloq
. Garnlisck-Purtorkirckeu
19Jb.
Welker.
l3runs-
wick. 19%. Proc. Illd lut. Conf. Semicond. Rochester 1938. Pkys. Chem. Solrds. 8 (1959) Ptac. Itzttrn. CoHf. Srmlcomf. pkyrr., Pmgue 1960, Publ. House. Czech. Xkad. Sci.. Prague. 1961. 7962. Inst. Physics and The Physical Sot.. Proc. Intern. Con/. Phys. ,I’emico#Jd., Exeter. London. 1962. The
following book isof special inter-z&: MANY, Y. GOLDSTEW ASD S. U. GROVER. Semrconductor
A.
Surfaces,
Amsterdam,
1965.
Neview articles G. L. PS~ARSON ASD \V. 2-L.BRATTAI~. History of Semiconductor Research, Proc. I.R.E.. 13 (1955) 1794. b R. H. K~XGSTON. Review of Germanium Surface Phenomena. J. Appt. r.5ys.. 27 (1956) 101. Bl. GREEN, Electrochemistry of the Semrconductor-Electrolyte Interface. MO& Asptict~ c Eiectrochem.. Z (1959) 343. _-1dv. Electrorheru. Electrochem. L’ng.. II. GERISCHER, Semiconductor Electrode Reactions, d 1 (1961) 139. e T. 13. War~rss, The Electrical Properties of Semiconductor Surfaces, in Progress an Semiconductors. Vol. 5. Hevwood and Co.. London, 1960. p. 1. f G. HEILASD, Herstellung und Eigenschaften reiner Halbleiteroberflachen. I;ortschr. Pkysik. 9 (1961) 393 11. FLIETNER, Eigenschaften gewohnhcher Halbleiteroberflrichen. Pkyslka Stutus Soltdr, g 2 (1962) 221. h P. J, BODDY. Structure of the semiconductor-electrolyte interface. /. Electvoanal. Chew.. 10 (1965) 199. i See also review articles in Electrockewc. Semicor-duct.. t (1962).
a
Reinhold 1959 (a compilation of revlexv papers). N. B. HANMAY (ed.). Semicoprdrcctors. See for example: a) \V, SHOCKLEY. Electrons aHd Holes ZH Semico?zdrtctors, Xran Sostrand. Prmceton, 1950; b) E. SPENKE, ElectroGsche Halblemr, Springer, Berhn. 1955; c) .A. F. Josre. Firika Potypravodnikov. JXoscow. 1957: d) C. H. \\‘ASNIER. SolrJ state tkeory, Cambridge University Press, London, 1960. See also ref 1. 3 J. R. IfacDoNarD ASD C. A. BARLOW JR.,J. Chem. Phys.. X (1962) 306%. 4 J. LYKLEMA. Thesis, Utrecht. 1957. 5 A. N. FRUMKKIN. Russ. J_ of Phys. Chem. (EqCish Transl.). 35 (1961) 1064. 6 See for example ref. 1 and: 31. S BROOKS .CLSD J. K. KESS~D~ (eds.). Semiconductor MateGats, JLacXillan. London, 196% Researches in Electricity, I. 1839. in ,!Zuev)murr’s Library 1922. 7 M. FARADAY, Experimentul p. 44. 8 G. L. PEARSOX AND W. H. BRATTAIH. PYOG. Z.R.E., 43 (1955) 179J. 9 L. GALVANI. De virrbus etectricitafrs in tnotu nrusculari comme~rtuvirrs, Prague. 1791, (Handwarterbuch der _B7atrrrwisserlchaftelr. IV. 462. Jena. 1913). 10 F. MASSARDI, his pile L-opera di Atessandro Volta, Jfrlano, 1927. p. 155 (He constructed in 1799). H. HELMHOLTZ. Pogg. Ann., LXRXIX. (1853) 211. :: I. TAMX, PhysiR. Z. Soujctunion. 1 (19%) 733. 13 W. SHOCKLILY. Phys. Rev.. 56 (1939) 317. 1-l V. HEINE, Surface Sci.. 2 (1964) 1: J_ KOUTECKY. J. Whys. Ghem. Solids, 14 (1960) 233; T. B. GRIMLEY. J. Phys. Cirem. Solids, 1-I (1960) 227. and private communication; 51. TOM ASEK, Surface Sci., 2 (1964) 8. 15a P_ HANDLER. in Semicond. Surface Fhys. Con/.. Pheladelphia. 7956. Univ. of Pennsylvania Press, Philadelphia, 1957; R. MISSJ~AX AWD P. HANDLER. J. Phys. Chews. Solids. 8 (1958) 109: G. HEILAND AXD P. HANDLER, J. Appl. Phys.. 30 (1959) 446. 15b A. H. BOO~ZSTRA. J. VAN RULER AND M. J. SPARNAAY, Proc. Konbkl. Ned. Rkad. Wetenschap., B 66 (1963) 70: ?Lf_J_ SPARNAAY, A. H. B~ONSTQA AND J_ VAN RULER. Surface Sci.. . 2 ( 1964) 56. 16 G. HEZLAND AND H. LAMATSCH. Surfncs Sci.. 2 (196.l) 18. 17 J. BARDEEN, Phys. Rev.. 71 (1947) 717. 18 J- H. BRAI-TAIN 15~~ J_ BARDZEN. Bell System TecRn. J.. 32 (1953) 1. :
Advan.
Colloid Interf’e
Sci..
1 (1967)
277-333
MI. J. SPARNAAY,
332
SEMICONDUCTOR
SURFACES
19 I<. ~jOIiNEKK.%MP AXD fi. J. &W.XLL, %. i?~CC~rOCilUH., 61 (1957) 1184. 19a H. GERISCHER. in P. I)ELAHAY led.), Rdv. in Elcclruchewislry, Vol. 1. lntersclence. Kew York. 1961. ch. 4. 20 f). TURXER. J. IZCectrochon. Sot.. IO3 (1956) 251. 21 W. H. BRATTAIZU AND C. c. B. GARRETT. Bell Sysleru Tedwc. J_, 34 (1955) 129. 22 C. HERRISG AND M. 14. S~cnom, A’eu. !Uod. Phys., 21 (l949f 185. 23 IZ. El. FOWLER AHD I,.W. NORDHEIM, Pmt. Roy SW. London. A119 (1926) 173. 24 1;. G. _h.te.~ AND G. \V. Goserr. Ph_l~. Rev., 127 (1962) 190; .-f~n. S.Y. dcod. Sci.. 101 (1963) 647; Proc. IM. Co~f. Phys. Senricond. Exctcr, 1962. p 815. 25 J. VAN Laa~ I% J. J_ SCHEER, PYOC. Intern. Cobtf. Phys. Srmtcotrd. Exeter. 1962. p. 827; J. SCHEER ASD J_ v_+s La~=, Pkys. Lcffers, 3 (1963) 31. ‘2% 25b
C. A. 3fr_~in, Solrd-skde Eleclrorrrc,p, 3. VAN L.&AR ASD J_ SCXEER, to be
9 (1966)
1023.
pubhshed in -~urfuce
.%iewe.
26 G_ GOUY. .-fn~. C/rim. Physique, (8) 8 (1908) 391; a review in .+IM~. Pkvs. (Paris), (9) 9 (1917) 129. 26a D. C. CET=.PMA~, Phil. Jfag . (6) 25 (1913) 175. 27a J. T. G. OITRBEEK, in H. R. KRUYT (en.). Colfoid Science. Vol. T, Elsevier, Amsterdam,
lk%?;
28 29 30 31
b) T).
v_ DERVACXS
ASD
L.
LASDA; . .-icta
Physicockim.
fYR.SS,
14 (I 94 1) 633;
J. E.r@.
‘I’Aerw. (Russ ), 11 (1941) 803; c) iI. J, \V. VERWEY MD J_ T. G. O~E%BEE#, Theory offhe 1948; d) B. J_ \c’. X'ERW'EY, Ckcm. Stabtlily of I_sopkabic c-olloids, Elsevirr. .imsterclam. Il’eekllad, 39 (19-E) 563. I-. ONSAGER, Chews, Recks.. 13 (1933) 73: J. G, ~CIRKW~OD. /_ Ckem. Pkys., f (1933) 767 \V. I?. \I'ILLI~MS.Proc. Roy. Sot. /Londo>:), A 66 (1953) 372, see also F. P. DVFB AXD I?.H. STILLXSGER, J. Ckcm.I’kys ., 39 (1963) 1911; H. L FRIEDMAN. Ionic Suiufio~ Theor> Based o)t Cjusier Brpanslou Jrethods. Interscience. Xew York, 1962. 12. Sl?IWilZ AND >f. GREEN. /. rtppr. Pkys.. 29 (1958) 103-1. q7-.Y. ST. J. MURPHY. Surface Ser.. 2 (1964) I;: DZWALD. Ann. .V. Y. .-fcad. Sci.. 101 ( f963) 87,. 86. 51. J. SPARNAAY,
Rev. ‘Tract. Chim.. 77 (1958) 872. 1’. S. KRYLOV. Dokl. .-lkad. .Vauk SSSR, I44 (1962) 155. E. H. KINGSTON ~SD S. F. SEUSTADTER. /_ Appf. Phys., 26 (1955) 718: GARRETT AND 1%‘. H. DRATTAIS. Pkys, Rev.. 99 (1955) 376. 35 h’. I;. MOTT, Proc. Roy. Sot. ILotrdon J. A 171 (1939) 27. 2. Pkysik, I13 (1939) 367; 118 (1943) 539; IV. SCHOTTKY 36 W. SCIIOTTKY,
see also G. G. B.
AXD
R. SPESKE.
lYiss.Ycrorffcrrt~. Siemctts,18 (1939) 3.
J. T. 39 a) in 40
41 42 43
44 45 45 46 47 47 48 48 49 50
51 52
F. DEWAX_D. SelC System Tcchm. J.. 39 (1960) 615. A. J_ P~QESS, Pirrlips Res. Repts., 10 (1955) 425. D. C. GRAHAVE. Chem. Revs., 4 1 (1947) 44 1: b) R. PARSOXS m P. DEL~HAY (ed.). A duanccs EZectrochcmisfry. Vol. 1. Interscience. Xew York, 1961. R. DE LEVIE, Efecbothim. Acta. 10 (1965) 1136. E. A. EPIMOV AXD I. G. ERUS~LIMCHIK, Zh. Fiz. Klriw~.33 (1959) 4-11: and book. ~~ecfroclewgistry o/ Gerwutriur>~ a>td Stlicorr, Sigma Press, 1963. 109 (1962) 571: 110 (1963) 570. a) W. H. BRATTAM AND P. J. BODDY.~. Electrochcm.Soc.. b) P_ J_ BODDY AND \V_ I-l. BRATTAIS. Surfuce Sci., 3 (1965) 348. b1. D. KROTOVA AND Yu. 1'. PLESKOV. Phvsika Status Solidr. 2 (1962) 111. a) P-J. BODDY AND W. H. BRATTMN. ].~‘lcctrocherrr.Soc.. 109(1962)812:b) R.MEMMING. Surface Sci.. 2 (1964) 436. a) Yv. V. PLYZSICOV,Dokl Akad. Nauk SSSR, 126 (1959) 111. b) IV. IV. HARVIZY. J. Phys. Chew Solids, 15 (1960) 82: W. 1’. HARVEY AND H. C. GATOS. /. AppI. Phys., 29 (1958) 1267. P. J. BODDY, /_ ElectroaHal. Ckem_. 10 (1965) 199. H. GEREXHER.M.HOPFMAN-PEREZ AND \V. ,MINDT, Ber.BunsengeseClsckaft. 64 (1965) 136. a) R. MHHMING AND G. SEUMANN. Phys. Lefters. 24X (1967) 19. J. W. GISBS. The Collected Works. Vol. 1. Yale Univ. Press. 1948. a) E. A. GUGGENHEIM. /. Phys. Chum., 33 (1929) 842: Thermodynamics, North Holland Publ. Co., 1959. 4th ed. J, FRENKEL. Ki#dic Theory o/Ltqrrids, Oxford Univ. Press. London, 1955 (first Eagllsh ed.1 ch. 6 (especially p. 361). D. H. EVERETT. PYOC. Chcm. Sot.. (1957) 38. A. VRU, GrensCaagvcrsc~rjnseZor, Coffoq. Koninki. Vlaamse Acad. Wetenschap.. Dru=els. 1965, p. 13 (in English). J. ~YKI.ehfA, Tmns. Faraday SOG.. 59 (1963) 418. Advan.
Colloid Interface
SC&. 1 (1967) 277-333
333
REFEREXCES
58 59 60 61 62 63 63 6-l 65 66 67
a) El. GERKCHER ASD \V. MINDT, Surfrrce Sci.. -l6 (1966) 440. G;. BROUWER. to be prtbhshed. 31. 1. SPARNAAY. Sur/uce Ser., L (1964) 102. u.
Advan.
Colloid
Inter-ace
Sci..
1 (1967)
277-333