TWO DIMENSIONAL SYSTEMS IN SOLID STATE AND SURFACE PHYSICS : STRONG ELECTRIC AND MAGNETIC FIELDS EFFECTS

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TWO DIMENSIONAL SYSTEMS IN SOLID STATE AND SURFACE PHYSICS : STRONG ELECTRIC

AND MAGNETIC FIELDS EFFECTS

R. O’Connell

To cite this version:

R. O’Connell. TWO DIMENSIONAL SYSTEMS IN SOLID STATE AND SURFACE PHYSICS :

STRONG ELECTRIC AND MAGNETIC FIELDS EFFECTS. Journal de Physique Colloques, 1982,

43 (C2), pp.C2-81-C2-96. �10.1051/jphyscol:1982207�. �jpa-00221817�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-81

TWO DIMENSIONAL SYSTEMS IN SOLID STATE AND SURFACE PHYSICS : STRONG ELECTRIC AND MAGNETIC FIELDS EFFECTS

R . F . O ' C o n n e l l

Department of Physios and Astronomy, Louisiana State Un-iveraity, Bâton Rouge, LA 70803, U.S.A,

Résumé - Les systèmes bidimensionnels en physique de la matière condensée sont une source d'information importante pour l'étude des effets de champs électriques et magnétiques intenses. Nous passerons en revue l'état de ces questions pour trois de ces systèmes : les couches charge d'espace-électrons, à la surface d'un semi-conducteur dans un système (MOS) métal-oxyde-semi- conducteur, dans les super-réseaux des hétéro-jonctions, et à la surface de l'hélium liquide. En outre, nous présentons des résultats théoriques nouveaux sur ce dernier système, obtenus en collaboration avec J.A.C. Gallas.

Abstract - Two-dimensional condensed matter systems provide a fertile ground for the study of strong magnetic and electric field effects.

We review the present status of studies in three such systems - the electron space-charge layer at the semiconductor surface in a metal- oxide-semiconductor (MOS) system, in heterojunction superlattices, and at the surface of liquid helium. In addition, we present the results of new theoretical calculations on the latter system, which we obtained in collaboration with J. A. C. Gallas.

1. Introduction - During the past decade there has been a burgeoning interest in what is commonly referred to as strong-field problems i.e., problems in which more than one field is playing a leading role in determining the dynamics. For the most part, these problems involve external magnetic and/or electric fields combined with a Coulomb field such that all fields make comparable contributions to the basic forces and energies. It is now conventional to refer to such external fields as strong magnetic and strong electric fields. Thus, we are dealing with situations for which conventional perturbation theory is no longer adequate. Instead one must resort to such methods as numerical integration, variational calculations, semi-classical methods and WKB techniques.

It is of interest to trace the history of strong-field problems

particularly as textbooks invariably treat problems in which only one field is dominant, a particular favorite being the linear Zeeman effect in which a weak magnetic field perturbs the atomic Coulomb field. However, in 1939 it was recognized by Schiff and Snyder [1] that for highly-excited states of atoms, it was possible to achieve a strong-field situation. In other words, because the ratio of the Coulomb and quadratic Zeeman magnetic contributions decrease roughly C2D as n , where n is the principal quantum number of the atom, it is

possible to use laboratory-produced magnetic fields to obtain magnetic and Coulomb forces of comparable strength if one studies electron in high n orbits.

The theoretical analysis of Schiff and Snyder was motivated by the pioneering experimental investigations of Jenkins and Segre [3], who examined sodium and potassium absorption lines for n values in the range n = 10 to 35.

Strong-field problems in atomic physics lay dormant for a long time but in the meantime the subject was taken up by solid-state investigators E4-6D. The interest here involved electrons and excitons in semi-conductors. Because of the small effective mass and large dielectric constant associated with semi-conductors, the Coulomb field is relatively weak and the magnetic field relatively strong so that magnetic forces can be as large as Coulomb forces even for the ground state. This is now a mature but still active area of research, dealing with "three-dimensional" systems as distinct from the "two-dimensional"

systems discussed below.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982207

C2-82 JOURNAL DE PHYSIQUE

The n e x t impetus f o r such s t r o n g - f i e l d s t u d i e s came from an unexpected s o u r c e v i z . a s t r o p h y s i c s . I n 1967 p u l s a r s were discovered [71 and i n 1969 Gold proposed C81 what i s now commonly accepted a s t h e c o r r e c t theory of such o b j e c t s v i z . t h a t t h e y c o n s i s t of n e u t r o n s t a r s w i t h d e n s i t i e s of t h e o r d e r of n u c l e a r d e n s i t i e s and having magnetic f i e l d s of t h e o r d e r of 1013 G i . e . , about e i g h t o r d e r s of magnitude l a r g e r t h a n laboratory-produced f i e l d s . It was c l e a r t h a t , f o r atoms on t h e s u r f a c e of s u c h n e u t r o n s t a r s , one was d e a l i n g w i t h a

s t r o n g - f i e l d problem even f o r e l e c t r o n s i n t h e ground-state. This gave r i s e t o a f l u r r y of t h e o r e t i c a l i n v e s t i g a t i o n s by many groups C9,lOl. I n t e r e s t i n such s t u d i e s was f u r t h e r heightened by t h e f i r s t o b s e r v a t i o n a l c o n f i r m a t i o n of t h e e x i s t e n c e of l a r g e magnetic f i e l d s on dense bodies v i z . t h e d i s c o v e r y by Kemp C 1 1 1 i n 1970 of a magnetic f i e l d = 1 0 8 G i n t h e white dwarf s t a r Grw + ^{70' 8247,}

and t h e subsequent d i s c o v e r y of comparable f i e l d s i n o t h e r white dwarfs C121.

I n 1969 t h e pendulum swung back t o t h e realm of atomic p h y s i c s when Garton and Tomkins C131 c a r r i e d o u t t h e i r c l a s s i c i n v e s t i g a t i o n of t h e a b s o r p t i o n spectrum i n t h e p r i n c i p a l s e r i e s of Ba I f o r n v a l u e s a s h i g h a s n - 75, i n a magnetic f i e l d of 2'4 x 1 0 5 G. The s p e c t r a o b t a i n e d by t h e s e a u t h o r s gave a s i g n i f i c a n t s t i m u l u s t o t h e o r e t i c a l i n v e s t i g a t o r s s i n c e i t was c l e a r t h a t t h e v a r i a t i o n a l techniques h e r e t o f o r e a p p l i e d s o s u c c e s s f u l l y f o r ground-state problems were n o t s u i t a b l e f o r h i g h l y e x c i t e d s t a t e s i t u a t i o n s . As a r e s u l t t h e use of s e m i - c l a s s i c a l C141 and WKB techniques C15-161 came t o t h e f o r e . I n a d d i t i o n , t h e scope of t h e experimental i n v e s t i g a t i o n s was c o n s i d e r a b l y extended by o t h e r groups, i n v o l v i n g now t h e e f f e c t of i n t e r n a l l y generated e l e c t r i c f i e l d s C171.

I n r e c e n t y e a r s s t r o n g - f r e e e f f e c t s i n s o l i d - s t a t e and s u r f a c e p h y s i c s have moved t o c e n t e r s t a g e . The e f f e c t s we r e f e r t o occur i n so-called

"two-dimensional systems" i.e., systems f o r which e l e c t r o n i c motion i s quantized i n one d i r e c t i o n , s a y z , whereas i n t h e o t h e r two d i r e c t i o n s t h e e l e c t r o n motion may b e f r e e (but we a l s o examine s i t u a t i o n s i n which t h i s two-dimensional freedom may b e wiped o u t by t h e a p p l i c a t i o n of a n a d d i t i o n a l magnetic f i e l d ) . An e x c e l l e n t encyclopedic review of t h i s g e n e r a l a r e a h a s j u s t appeared C181.

Here we c o n c e n t r a t e on t h e s a l i e n t f e a t u r e s which i n v o l v e s t r o n g magnetic and e l e c t r i c f i e l d s . However, under t h i s umbrella, we a l s o wish t o i n c l u d e s i t u a t i o n s i n which t h e temperature i s s o low t h a t magnetic f i e l d e n e r g i e s a r e g r e a t e r t h a n thermal e n e r g i e s . T h i s g i v e s rise t o many i n t r i g u i n g o s c i l l a t o r y e f f e c t s i n v a r i o u s p h y s i c a l q u a n t i t i e s , such a s t h e quantized H a l l e f f e c t and t h e Shubnikov-de Haas o s c i l l a t i o n s of t h e l o n g i t u d i n a l magnetoresistance.

I n S e c t i o n s 2 - 4 we w i l l examine i n d e t a i l t h e most commonly s t u d i e d two-dimensional systems:

( i ) The e l e c t r o n space-charge l a y e r a t t h e semiconductor s u r f a c e i n a

metal-oxide-semiconductor (MOS) systems. Here t h e dominant f i e l d i s t h e applied e l e c t r i c f i e l d which g i v e s r i s e t o t h e space-charge l a y e r . Coulomb f i e l d s come i n t o p l a y v i a t h e s t r o n g many-body e f f e c t s of e l e c t r o n - e l e c t r o n i n t e r a c t i o n s . The a p p l i c a t i o n of a magnetic f i e l d t i l t e d w i t h r e s p e c t t o t h e semiconductor s u r f a c e g i v e s r i s e t o combined resonance t r a n s i t i o n s [ I 9 1 and t h e a p p l i c a t i o n of a magnetic f i e l d p e r p e n d i c u l a r t o t h e s u r f a c e g i v e s use t o t h e now-famous quantized H a l l measurements, which were pioneered by Von K l i t z i n g C201 and form t h e b a s i s of a new r e s i s t a n c e s t a n d a r d . Strong magnetic f i e l d e f f e c t s a r e a l s o of i n t e r e s t i n f a c i l i t a t i n g t h e formation of a Wigner l a t t i c e i n a MOS

system C211.

( i i ) A l t e r n a t i n g u l t r a t h i n l a y e r s of two semiconductors t h a t c l o s e l y match i n

l a t t i c e c o n s t a n t s , r e f e r r e d t o a s a h e t e r o j u n c t i o n s u p e r l a t t i c e , give r i s e t o

p o t e n t i a l w e l l s f o r e l e c t r o n s and h o l e s s o t h a t t h e e l e c t r o n i c quantum s t a t e s

a r e e s s e n t i a l l y two-dimensional i n n a t u r e . The a p p l i c a t i o n of a magnetic f i e l d

can g i v e r i s e t o a quantized H a l l e f f e c t L221 and a l s o evidence h a s been

presented f o r t h e e x i s t e n c e of a h i g h l y c o r r e l a t e d s t a t e such a s a charge d e n s i t y wave o r a Wigner l a t t i c e C231.

( i i i ) E l e c t r o n s a t t h e s u r f a c e of l i q u i d helium C24-263 c o n s t i t u t e perhaps t h e

"cleanest" two-dimensional system which can be s t u d i e d . A s d i s t i n c t from t h e o t h e r systems, here we a r e d e a l i n g with a non-degenerate e l e c t r o n gas, f o r which

- i n p a r t i c u l a r - t h e complicating e f f e c t s due t o many-body i n t e r a c t i o n s a r e n e g l i g i b l e .

In t h i s system we a r e dealing with a one-dimensional Coulomb p o t e n t i a l i n t h e presence of s t r o n g e l e c t r i c and magnetic f i e l d s . Here t h e Coulomb f i e l d i s considerably smaller than hydrogenic f i e l d s because t h e " e f f e c t i v e 2 ' ' i s small due t o t h e small d e v i a t i o n from u n i t y of t h e d i e l e c t r i c constant of helium. W e w i l l survey t h e experimental and t h e o r e t i c a l r e s u l t s a l r e a d y obtained f o r t r a n s i t i o n s i n t h i s system and we w i l l a l s o comment on t h e s t r o n g evidence f o r t h e e x i s t e n c e of a Wigner l a t t i c e C 271 .

In a d d i t i o n , we w i l l present t h e r e s u l t s of new t h e o r e t i c a l c a l c u l a t i o n s which were obtained by WKB techniques, i n c o l l a b o r a t i o n with J. A. C. Gallas.

2. The Metal-Oxide-Semiconductor (MOS) System - When a semiconductor s u r f a c e i s brought i n t o contact with another substance - u s u a l l y a metal, i n s u l a t o r , o r another semiconductor - a space-charge l a y e r a r i s e s on t h e s u r f a c e . Such a s u r f a c e space-charge l a y e r i s perhaps t h e b a s i c key ingredient of t h e microelectronics industry.

It i s generally agreed t h a t t h e metal-insulator-semiconductor (MIS) system

- f i r s t proposed by Moll C281 and by Pfann and G a r r e t t C291 i n 1959 a s a v o l t a g e v a r i a b l e c a p a c i t o r - i s t h e most u s e f u l of t h e p l e t h o r a of devices which e x i s t . Since by f a r t h e m s t common i n s u l a t o r used i s s i l i c o n dioxide, most s t u d i e s and a p p l i c a t i o n s make use of t h e metal-oxide-semiconductor (MOS) system, a s depicted i n Fig. 1.

The MOS system i s of i n t e r e s t f o r both technological and b a s i c s c i e n t i f i c reasons. F i r s t of a l l , t h e MOS system i s t h e gate s t r u c t u r e f o r most of t h e insulated-gate f i e l d - e f f e c t t r a n s i s t o r s (IGFET) o r t h e metal-oxide-semiconductor f i e l d - e f f e c t - t r a n s i s t o r (MOSFET). Secondly, t h e MOS system i s of i n t e r e s t from t h e viewpoint of b a s i c physics research because most of t h e important parameters can be varied simply by "turning a knob" [18,307, e.g., by simply changing t h e g a t e v o l t a g e ( s e e below) from i t s threshold value up t o - 10 ~ / c m , t h e charge 6 c a r r i e r d e n s i t y can b e v a r i e d continuously by about many o r d e r s of magnitude. -

(1015 - ^{3 x} lo1' c m 3 ) . This corresponds t o a s u r f a c e charge-density range of - from 10'~ - 1013 ( t h e l a t t e r v a l u e corresponding t o t h e maximum t o t a l induced charge beyond which t h e i n s u l a t o r breaks down). Thus, we have t h e p o s s i b i l i t y of studying many-body e f f e c t s under c o n t r o l l e d conditions.

Let us consider t h e d e t a i l s of a MOS system. To be s p e c i f i c , consider a l a y e r of p-type S i = cm thick. By oxidizing t h e s u r f a c e of the S i a t high temperatures a t h i n (1,000 - 5,000 1() i n s u l a t i n g l a y e r of S i O2 i s formed. By use of a metal l a y e r of t h i c k n e s s 20-50 8, c a l l e d t h e gate, a g a t e voltage V i s

R applied. I f a p o s i t i v e voltage i s applied a t t h e gate then a negative charge i s induced a t t h e s u r f a c e of t h e S i i n t e r f a c i n g with t h e i n s u l a t o r .

The e f f e c t of a p o s i t i v e g a t e v o l t a g e (V >O) i s t o bend t h e conduction and IL

valence bands of t h e semiconductor downwards and t o d e p l e t e t h e majority

c a r r i e r s (holes) i n a region, c a l l e d t h e d e p l e t i o n l a y e r , which extends i n t o t h e semiconductor about lo4 8. I f V_>O i s l a r g e enough such t h a t t h e conduction

6

band edge Ec c r o s s e s over t h e Fermi l e v e l EF, t h e number of minority c a r r i e r s

( e l e c t r o n s ) a t t h e i n t e r f a c e becomes l a r g e r than t h a t o f t h e majority c a r r i e r s

and t h e r e f o r e t h e s u r f a c e i s inverted. This reglon i s c a l l e d an i n v e r s i o n l a y e r

C2-84 JOURNAL DE PHYSIQUE

and extends E311 into the semiconductor to about 100 2. The case in which we are concerned is that of an n-type inversion layer in p-type Si, as shown in Fig. 1.

vg

? METAL

OXIDE

SEMICONDUCTOR

Fig. 1: (a) Metal-oxide- semiconductor (MOS) system. (b) Energy band diagram of a MOS system with p-type semiconduc- tor and a positive gate voltage.

/INVERSION LAYER

Fig. 2: Quantized energy levels for the motion of the inversion layer electrons perpendicular to the semi- conductor interface.

4

evg '0 7 -

M

2.1 Electric Subbands - In 1957, Schrieffer C321 proposed that the electric field associated with an inversion layer was strong enough to produce a

potential well whose width perpendicular to the interface, which we take as the z direction, was small compared to the wavelengths of the carriers. This one-dimensional potential well is responsible for the quantization of the energy levels of the inversion layer electrons. These energy levels are grouped into what are called electric subbands, each corresponding to a quantized level for motion in the z direction and no restrictions on the motion in the xy-plane, the plane parallel to the interface (see Fig. 2 ) . This two dimensional structure of the electron gas was confirmed experimentally in 1966 by Fowler et al. C331 using cyclotron resonance techniques.

The most extensive calculation of energy subband levels En has been carried

-

d 0

--

out by Stern and Howard C311 using a self-consistent calculation in which the surface layer charge is used in conjunction with Poisson's equation to determine the profile of the surface potential. The latter in turn is substituted into Schrodinger's equation to give the wave-function from which the surface charge can be determined - and so on. As it turns out a good estimate of the relevant parameters - with an attendant physical insight - may be obtained if we follow Stern C341 and use the triangular-potential approximation i.e., V(z) = eFz for z > 0, with an infinite barrier for z > 0. The corresponding wave- function is an Airy function, from which it readily follows that C341

/

, f snv

g z d r p *

- c

EF

E v

/-DEPLETION L

S(p-type)

and

where F i s t h e f i e l d a p p l i e d normal t o t h e semiconductor v o l t a g e by means of t h e g a t e v o l t a g e , and where m, = 0.9160~1 i s t h e e f f e c t i v e mass i n t h e d i r e c t i o n normal t o t h e s u r f a c e , m denoting ?he f r e e e l e c t r o n mass. Thus, i n atomic u n i t s (H = mo = e = 1, s o ? h a t t h e u n i t of energy i s 27.18 eV and t h e u n i t of l e n g t h i s t h e Bohr r a d i u s a and hence t h e atomic u n i t of e l e c t r i c f i e l d is F

e / a 2 = 5-142 x 10 Vlcm), 9

Thus, f o r a t y p i c a l f i e l d of 10 V/cm 5 = 1.945 x FOB i t f o l l o w s t h a t E o z 63 meV. S i m i l a r l y , t h e confinement l e n g t h i s given by

Correspondingly, t h e subband s p l i t t i n g s a r e t y p i c a l l y 10-100 meV and s o t h e s p e c t r o s c o p i c range of i n t e r e s t is i n t h e i n f r a - r e d . These e n e r g i e s a r e , of c o u r s e , much s m a l l e r t h a t t h e S i band-gap energy of 1.11 eV.

1 2 2 I n a d d i t i o n , we n o t e t h a t a t y p i c a l s u r f a c e d e n s i t y of 10 /cm

i m p l i e s a Fermi energy = 10 meV and hence, a t low temperatures, t h e system i s a d e g e n e r a t e Fermi gas.

Observation of i n t e r b a n d o p t i c a l t r a n s i t i o n s i s t h e b e s t means of o b t a i n i n g i n f o r m a t i o n on t h e subband s t r u c t u r e . However, i n t h i s system many-body

c o n t r i b u t i o n s a r e v e r y s i g n i f i c a n t s o t h a t , i n g e n e r a l , t h e r e s o n a n t energy does n o t correspond t o t h e a p p r o p r i a t e subband energy s e p a r a t i o n . I n p a r t i c u l a r , t h e r e i s a d e p o l a r i z a t i o n (resonance s c r e e n i n g ) e f f e c t [351 and a l o c a l - f i e l d c o r r e c t i o n ( e x c i t o n - l i k e e f f e c t ) C361. The former i n c r e a s e s t h e resonance energy whereas t h e l a t t e r d e c r e a s e s i t s o t h a t o v e r a l l t h e s e e f f e c t s tend t o c a n c e l o u t C361. Because of t h e l a t t e r and c o g n i z a n t of t h e f a c t t h a t t h e i n c l u s i o n of such e f f e c t s must b e t r e a t e d n u m e r i c a l l y , - w i t h a corresponding l o s s i n p h y s i c a l i n s i g h t - we i g n o r e such e f f e c t s from h e n c e f o r t h .

2.2 Combined subband-Landau l e v e l t r a n s i t i o n s - We w i l l now t u r n t o a c o n s i d e r a t i o n of t h e magnetic f i e l d e f f e c t s . I f B = B ( i . e . i f t h e magnetic f i e l d i s a p p l i e d along t h e same d i r e c t i o n a s t h e e l e c t r i c f i e l d ) then each subband i s f u r t h e r quantized i n t o d i s c r e t e Landau l e v e l s s o t h a t t h e energy becomes E

En + ^(N + $ ) b e , where wc = (e~lcrn,, ) i s t h e c y c l o t r o n frequency.

n, N

A more i n t e r e s t i n g s c e n a r i o i s t h a t of " t i l t e d magnetic f i e l d s . " I n p a r t i c u l a r , Beinvogl and Koch C191 i n v e s t i g a t e d e l e c t r o n s on Si(l,O,O), i n t h e presence of a magnetic f i e l d t i l t e d w i t h r e s p e c t t o t h e sample s u r f a c e (B and B components), and observed combined resonance t r a n s i t i o n s because of a

& u p l i n g of Landau l e v e l s and subband s t a t e s . S u r p r i s i n g l y , t h e y found t h a t t h e sum of t h e s e p a r a t i o n s f o r t h e AN = 1 and AN = -1 t r a n s i t i o n s is {0.7 - 1 . 6 ) h c

= 2 . 3 h c i . e . 15% h i g h e r t h a n t h e expected r e s u l t of 2Huc.

C2-86 JOURNAL DE PHYSIQUE

P r i o r t h e o r e t i c a l work on t h i s problem by Ando C371 reached t h e conclusion t h a t

Thus we get t h e c h a r a c t e r i s t i c energy changes of (hC)AN corresponding t o AN t r a n s i t i o n s . As shown by Ando E371 t h i s conclusion i s not affected by the i n c l u s i o n of many-body and o t h e r e f f e c t s . The l a t t e r a f f e c t s t h e d i f f e r e n c e i n t h e p o s i t i o n s of the main (ANSO) and a combined (AN # 0) resonance peak b u t does not a f f e c t t h e d i f f e r e n c e i n t h e p o s i t i o n s of two combined resonance peaks. A b a s i c assumption made by Ando was t o t r e a t B a s a p e r t u r b a t i o n so t h a t i t s

Y

i n f l u e n c e on t h e z-part of t h e wave f u n c t i o n - i s neglected. Recently Ando f3Sl c a r r i e d o u t a more d e t a i l e d i n v e s t i g a t i o n without t h i s r e s t r i c t i o n but i t i s c l e a r ( s e e f i g . 9 of r e f . 38) t h a t discrepancies between experiment and theory s t i l l e x i s t . Ando used an approximation based on t h e local-density-functional theory, taking i n t o account 5 subbands and 20 Landau l e v e l s f o r the inversion l a y e r .

2.3 Wigner L a t t i c e - We t u r n now t o another phenomenon i n which t h e presence of a s t r o n g magnetic f i e l d plays a s i g n i f i c a n t r o l e ( e s s e n t i a l l y by l o c a l i z i n g t h e e l e c t r o n s i n t h e i r quantized o r b i t s ) . This concerns t h e very recent evidence f o r t h e existence of a highly c o r r e l a t e d o r c r y s t a l l i z e d ground s t a t e - ^{a Wigner}

l a t t i c e I 3 9 1 - i n S i i n v e r s i o n l a y e r s i n t h e extreme quantum l i m i t . This evidence came from i n f r a r e d measurements of t h e cyclotron resonance i n t h e two-dimensional e l e c t r o n gas, which revealed a remarkable l i n e narrowing and s h i f t i n t h e resonance frequency t o higher values. A s emphasized by Wilson e t a l . C211 "...the strong l i n e narrowing i s not a f e a t u r e of the one-electron t h e o r i e s of Ando...". Q u a l i t a t i v e l y speaking, t h e formation of a l a t t i c e i s f a c i l i t a t e d by having low inversion-layer e l e c t r o n concentrations ns, low temperatures T, and high magnetic f i e l d s B. I n t h e experiments of Wilson e t a l .

11 2

E211, t y p i c a l values used were n z 10 /cm , T = 1.2

and B values of 6.15 T and 7.69 T.

Several i n v e s t i g a t o r s C40,411 have pointed out t h a t a strong magnetic f i e l d I3 can induce Wigner c r y s t a l l i z a t i o n . I n p a r t i c u l a r , Alastuey and Jancovici l411 have made u s e of t h e Wigner - Kirkwood expansion t o c a l c u l a t e quantum

c o r r e c t i o n s t o t h e f r e e energy i n t h e f l u i d and s o l i d phase. In a d d i t i o n , they point out t h a t , i f B i s s u f f i c i e n t l y l a r g e so t h a t t h e cyclotron r a d i u s & ' l e s s than t h e de Broglie wavelength A, then L replaces h a s a c h a r a c t e r i s t i c quantum l e n g t h s c a l e and t h e motion t r a n s v e r s e t o t h e magnetic f i e l d has a l a r g e e f f e c t i v e mass associated with i t . As a r e s u l t , t h e quantum f l u c t u a t i o n s a r e quenched and a degenerate system such a s t h e electrons i n t h e MOS inversion l a y e r becomes more c l a s s i c a l . This leads t o t h e aforementioned conclusion t h a t a strong B i n c r e a s e s t h e s i z e of t h e solid-phase domain.

F i n a l l y , w e s p e c u l a t e t h a t t h e e x i s t i n g discrepancies between t h e o r e t i c a l

expectations and t h e observations of Beinvogl and Koch might be due t o t h e

complete neglect of c o l l e c t i v e e f f e c t s i n t h e t h e o r e t i c a l a n a l y s i s , s i n c e

one-electron t h e o r i e s a r e t h e b a s i s of a l l t h e e x i s t i n g t h e o r e t i c a l

i n v e s t i g a t i o n s of t r a n s i t i o n e n e r g i e s . A s we have p r e v i o u s l y noted C421, t h e v a l u e s of ns and B i n t h e Wilson e t a l . experiments a r e n o t very d i f f e r e n t t h a n

t h o s e used by Beinvogl and Koch C191, b u t t h e l a t t e r a u t h o r s used T = 4.2

Now Wilson e t a l . C211 p o i n t o u t t h a t t h e c y c l o t r o n resonance broadens and s h i f t s t o lower frequency ( i . e . t h e Wigner l a t t i c e s t a r t s t o disappear) a c r o s s t h e temperature range of 5 - ²⁰

and that t h e temperature dependence i s a p p a r e n t l y independent of t h e v a l u e of e l e c t r o n d e n s i t y , e l e c t r i c f i e l d , o r magnetic f i e l d . The T v a l u e of 4.2

used by Beinvogl and Koch i s thus seen t o b e on t h e b o r d e r l i n e and perhaps t h e system i t s e l f i s b o r d e r l i n e between t h e extremes of a e l e c t r o n gas and a Wigner l a t t i c e .

2.4 Magneto-Optical Experiments - O p t i c a l experiments can b e divided n a t u r a l l y i n t o two d i s t i n c t c l a s s e s , v i z . interband and i n t r a b a n d e f f e c t s . The i n t e r b a n d t r a n s i t i o n s i n v o l v e quantum s t a t e s i n two d i f f e r e n t energy bands w h i l e i n t r a b a n d t r a n s i t i o n s , or simply f r e e c a r r i e r e f f e c t s , involve only a s i n g l e energy band.

Our a t t e n t i o n h e r e w i l l be confined t o intraband e f f e c t s and s i n c e t h e energy band gap C42al of S i a t 0

i s 1.16 eV (and 1.12 eV a t 300 OK), we a r e t h e r e f o r e r e s t r i c t e d t o photon e n e r g i e s i n t h e range 1 0 -1 eV which is t h e -4 mid-infrared, f a r i n f r a r e d and microwave r e g i o n s of t h e electromagnetic spectrum.

Cyclotron resonance has been t h e most u s e f u l and most f r e q u e n t l y used magneto-optical t o o l s i n c e i t s discovery i n t h e e a r l y f i f t i e s C431. I f a n electromagnetic wave of angular frequency w i s s e n t through t h e system, t h e e l e c t r o n w i l l o s c i l l a t e a t t h e f r e q u e n c i e s w and wc simultaneously. When o i s a d j u s t e d t o e q u a l wc, r e s o n a n t a b s o r p t i o n occurs and t h e e l e c t r o n w i l l move i n a n o r b i t of i n c r e a s i n g r a d i u s u n t i l i t c o l l i d e s a f t e r a time r , c a l l e d t h e c o l l i s i o n time, w i t h t h e l a t t i c e . This method has been used e x t e n s i v e l y t o o b t a i n information on t h e e f f e c t i v e mass of t h e i n v e r s i o n l a y e r e l e c t r o n s i n MOS systems and p a r t i c u l a r l y i t s dependence on T , B , w and ns C18,441.

Another u s e f u l t o o l , which b o t h complements and supplements t h e c y c l o t r o n resonance work i s t h e o b s e r v a t i o n of Faraday r o t a t i o n , f i r s t i n v e s t i g a t e d i n 1845 by Faraday C451. It r e f e r s t o t h e r o t a t i o n of t h e p o l a r i z a t i o n d i r e c t i o n of a l i n e a r l y p o l a r i z e d electromagnetic wave propagating p a r a l l e l t o t h e

magnetic f i e l d d i r e c t i o n . Faraday r o t a t i o n measurements have only r e c e n t l y been c a r r i e d on t h e MOS system C461 and they should provide information n o t only on e f f e c t i v e mass v a l u e s b u t a l s o on c o l l i s i o n times C471, a s w e l l a s possibly throwing some l i g h t on t h e so-called c y c l o t r o n resonance dilemna C481. I n a d d i t i o n , we have r e c e n t l y pointed o u t t h e importance of i n c l u d i n g m u l t i p l e r e f l e c t i o n e f f e c t s i n t h e i n v e r s i o n l a y e r C491 r e t h e i n t e r p r e t a t i o n of Faraday r o t a t i o n and e l l i p t i c i t y measurements.

2.5 Quantized H a l l E f f e c t - Condensed m a t t e r systems a r e o f t e n considered t o b e

" d i r t y " i n t h e sense t h a t t h e i r complexity o f t e n m i t i g a t e s a g a i n s t o b t a i n i n g a v e r y a c c u r a t e experimental number. A counter example of course is t h e

determination of e/h from t h e a c Josephson e f f e c t . A f u r t h e r example h a s r e c e n t l y emerged from measurements of t h e quantized H a l l e f f e c t C20i. This provides a new method f o r a p r e c i s o n d e t e r m i n a t i o n of t h e f i n e - s t r u c t u r e c o n s t a n t cr o r , a l t e r n a t i v e l y , i f a i s assumed t o be known from some o t h e r experiments ( a s , f o r example, measurements of t h e a c Josephson e f f e c t and t h e gyromagnetic r a t i o ) , t h e n we a r e provided with an improved standard r e s i s t a n c e .

In t h e pioneer quantized H a l l experiment, von K l i t z i n g e t a l . a p p l i e d a magnetic f i e l d of 150 kG normal t o t h e oxid$-silicon s u r f a c e of t h e MOS system

( z d i r e c t i o n ) , a t a low temperature of 1.5 K. Next, a n e l e c t r i c f i e l d was a p p l i e d i n t h e x d i r e c t i o n g i v i n g r i s e t o a l o n g i t u d i n a l c u r r e n t j = a Ex and

X

a H a l l c u r r e n t j = o E , where o denotes t h e c o n d u c t i v i t y t e n s o r . Then

Y y x x i j

t h e g a t e v o l t a g e was Garied from 0 to-25 V r e s u l t i n g i n an i n c r e a s e i n n with a

C2-88 JOURNAL DE PHYSIQUE

r e s u l t a n t approximate l i n e a r i n c r e a s e i n t h e Fermi energy = ⁶ m e ~ / 1 0 ~ ~ e l e c t r o n s cm -2 , r e l a t i v e t o t h e lowest e l e c t r i c subband Eo. The remarkable r e s u l t found was t h a t t h e H a l l r e s i s t a n c e , % say, d i s p l a y e d quantized s t e p s a s a f u n c t i o n of V ( s e e Fig. 3) and a t each s t e p % i s given i n terms of t h e fundamental

g

c o n s t a n t s t o an accuracy of about 1 ppm. S p e c i f i c a l l y ,

where n i s t h e number of Landau l e v e l s l y i n g below t h e F e d energy.

GATE VOLTAGE (V) Fig. 3: H a l l v o l t a g e a s a f u n c t i o n of t h e MOS g a t e v o l t a g e .

W e t u r n now t o a n e x p l a n a t i o n of t h i s r e s u l t . The key requigements a r e temperature and s t r o n g magnetic f i e l d . Since a temperate of 1.5 K i m p l i e s a kT v a l u e of only 0.13 meV, t h e s e p a r a t i o n between E and E i s such t h a t f o r n <

3 -

" 1

6 x loLL ^{~ m - ~} o n l y t h e En l e v e l i s occupied ( e l e c t r i c quantum l i m i t ) . I n t h e presence of t h e magnetic f i e l d Eo s e p a r a t e s i n t o Landau l e v e l s and f o r a B f i e l d v a l u e of 150 kG and a n e f f e c t i v e mass of 0 19 Mo ( a p p r o p r i a t e t o S i ) t h e Landau l e v e l s e p a r a t i o n s & e q u a l s 9.13 meV.

I n g e n e r a l , when kT << hac one can obsercre o s c i l l a t o r y behavior, p e r i o d i c

i n B - ~ , i n v a r i o u s p h y s i c a l q u a n t i t i e s as, f o r example, t h e Shubnikov-de Haas

o s c i l l a t i o n s i n t h e l o n g i t u d i n a l c o n d u c t i v i t y uXX. For t h e i n v e r s i o n l a y e r ,

t h e s e o s c i l l a t i o n s can be s e e n n o t only i n t h e u s u a l manner by v a r i a t i o n of t h e

magnetic f i e l d b u t a l s o by varying t h e s u r f a c e c o n c e n t r a t i o n , t h e l a t t e r being

achieved simply by changing t h e g a t e v o l t a g e . It t u r n s o u t t h a t t h e period i s

independent of t h e e f f e c t i v e mass b u t i n c r e a s e s when e l e c t r o n s populate t h e

h i g h e r subbands, t h u s providing a method f o r t h e determination of t h e e l e c t r o n

d e n s i t y i n each subband. It i s a l s o of i n t e r e s t t o n o t e t h a t t h e s e o s c i l l a t i o n s

do n o t appear when B i s a p p l i e d p a r a l l e l t o t h e s u r f a c e , hence p r w i d i n g a

s t r o n g confirmation f o r t h e two-dimensional n a t u r e of t h e e l e c t r o n i c space-charge system.

The g a t e v o l t a g e i s now adjusted t o s e t E between two Landaulevels so t h a t F

a l l t h e Landau l e v e l s below E a r e completely occupied and t h o s e above a r e F empty. A s a r e s u l t a= (and hence j ) e s s e n t i a l l y drops t o z e r o because t h e

X

e l e c t r o n s have no p l a c e t o s c a t t e r i n t o due t o t h e l a r g e gap of hc between t h e f i l l e d and empty Landau l e v e l s . Furthermore, o = n ec/B and n -

yx s - n gn where

g is t h e degeneracy of t h e nth Landau l e v e l . However, s i n c e g eB/hc i t

n 2 n

f o l l o w s t h a t o = ne /h i . e . t h e H a l l c o n d u c t i v i t y (and hence t h e H a l l

YX

r e s i s t a n c e ) does n o t depend on any of t h e MOS parameters b u t only on fundamental p h y s i c a l c o n s t a n t s .

3 . Heterojunction S u p e r l a t t i c e s - A l t e r n a t i n g u l t r a t h i n l a y e r s of two semi- conductors t h a t c l o s e l y match i n l a t t i c e c o n s t a n t s a r e r e f e r r e d t o a s a h e t e r o j u n c t i o n s u p e r l a t t i c e . S t u d i e s s o f a r have concentrated on two systems, one being made of GaAs-A1As o r G ~ A S - G ~ ~ - ~ % A ~ C501 and t h e o t h e r i s of

InAs-GaSb o r Inl-xGaxAs-GaSbl-yAsy C511. Most i n v e s t i g a t i o n s have been c a r r i e d o u t on t h e former system and f b r t h i s reason o r remarks w i l l b e confined t o i t . We n o t e that Gal-x Al A s i s a s i n g l e c r y s t a l m a t e r i a l i n which A 1 atoms have

X

r e p l a c e d a f r a c t i o n x of t h e gallium atoms. This three-atom system possesses a h i g h e r energy gap and a s m a l l e r index of r e f r a c t i o n t h a n Ga A s .

One can regard t h e s u p e r l a t t i c e a s e s s e n t i a l l y "....nothing more t h a n a s t r i n g of back-to-back s i n g l e i n t e r f a c e s . . . " C501. A s e l e c t i v e l y doped Ga A s -

A l Ga A s system can r e s u l t i n an energy l e v e l diagram d e p i c t e d i n Fig. 4. This comes about from t h e demand t h a t t h e Fermi l e v e l be c o n s t a n t a c r o s s t h e system w i t h t h e r e s u l t t h a t e l e c t r o n s move a c r o s s t h e i n t e r f a c e . This in t u r n c r e a t e s a s t r o n g i n t e r n a l e l e c t i v e f i e l d which causes a s i g n i f i c a n t bending of t h e bands a t t h e i n t e r f a c e . As a r e s u l t , we s e e t h a t t h e r e i s a s t r o n g confinement of e l e c t r o n s i n a one-dimensional p o t e n t i a l w e l l perpendicular t o t h e s u r f a c e . T h i s region i n Fig. 4 i s marked "2 DEG" r e f e r r i n g t o t h e two-dimensional e l e c t r o n gas, s i n c e e l e c t r o n i c motion along t h e two-dimensional i n t e r f a c e i s unhindered.

F i g . 4: Energy l e v e l diagram f o r a

E ~ 2

GaAs - AlGaAs s u p e r l a t t i c e showing

t h e p o t e n t i a l w e l l which g i v e s r i s e

t o t h e two-dimensional e l e c t r o n gas

(2 DEG).

JOURNAL DE PHYSIQUE

Thus we have a s i t u a t i o n v e r y s i m i l a r t o t h a t which r e s u l t s i n an i n v e r s i o n l a y e r i n t h e MOS system, with a corresponding range of s t u d i e s . However, t h e d e n s i t y of two-electrons e l e c t r o n s i n Ga As cannot be v a r i e d e a s i l y ( i n c o n t r a s t t o t h e MOS system) s i n c e t h e e l e c t r o n s a r i s e from donor i m p u r i t i e s placed i n s i d e t h e A l x Gal-x As c r y s t a l . On t h e o t h e r hand, one advantage of t h e s u p e r l a t t i c e systems i s t h e f a c t t h a t we a r e d e a l i n g with r e l a t i v e l y low t r a n s v e r s e e f f e c t i v e masses v i z . 0.07 m f o r Ga A s and 0.023 mo f o r I n A s ( t o b e compared t o 0.1905 m f o r t h e MOS sys?em). A s a r e s u l t , s t r o n g f i e l d regimes s t a r t a t lower f i e l d v a l u e s . To c i t e two examples, we n o t e t h a t w i s p r o p o r t i o n a l t o ,. m-' and t h a t energy l e v e l s i n a n e l e c t r i c f i e l d a r e p r o p o r t i o n a l t o m-L ( s e e f o o t n o t e i n r e f . 52).

I n p a r t i c u l a r , s i n c e t h e e f f e c t i v e mass i n Ga A s i s only about 113 of i t s v a l u e i n S i , i t was p o s s i b l e t o c a r r y o u t quantized H a l l measurements i n t h e Ga A s - ^{A l Ga A s} system i n a f i e l d of 80 kG and a t a temperature of 4

C221 which, of course, makes t h i s h e t e r o j u n c t i o n s u p e r l a t t i c e a prime c a n d i d a t e a s a p o t e n t i a l s t a n d a r d of a b s o l u t e r e s i s t a n c e i n l a b o r a t o r i e s throughout t h e world, p a r t i c u l a r l y s i n c e i t i s s i g n i f i c a n t l y cheaper t o produce magnetic f i e l d s of t h i s v a l u e vis-a-vis t h e 150 kG f i e l d s r e q u i r e d i n MOS systems.

A s emphsized by Tsui e t a l . C221, t h e quantum regime can b e reached with even s m a l l e r B v a l u e s i n narrow-gap semiconductors, p o i n t i n g o u t a s a n example t h e r e a l t i v e l y low v a l u e of 10 kg r e q u i r e d t o o b t a i n t h e quantized H a l l e f f e c t u s i n g t h e two-dimensional e l e c t r o n system i n I n Sb. I n f a c t , Tsui e t a l . C221 have determined a-' t o 0.17 ppm, i n e x c e l l e n t agreement with t h e 0.11 ppm v a l u e obtained from t h e gyromagnetic r a t i o of t h e p r o t o n combined w i t h t h e Josephson e f f e c t .

F i n a l l y , we would l i k e t o mention t h a t magnetotransport s t u d i e s of two- dimensional e l e c t r o n s i n t h i n Ga A s - ^Al Ga A s m u l t i l a y e r s g i v e evidence f o r

x 1-x

t h e formation of e i t h e r a Wigner l a t t i c e o r a charge-density wave C231.

Magnetic f i e l d s a s high a s 210 kG were used and temperatures from 2.4 t o 4.2

4. E l e c t r o n s a t t h e s u r f a c e of l i q u i d Helium - The study of image-potential- induced s u r f a c e s t a t e s f o r e l e c t r o n s o u t s i d e of l i q u i d helium h a s a t t r a c t e d much i n t e r e s t i n r e c e n t y e a r s C24-261. This system i s of i n t e r e s t due t o t h e f a c t t h a t t o a good approximation t h e e l e c t r o n s form a two-dimensional e l e c t r o n gas and a l s o because of t h e v e r y l a r g e m o b i l i t y of t h e e l e c t r o n s along t h e s u r f a c e . I n a d d i t i o n , it provides information on t h e n a t u r e of t h e helium liquid-vapor i n t e r f a c e and, most s i g n i f i c a n t l y , i t appears t o be t h e f i r s t system i n which t h e e x i s t e n c e of a Wigner l a t t i c e C401 has been experimentally v e r i f i e d C271.

These v e r y i n t e r e s t i n g p r o p e r t i e s o r i g i n a t e from t h e circumstance of t h e d i e l e c t r i c c o n s t a n t

of helium being v e r y s l i g h t l y g r e a t e r than 1 ( s

1.0572), a s was f i r s t pointed o u t independently by Cole and Cohen C241 and by ShikinC251.

I n t h e c a s e of a t y p i c a l d i e l e c t r i c t h e e l e c t r o n s a r e a t t r a c t e d t o t h e s u r f a c e by a n e l e c t r o s t a t i c image f o r c e and t h e n become bound t o s p e c i f i c atoms o r d e r e c t s . However, i n t h e c a s e of l i q u i d helium, t h i s f o r c e , which i s

p r o p o r t i o n a l t o E-1, i s very weak. Also, t h e r e i s a p o t e n t i a l b a r r i e r a t t h e s u r f a c e of 1 eV which p r e v e n t s t h e e l e c t r o n p e n e t r a t i n g i n t o t h e s u r f a c e . As a r e s u l t t h e e l e c t r o n s a r e trapped i n a one-dimensional w e l l , g i v i n g r i s e t o quantized s t a t e s i n t h i s d i r e c t i o n (x s a y ) . The p o t e n t i a l is given by

where C531 Z = % ( E - 1 ) / ( ~ + 1 ) = 6.951 x I f we now assume t h a t t h e b a r r i e r

i s i n f i n i t e a t t h e o r i g i n we a r e l e d t o a "hydrogenic" spectrum f o r e l e c t r o n s

trapped a t t h e s u r f a c e . Some i n t e r e s t i n g c h a r a c t e r i s t i c s of t h i s spectrum were

d i s c u s s e d by Grimes and Brown 1541. I n p a r t i c u l a r , t h e ground s t a t e energy i s

0.6 meV and t h e a s s o c i a t e d Bohr r a d i u s i s 114 %. The t h e o r e t i c a l t r a n s i t i o n e n e r g i e s from t h e n = 1 ground s t a t e t o t h e n = 2 and 3 e x c i t e d s t a t e s a r e 119.3 GHz, and 141.3 GHz r e s p e c t i v e l y , whereas t h e experimental v a l u e s C551 a r e 125.9 f0.2 and 148.6t 0.3 GHz. Also, because t h e s p l i t t i n g s a r e 5.7

and 6.8

i n temperature u n i t s , only t h e ground s t a t e i s s i g n i f i c a n t l y populated a t t h e temperature 1

a t which t h e experiments a r e performed.

To improve agreement between t h e o r y and experiment, and a l s o t o provide a more r e a l i s t i c c a l c u l a t i o n , Cole and Cohen C241 argued that t h e 6' p o t e n t i a l should be t r u n c a t e d n e a r t h e i n t e r f a c e and i n f a c t t h e y assumed a c o n s t a n t p o t e n t i a l -Ze / b f o r 0 _< 2 x _< b ( s e e Fig. 5 a ) , o b t a i n i n g agreement with experiment f o r a 1 eV r e p u l s i v e b a r r i e r and a b v a l u e

10%. However, such a l a r g e v a l u e of b is i n c o n s i s t e n t w i t h t h e premise of t h e model t h a t Eq. (9) i s r e a s o n a b l e f o r v a l u e s of x g r e a t e r t h a n t h e inter-atomic spacing i n t h e l i q u i d , which i s 3.6 8. Next, Grimes e t a l . C251 proposed a p o t e n t i a l ( s e e f i g . 5b) V(x) = 1.0 eV f o r x 5 0 and v ( x ) = -Ze /(x+B) f o r x>O, which i s e q u i v a l e n t t o assuming t h a t t h e image p o t e n t i a l has i t s o r i g i n a small d i s t a n c e i n s i d e t h e l i q u i d . They found agreement with experiment f o r 6 = 1.04 8. ^Other

c a l c u l a t i o n s C561 have a l s o taken account of t h e f a c t t h a t t h e helium d e n s i t y does not change a b r u p t l y from t h e b u l k l i q u i d v a l u e t o t h e vapor v a l u e .

These s t u d i e s took on a new dimension by t h e a p p l i c a t i o n of an e l e c t r i c f i e l d F. As emphasized by Grimes e t a l . C551, s i n c e t h e s p l i t t i n g between energy l e v e l s i s i n t h e mm-wave region i t i s convenient from an experimental p o i n t of view t o work w i t h a f i x e d frequency and t o v a r y t h e energy s e p a r a t i o n s by means of t h e e l e c t r i c f i e l d . The e l e c t r i c f i e l d s e r v e s t o p r e s s t h e

e l e c t r o n s toward t h e s u r f a c e with a consequent compression i n t h e wave f u n c t i o n s . Grimes e t a l . analyzed t h e i r experimental r e s u l t s u s i n g a v a r i a t i o n a l c a l c u l a t i o n and employing t h e a d j u s t a b l e parameter B. More r e c e n t l y , Lambert and Richards [57], u s i n g f a r - i n f r a r e d l a s e r s p e c t r o s c o p i c techniques, observed t r a n s i t i o n s a t v a r i o u s e l e c t r i c f i e l d on t h e same system.

To model t h e i r o b s e r v a t i o n s they used t h e one-dimensional Hamiltonian

where p i s t h e momentum i n t h e x d i r e c t i o n , Z

6.95 x and F i s t h e a p p l i e d e l e c t r i c f i e l d . By numerically i n t e g r a t i n g e q . ( 1 0 ) they were a b l e t o show t h a t t h i s model can a c c u r a t e l y reproduce t h e o b s e r v a t i o n s .

I n a previous work C581 we have s t u d i e d t h e dependence of t h e t r a n s i t i o n f r e q u e n c i e s a s a f u n c t i o n of t h e e l e c t r i c f i e l d , i n t h e "hydrogenic" model, using two d i f f e r e n t approaches : a s e m i c l a s s i c a l [58,591 and a WKB one C581.

The WKB approach i s p a r t i c u l a r l y simple t o apply s i n c e t h e corresponding q u a n t i z a t i o n r u l e can b e reduced t o standard e l l i p t i c i n t e g r a l s which, i n t h e i r t u r n , can be evaluated even with programmable pocket c a l c u l a t o r s . The

f i r s t - o r d e r WKB q u a n t i z a t i o n r u l e corresponding t o t h e Hamiltonian i n Eq. (10) i s given by ( u s i n g atomic u n i t s )

where

and where E i s t h e energy. The t u r n i n g p o i n t a i s t h e p o s i t i v e r o o t of t h e

r a d i c a l of Eq. (121, namely

JOURNAL DE PHYSIQUE

LIQUID nT- ^{LIQU'D HELIUM d -p 0 VACUUM -x-}

I I I I

I I I I I I I

Fig. 5: Model of the potential seen by electrons at the surface of liquid helium, as proposed by (a) Cole and Cohen (24) and (b) Grimes et al. (55).

Eq. (12) may be then rewritten as

where K(k) and E(k) are the complete elliptic integrals of the first and -

second kind, respectively, and k" ^{= a/(a-b).} A closely related integral is

d~/dn, the energy spacing between the energy levels. By differentiating

Eq. (11) with respect to n and integrating one obtains

Field (Vlcm)

Field (Vlcm)

Electric Field ( V l c m )

0 . . .

0 500 1000 1500 2000 Electric Field (Vlcm)

Fig. 6: P l o t of t r a n s i t i o n f r e q u e n c i e s v s e l e c t r i c f i e l d f o r t h e bound e l e c - t r o n s t a t e s a t t h e s u r f a c e of l i q u i d helium. The c r o s s e s a r e t h e measured d a t a p o i n t s of Grimes e t a l . (55) w h i l e t h e s o l i d c u r v e s a r e t h e r e s u l t of a WKB c a l c u l a t i o n based on t h e model Hamiltonian given by E q . (10) i n t h e t e x t .

Fig. 7: P l o t of t r a n s i t i o n f r e - q u e n c i e s v s e l e c t r i c f i e l d f o r t h e bound e l e c t r o n s t a t e s a t t h e sur- f a c e of l i q u i d helium, correspond- i n g t o t h e f i r s t 9 t r a n s i t i o n s from t h e ground s t a t e r e c e n t l y ob- served by Lambert and Richards C571. The c u r v e s a r e t h e r e s u l t of a WKB c a l c u l a t i o n based on t h e model Hamiltonian g i v e n by Eq.

(10) i n t h e t e x t . The bottom

curve is simply a blow-up of a

p o r t i o n of t h e t o p curve.

C2-94 JOURNAL DE PHYSIQUE

The e l l i p t i c i n t e g r a l s K(k) and E(k) needed i n t h e s e e x p r e s s i o n s can be

computed very e f f i c i e n t l y . From t h e s e v e r a l algorithms t o t h i s end a v a i l a b l e i n t h e l i t e r a t u r e C601 we c a l l a t t e n t i o n t o t h e ones by Carlson C611, which, a s p r e v i o u s l y mentioned, may e a s i l y be implemented i n programmable pocket c a l c u l a t o r s .

Using Eqs. ( l l ) , (14) and (15) we c a l c u l a t e d t h e f i e l d dependence of t h e frequency f o r t h e 1-t 2 and 1 + 3 t r a n s i t i o n s . These r e s u l t s a r e given i n Fig. 6 along w i t h t h e experimental v a l u e s of Grimes e t a l . L551 a s read from t h e i r Fig. 2. A s can be seen from Fig. 6, t h e WKB r e s u l t s a r e i n good agreement w i t h t h e v a r i a t i o n a l c a l c u l a t i o n of Grimes e t a l . A s i n t h e c a l c u l a t i o n of Grimes e t a l . , t h e agreement with t h e measured v a l u e s d e g e n e r a t e s a s t h e f i e l d i n c r e a s e s . S i n c e Fig. 6 was o b t a h e d i n t h e "hydrogenic" model, f o r which t h e c a l c u l a t e d v a l u e s a r e s l i g h t l y s m a l l e r than t h e measured ones, b o t h curves were normalized t o a g r e e a t t h e f i r s t measured p o i n t .

I n Fig. 7 we g i v e t h e f i e l d dependence of t h e frequency f o r t h e f i r s t 9 t r a n s i t i o n s which were r e c e n t l y observed by Lambert and Richards 5 7 1 . Our curves were o b t a i n e d from t h e WKB approximation, u s i n g Eq. (14), and Fig. 7b should be compared with t h e corresponding curves i n Fig. 5 of Ref. 57, obtained through a numerical i n t e g r a t i o n of Eq. (10). The o v e r a l l agreement i s very good i f one n o t e s t h a t Fig. 7b h a s n o t been r e s c a l e d t o t a k e t h e "non-hydrogenic"

c h a r a c t e r i n t o account.

Actually, s i n c e we a r e using an i n f i n i t e b a r r i e r (which i s a good approximation s i n c e t h e a c t u a l b a r r i e r of 1 eV i s more t h a n lo3 times t h e t y p i c a l binding e n e r g i e s ) , t h e n u s i n g an argument due t o Langer C621 ( s e e a l s o M i l l e r and Good C631 and Adams and M i l l e r C641) Eq. (12) f o r - I must b e modified (by adding a -1 /8xL term under t h e r a d i c a l ) f o r u s e i n Eq. (11) s o t h a t i n t h e l i m i t F = 0 we o b t a i n t h e c o r r e c t q u a n t i z a t i o n r u l e E

n -2 . However, i t

n

t u r n s o u t t h a t t h i s term does n o t make a s i g n i f i c a n t c o n t r i b u t i o n t o t h e energy s p a c i n g s except f o r small F v a l u e s C651.

We would a l s o l i k e t o p o i n t o u t t h a t t h e WKB approximation provides an easy way t o study t h e f i e l d dependence of t h e t r a n s i t i o n f r e q u e n c i e s f o r a l t e r n a t i v e models, e.g. t h e "0-shifted" model of r e f . (55). These t o p i c s , however, w i l l b e discussed elsewhere C657 .

Acknowledgements - The new t h e o r e t i c a l r e s u l t s on l i q u i d helium, which were presented i n S e c t i o n 4 , were obtained i n c o l l a b o r a t i o n w i t h J. A. C. G a l l a s , t o whom t h e author would l i k e t o e x p r e s s h i s thanks.

This r e s e a r c h was p a r t i a l l y supported by t h e Department of Energy, D i v i s i o n of M a t e r i a l s Science, under c o n t r a c t no. DE-AS05-79ER10459.

The author i s very g r a t e f u l t o Mrs. Gay Sutton f o r t h e g r e a t c a r e s h e took i n t y p i n g t h e manuscript and l a y i n g i t o u t i n a camera-ready form.

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