Calculation of surface quantum levels in tellurium inversion layers

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Calculation of surface quantum levels in tellurium inversion layers

J. Bouat, J.C. Thuillier

To cite this version:

J. Bouat, J.C. Thuillier. Calculation of surface quantum levels in tellurium inversion layers. Journal

de Physique, 1978, 39 (11), pp.1193-1197. �10.1051/jphys:0197800390110119300�. �jpa-00208859�

CALCULATION OF SURFACE QUANTUM ^LEVELS

IN TELLURIUM INVERSION LAYERS

J. BOUAT and J. C. THUILLIER (*) E.S.P.C.I., 10, ^rue Vauquelin, ⁷⁵⁰⁰⁵ Paris, ^France.

(*) ^Faculté des Sciences Mirande, Université de Dijon, ^on leave from Groupe ^de Physique ^{des Solides} (**), E.N.S., 24, ^rue Lhomond, ⁷⁵⁰⁰⁵ Paris, ^France

(Reçu ^le 26 janvier 1978, accepté ^le 20 juillet 1978)

Résumé.

Les énergies des niveaux quantifiés dans les couches d’inversion sur des plans [0001]

en surface du tellure, calculées à partir ^{de mesures Shubnikov de Haas, sont} comparées ^{avec des}

valeurs théoriques ^{obtenues avec} l’hypothèse ^d’un puits ^de potentiel triangulaire ^en surface. Dans la gamme expérimentale ^de concentration, les résultats s’expliquent par l’existence d’une seule ^sous- bande, séparée ^en deux niveaux du fait de la conformation en deux nappes de la bande de conduction du tellure.

Abstract.

The energies of surface quantum levels of inversion layers ⁱⁿ [0001] planes ^{at the}

surface of tellurium, calculated from Shubnikov de Haas measurements, ^are compared ^{with theo-}

retical values obtained with the assumption ^{of a} triangular ^surface potential well. In the experimental

concentration range, results ^are explained by the existence of a single ^sub-band, separated ^{in two}

levels due to the constitution in two sheets of the conduction band of tellurium.

Classification

Physics ^Abstracts

72.80C - 73.25 - 73.40N

1. Introduction.

When an inversion layer ^{is creat-}

ed at a semiconductor surface, ^for example by applying

a strong electric field perpendicular ^{to the} surface,

the motion of carriers in the corresponding potential

well is quantized ^and they ^{behave as a} two-dimensio- nal _{gas. The} energetic spectrum ^is composed ^of

discrete sub-bands and becomes completely ^discrete whén a magnetic ^{field is} applied perpendicular ^{to the}

surface. Numerous experiments, associated with theo- retical calculations, has been made on silicon, ^the band-structure of which is well known. For tellurium,

which is always p-type ^{at low} temperature, ^{it is} only recently ^that calculations of the conduction dispersion

relation were performed [1] and the conduction band appears ^to have a rather complicated structure. So the study of surface oscillatory magnetoresistance ^on

inversion layers ^{is of} particular ^interest.

Shubnikov de Haas (S ^d H) experiments ^have already ^been reported ^on tellurium inversion layers,

created in M.I. S. structures, the surface being [0001] [2]

or [1100] [3] planes. ^{In the last case, for which the}

electric and the magnetic ^{fields are} parallel ^{to the} trigonal (c-) axis, ^a simple interpretation ^{of the}

oscillation patterns ^{has been} given, which allows

graphic determination of the energetical position ^{of the}

(**) Laboratoire associé n° 17 au C.N.R.S.

single quantized sub-band with respect ^{to the Fermi} level. We present here detailed experimental results, with a more realistic interpretation ^and emphasize ^the

theoretical problem associated with the band structure of tellurium. Another way of determination of the

energies of electric field quantized ^{levels is} developed

and the results are cômpared ^{with a} theoretical calculation assuming ^a triangular potential well ;

the agreement ^{is rather} satisfactory, taking ^into

account the complexity ^{of the} conduction band of tellurium.

2. Experimental ^results.

Figure ¹ shows twice differentiated magnetoresistance ^{curves at various sur-}

face carrier concentrations induced by ^the applied gate

voltage _Vg. ^Two periods Pl ^and P2 ^{of S d H oscilla-}

tions can be measured, ^{and each extremum of} magne- toresistance is split ^{in two. The} position ^{in H -1 1 of}

each minimum is proportional ^{to the} period by ^a

coefficient (n ⁺ y). n ^{is an} integer ^{related to the} quantum ^{order of the} Landau level which induces the

oscillation, ^the phase ^factor y is about 0.25 for mini-

ma labelled n and 0.75 for those labelled n.

The periods depend ^{on the} energetic position ^{of the}

levels in the usual way :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390110119300

1194

FIG. 1.

Second derivative of magnetoresistance for various

applied gate voltages.

and the Fermi statistic at T

0 K in a two dimensio- nal electron _gas gives the total number of carriers in the inversion layer ^as

g, and _g, are the spin ^and valley degeneracies ; ^{for the}

conduction band of tellurium, ^{each one is} equal ^to

2. 0(x)=xifx > 0 ; 8(x) = 0 if x 0. _md is the

density ^{of states mass in the} plane parallel ^{to the} surface, equal ^{in modulus to the} cyclotron ^mass,

so that when two periods ^{are measured :}

Taking ^{into account this} relation, the inverse of S d H periods, ^{related to} energies by (1), ^are plotted

versus carrier concentration (Fig. 2) for different

samples.

3. Interprétation.

The conduction band of tellu- rium is described by the following dispersion ^{relation :}

kll ^{and ki 1 are} associated with motion parallel ^or perpendicular ^to the c-axis. The parameters ^{in this} relation are given by Blinowski et al. [3] ^to interpret

FIG. 2.

Inverse of oscillatory ^{S d H} periods ^{as a} function of carrier concentration.

magneto-absorption measurements. It can be easily

seen that, except very close to the bottom of the band,

the isoenergetic ^{surfaces are} separated ^{in two} parts,

an external one and an intemal _one, so that the conduction band is constituted of an external sheet which we label 2, ^{and an} intemal sheet which we label 1

(Fig. 3b, full lines). ^We shall examine how this consti- tution generates ^two energetic ^{levels for each sub-band}

under an electric field.

Figure ^{3a shows the} energetic diagram ⁱⁿ real space in the direction ^z perpendicular ^to the surface of tellurium when only ^the electric field is applied, creating ^a potential well which is supposed ^{to be}

linear in the inversion layer. ^Another shape ^of poten- tial well does not modify ^the qualitative phenomena.

The Fermi level EF ^{is fixed near the} top of the valence band in the bulk. Due to the confinement (some ^tens

of angstrôm), ^the motion of the electrons in the direction z is quantized ^{that is to} say that the modulus of their wave vector k (z ^is parallel ^{to the} c-axis) ^is imposed by the width of the potential ^{well at the} energetic level of motion. This configuration ^is transposed ^{in the} reciprocal ^{space as shown in} figure ^3b.

The potential being linear, 1 kil ^{1 varies in an hyper-}

bolic _way as a function of _energy, becoming ^{infinite at}

the bottom of thé well, ^as drawn in dashed lines. In that plane kl

0, ^a sub-band is represented ^from [4], assuming ^{that its} position ^with respect ^{to the bottom} of the well is known. It can be seen that two energetic levels, El, ^{on the internal} part ^{of the} sub-band, E2

on the external part, ^{with wave vectors} k, ^and k2,

are induced by ^the potential ^well (the figure ^{is not to}

scale). ^In fact, El ^and E2 ^{must be} simultaneous solutions of the Schrôdinger and Poisson equations,

and are related to the Fermi level EF by (2).

We now look at the effect of a magnetic ^field applied ^{in the same direction as} the electric field. The motion of the electrons in the plane parallel ^{to the}

surface is quantized in Landau orbits. The energetic position ^of Landau levels as a function of k II in bulk material has been theoretically calculated by ^{Blinowski i}

FIG. 3.

Conduction band in a plane ^k

^0.

et al. [1], ^and already ^{used to} interpret ^{S d H surface}

oscillations. Figure ^{3c shows} the Landau levels

arising ^from figure ^{3b when a} magnetic ^{field H}

^{30 kG}

is applied. ^{Levels n+ are} generated by the internal part ^{of the} sub-band, ^{levels n- from the external} part, and consequently ^{the electric field} imposes ^that 1 k 1 = 1 k 1 for electrons with spin T, ad k 1 = I k2 1

for electrons with spin 1. ^An interpretation ^{of the}

oscillation patterns ^has already ^{been made} [2] ^assum- ing that ) 1 k, 1 = ( k2 1, ^{and even if this} assumption

is not quite realistic, the result remains that, ^{due to the} asymmetry of Landau levels around kjj ^{= 0, the}

oscillations are split ^{in two} by ^the sign ^of kjj. ^An

oscillation is indeed observed when, increasing ^the magnetic field, ^a Landau level crosses the Fermi level ;

it appears clearly ^on figure 3c that this crossing ^does

not occur at the same magnetic ^{field if} kll ^{is positive}

of negative. ^For example, ^{level 2+ is} just crossing EF

for H

30 kG if kl,

^{+ 1 kl 1, but has already}

crossed it if kll ^{= -} 1 kl 1. So, the oscillations of each period ^are split ^{in two. This} procedure ^allows

a good interpretation ^{of the} experimental ^data,

even with the assumption that I 1 k, 1 = k2 1, ^which

is perhaps ^not very far from reality ^{if in} figure ^3b

the sub-band intersects the hyperbola ^{in a} part ^where the slope ^is steep. ^{It is} justified ^a posteriori by ^{the fact}

that the total number of carriers, calculated from (3),

is exactly ^{the same as the} number of electrons theore-

tically ^induced by ^the applied voltage ^{over a} large

experimental range. Therefore, ^the corresponding

1196

physical process, which has never been encountered in other semiconductors, ^{is not} clearly understood : for fixed values of electric field (j k1 1 and 1 k2 fixed)

and magnetic field, ^the electrons with spin T ^which

are reflected by ^{a side of the} potential well, ^for example

their wave vectors becoming negative ^from positive,

lose energy in their motion parallel ^{to the} surface, while ^at the same time, the electrons with spin 1

increase their energy. Is there a total conservation of energy, and what is the exchanging mode between electrons with différent spins ? The theoretical _pro- blem is _open, but this interpretation, ^{based on mathe-}

matical results valid in the bulk, ^{fits the} experimental

surface data rather well.

So two S d H periods may be observed, ^{that is to} say

a single electric field induced sub-band, separated

in two levels with energies EF - El ^and EF - E2

referred to the Fermi level. When a magnetic ^{field is} applied, ^the spin degeneracy of the levels is lifted, ^the

electrons with spin T being transferred on the internal part of the sub-band EF - El, and those with spin 1

on the external part EF - E2. ^Moreover the oscilla- tions are split by ^the asymmetry of Landau levels in

reciprocal space.

4. Energies ^{of the levels.}

^The energies E F - E ^can

be calculated from the periods Pi by (1) ^{if the} cyclotron

mass mc of the motion in ^a plane parallel ^{to the surface}

is known. Some cyclotron ^resonance experiments

were made on inversion layers by ^von Ortenberg

and Silbermann [4], but the measured _masses, given ^as

functions of gate voltage, ^{do not allow us to} interpret

our S d H data. So we use cyclotron ^{masses calculated}

as functions of _energy from the dispersion ^relation by :

where A is the extremal area of the isoenergetic

surface. This surface being split ⁱⁿ two, ^{a mass 1 is} associated with the internal part ^{of the} band, ^{and a}

mass 2 with the extemal part. The results are shown

on figure ^{4. The} cyclotron ^{mass for H} // ^{c is called} m 1- because it is equal ^{in modulus to} the effective ^mass in the direction perpendicular ^{to c if} trigonal warping ^is

FIG. 4.

Masses as functions of energy.

neglected, which is done in the dispersion ^relation (4).

The cyclotron ^{mass 2 is of the same order of} magnitude

as ordinary ^measured [4] ^or calculated [5], ^and

decreases when energy increases. The cyclotron ^{mass 1}

is smaller, and varies in an opposite way with _energy.

The energetic positions ^{of the levels are calculated}

from (1) through ^{an iterative} process in the following

way : ^a cyclotron ^mass m_L 1, is used for Ef. - Ei and a cyclotron ^mass m2 for EF - E2. ^{As the two}

levels are not independent, ^{mll and m2 are taken}

at the same value of _energy, corresponding ^{to the}

Fermi level and approximately equal ^to EF - E2.

The experimental ^{results are} represented by ^crosses

on figure ^{5. These} energies ^differ from those deter- mined by ^direct comparison ^between experimental ^and

theoretical positions ^{of S d H} oscillations [2] by ^{1 to}

2 meV, showing ^{that the} assumption ^{that k is the}

same for the two levels is ^not _very far from reality,

another reason for discrepancy being ^the large ^incer-

titude on the value of S in (4), determined by magneto-

absorption measurements and to which the energetic

distance E2 - El between levels is proportional.

FIG. 5.

Comparison ^between experimental ^{results and theo-}

retical curves for the energies of the levels as functions of carrier concentration.

5. Comparison ^{with a} theoretical model.

As

already said, the surface potential ^{well is assumed to be}

linear for a first approach ^{to the} problem. ^The quan- tized _energy levels, ^{referred to} the bottom of the

potential well, ^are given by :

where m is the effective mass for motion perpendicular

co the surface (here parallel ^{to the} trigonal axis).

Gauss’ law relates the surface electric field F, to ^the total number of carriers _{per unit} area N, :

E is the dielectric constant of the semiconductor

perpendicular ^{to the surface.}

The effective mass m (parallel ^{to the} c-axis) ^is

determined from the dispersion ^relation by :

km being ^the maximum value of k in the direction where m is calculated. Figure ^{4 shows} the existence of two digèrent masses corresponding ^{to the two} parts of the conduction band, ^so that, by combination of (5) ^and (6), ^{two levels are} generated ^{from the same}

quantum ^{order i :}

with

and relation (2) ^{must be writen :}

The self-consistent solutions solve simultaneously (1)

and (8). Ei1 ^and Ei2 ^{are not} strictly independent

because the masses involved in these equations ^{must all}

be taken at the _energy of the Fermi level.

In the concentration _range of our experiments (N, ^{4 x} 1012 cm- 2), calculations show that

E12 - E,2 ^{never becomes} smaller than EF - Eo2,

that is to say that the second sub-band ^cannot be observed and it corresponds ^to experimental ^results.

The theoretical curves for the two levels of the first

sub-band, ^now simply ^called El ^and E2, ^are compared

with the experimental points ^on figure ^{5. The} agree- ment is better when the carrier concentration and the

energies increase, ^{which is} conceivable from the fact that in the high energy range, the variations of the

masses are small, ^{and the error in} their choice is minimized. Through the theoretical model is simpli- fied, ^{it fits the} experimental results rather well.

6. Conclusion. The magnetic ^field being parallel ^to

the c-axis, ^{S d H} experiments ^{on inversion} layers ⁱⁿ [0001] J planes of tellurium M.I.S. structures can be

interpreted by the existence of a single electric field

quantized ^sub-band split ^{in two} levels due to the

separation ^{in two} sheets of the conduction band.

The problem of conservation of _energy of the electro- nic system ^under magnetic field, typical ^of the asym- metrical Landau levels in the reciprocal space, is open.

The _{energy of} the lower level is about 30 meV at a

carrier concentration of 3.5 x 1012 cm-2, ^that

is to say that the Fermi level is rather far from the bottom of the sub-band, ^which may be of much interest for the study of the conduction band. Even with the use of a simplified model, assuming ^{a linear}

variation of the potential ^{in the surface} region, ^the dependence ^{of the} energy of the surface sub-band with the charge ^{is well} described, in view of the

complexity of the conduction band of tellurium.

References [1] BLINOWSKI, J., REBMANN, G., RIGAUX, C., MYCIELSKI, J.,

J. Physique ³⁸ (1977) ^1139.

[2] ^BOUAT, J., THUILLIER, ^J. C., Phys. Lett. A 62 (1977) ^523.

[3] SILBERMANN, R., LANDWEHR, G., ^Solid State Commun. 16 (1975)

1055.

[4] ORTENBERG, ^M. V., SILBERMANN, R., ^Solid State Commun. 17

(1975) 617.

[5] SHINNO, H., YOSHIZAKI, R., TANAKA, S., DOI, T., KAMIMURA, H.,

J. Phys. ^Soc. Jpn ³⁵ (1973) ^525.