A new general quantum transport model

SupeHattices and Microstructures, VoL 7, No. 2, 1990

103

A NEW GENERAL QUANTUM TRANSPORT MODEL G. Dorda Siemens AG, Corporate Research and Development, Munchen, Fed. Rep. Germany (Received 13 December 1989) A new universal quantum transport model is presented for discussion. It represents the specimen conductance as an e2/h value modified by quantum numbers referring to the limiting length valuele = 18.2 nm, which is characteristic for the two-dimensional electron gas. This value is derived from the reformulated Bohr postulate in which a quasi mass me = a2mo assigned to the coulomb energy unit is used to represent the electron mass. This reformulation permits the definition of the fine-structure constant a only in the standard form for QED. It will also be shown that the electron density 3.0x101s m-2 represents a boundary between localization and current conductivity. The plausibility of this model is verified on the basis of various data obtained from Si MOSFETs and (At,Ga)As/GaAs heterostructures. Its general applicability is confirmed by a suitable interpretation of the spectacular dependencies of the 2-dimensional electron gas on the magnetic field and temperature. In addition, the dimensional independence of the magnetoresistance at high magnetic fields, observed on Si MOSFETs, is shown here for the first time to be a significant property of the 2-dimensional electron gas and is interpreted successfully on the basis of our transport model.

1 Introduction A series of unexpected, fundamental results have come to light from transport measurements recently performed on 2-dimensional electron gas systems at low temperatures. It is known that the quantization in the magneto-transport in the quantum Hall effect (QHE) refers to the universal value h/e~ 1,2 Similar behavior was also found in conductivity measurements w i t h o u t a magnetic field at point contacts 3,4. Measurements at thin constrictions of GaAs heterostructures showed quantized stages in the resistance at the breakdown of the QHE, and these refer only to h/e 2 and not to the specimen material 5. Another new aspect of the relation between h/e2 and the charge transport became apparent when investigating the temperature dependence of the resistance in the magnetic field zone of the QHE. It was shown that in a relatively wide temperature range, the conductivity for (AI,Ga)As/GaAs heterostructures can be formulated by using the resistivity tensor in the relation o~x~ = Px./(h2/ne~)2 = e2/h exp(-&/T) 6; here n is a quantum number and & an activation constant depending on the magnetic field. For Si MOSFETs too, an adequate temperature dependence of the specific resistance was observed, the measured data being described by px~,, = hi(he) 2 exp(-A /-r) 7. This temperature behavior of the transport electrons can therefore also be regarded as an universal phenomenon. The value h/e2 clearly plays a marking role in the electron transport. This is also shown by the experimental result found on thin superconducting granular films, which shows that the superconducting property can occur only at resistances R~,~< h/4e~ 8.

(3-/49-6036/90/0201 03 + 11 S02.00/0

Further questions of interest arise when we try to present the spatial distribution of the current in a specimen with a 2-dimensional electron gas. The recently presented n o n c o n t a c t i n g m e t h o d based on t h e Pockels effect showed that the current flows not only along the specimen edges as postulated for the QHE regime, but also in the middle of the specimen, and more surprisingly - that the potential distribution does not differ significantly when it is measured in or outside the QNE region 9. The resulting conclusion of the similarity, if not the equivalence between the transport mechanism w i t h i n and outside the QHE zone is consolidated by measurements on Si MOSFETs, which show the independence of the magnetoresistance on the specimen dimensions within and outside the QHE condition 7. This new kind of effect will be discussed in detail at the end of this paper. The recently observed material-independent universality of the resistance behavior in the intermediate regions of the QHE, deduced from the temperature dependence of &G", a kind of inversely proportional half width, and of (Sp,y~/S8)rnax, a field derivation of the Hall resistivity p,y~ 10,1~,can be regarded as corresponding to our interpretation. If we now try to weigh up and evaluate all these e x p e r i m e n t a l results and d a t a , w e reach t h e conclusion that the charge transport in 2-dimensional systems, and perhaps in solids generally, is still not fully understood and correctly formulated. It is thus an obvious t h o u g h t to develop a generally applicable transport model whose formulation would be based on a more fundamental starting point, such as the OHE state, and which would represent every other current-conducting electron state, characterized by © 1 990 Academic Press Limited

104

Superlattices and Microstructures, Vo/. 7, No. 2, 1990

the given resistance, as a disturbance of this initial state. To be able to develop this idea, we will set up a new type of quantum transport model and present it for discussion. Encouraged by the e x p e r i m e n t a l findings of Bliek et al. 5, in which the specimen resistance at the breakdown of the OHE is given by Rs = (nRnL/n o i)h/ez, where nR, nL, no and i are quantum numbers, we use the Bohr quantum condition as a starting point. The conclusions from this model to be regarded as decisive for solid-state transport will be contrasted with various experimental results and discussed.

not to the quantizable rotation velocity v but to the constant velocity of light c linked to a quantizable coulomb force mass with the limiting value m e. In place of the Bohr radius, d e f i n e d a t n = 1 in SI units by 411~oh2 a°

=

m

e2

=

5,29"10 -11 m

(6)

o

we now have the fundamental reference length ae

2. The Transport Model ¢t4nr

a) The Coulomb Interaction

ae =

The energy unit referred to the coulomb force is eVo = 27 eV 12 This value differs from the intrinsic electron energy, given by its rest mass mo, by a2 -~ 137-2, where a has the meaning of the Sommerfeld finestructure constant, or coupling constant, i.e.

m° c

2

=

a

~2

_

=

7,25.10_

9 m

(7)

where ~o is the electric field constant and e the charge of the electron. This equation, regarded from the standpoint of (4), implies the relations

(8)

e2

tr

(1)

e V

h2

_ ° m ee 2

0

-4Uh

where c is the velocity of light. If we now assume as a hypothesis that in describing the coulomb interaction between t w o charges we may start only from the quasi mass me defined by

(2) me =

0 3 m°

i.e.

mec ~ =

/if

e

= eV °

° =

moo

c

a°

(3) =

acquires a new form, as expressed below

meC

(4)

ae=

Here in (2a) and (3) h is Planck's constant, fe the maximum frequency, ao the Bohr radius and Vo the velocity of the electron along the first Bohr orbit in the hydrogen atom. Instead of the thus designated "Bohr radius", we obtain the new value ae - J ae

:

¢t

o,tr

=

C

,

(9)

where etr and ~o,tr represent transformed quantities, respectively. Both these equations are physically meaningful and thus justified: Equation (8) is in full agreement with the formulation of a on the basis of ~ t r = 1 used in quantum electrodynamics (QED) 13, i.e.

(2a)

the Bohr quantum relation referred to the coulomb force, given bythe following expression for n = 1

movoa

C

(5) a°

which should be more appropriately regarded as a reference length assigned to the electron state. Equation (4)suggested here implies that we should understand the electron as a location-bound wave, assigned

e2 o

e2 o

4 I1

411 ~ tr

(10)

where eo is the observable charge of the electron. The identity of (8) with (10) becomes obvious after the allowed transformation eo2 --~ err2 and ~tr "~/~Equation (9) also corresponds to physical experience, as eo can be freely selected 14, but must, t o g e t h e r with the magnetic field constant Po, correspond to the Maxwell equations as a boundary c o n d i t i o n , which is satisfied in conjunction with Po tr = 1/c. It is evident that the statements in (8) ~'(10) require a transformation of the charge magnitude e and also of the field constants eo and ]ao. This is possible because in physics we have only three independent fundamental quantities - length, time and mass (or energy). But taking into account that it produces no physical contradiction in the event of these transformations not being performed, thus they need not be pursued further in this paper. We may therefore use the SI units in all following formulations, giving us a better overview. In contrast to (4) which has been reformulated here, equation (3), which was originally formulated by Bohr and which starts from a classical corpuscular concept of the electron, leads to the relation

Superlattices and Microstructures, Vol. 7, No. 2, 1990 % El :

(11)

-C

It is a conclusion which does not contain the basic idea of QED, according to which Q is a coupling constant between the electron and the photon. No coupling is expressed in (11), which means t h a t this e q u a t i o n misses any further justification. In concluding this section, let us again point out that the proposed change in mass and length formulated in (2) to (5) does not affect the magnitude of the electron energy, which in turn means that our model is invariant with respect to Schr6dinger's equation, i.e. the compatibility of wave mechanics w i t h our simple quantum model can be assumed. It will be shown below that in attempting to formulate a quantized resistance, the minimum length ae assigned to the coulomb force plays an important role in the interpretation of the quantum numbers.

105 is to be understood asatime-referred n u m b e r Here, meff is the effective mass and ~r the mean time of the free paths. By including mo/meff in the relation (14), the observed independence of the material, i.e. the possible universality of the transport phenomena, is antici-pated It should be noted that the (possibly quantum) number np, although referred to the invariant time parameter Te, is, due to (13), also related to the area 2nao2, which we recognize as a basic area of the 2-dimensional system. At first glance this presents a problem, because due to (5) the number np would not be invariant with respect to the selection of the basic area. But since we can represent the density N quantized with respect to the basic area, the product Np becomes invariant with respect to the selection of the magnitude of this area, i.e. e

p e

Np-

h

2

n

*2 h

n

e

b) The Ouantized Two-Dimensional Conductance

(16)

t~

npe h

-

e

where The two-dimensional resistance R~ is defined by ~lp

RE]-

n

(12)

L

eNp W

---- - De 112 e

n* = u 2 n

where N is the charge carrier density per m-2, p the mobility, L the length and W the width of the specimen. in the following, we will try to relate the resistance R. to h/e2 so that we can show that the experimental-results of Bliek et al. 5, Clark et al. 6, Wittmann et al. 7, van Wees et al. 3 and others are in agreement with our model. As in the Bohr postulate, we assume that the length ae can be increased only in quantum iumps. This premiss allows a simple and easily understood solution of the problems involved to be obtained. The mobility # can be reformulated with the aid of (2) and (4) to become m iJ

=

e

meff

e

h

,t, cl 2

o

- mefr

m o

"17 t t 2

reel T

Te

m e

p

I1

=

n

N:

• (.~,,.") ,

e

e

(17a)

e

' : ( .'2'.,,.~I o

\

e

'

(18)

e /

Here nt4e, np* and he* are possibly quantum numbers and n e is a number. The quantum number npe, which characterizes the product Np, is referred exclusively to the invariant time and is thus physically incontestable. On the basis of our analysis therefore, splitting up the magnitude Np among N and p appears to be solely a theoretically formal but physically unfounded procedure. We can therefore rewrite (12) as I

--e ~11~t,)o P h

R~

where np

n*2 = El2n2

p

and

(13) 2/la 2 e

(17)

!1" I1 -n*2 e

:

-

n/

ripe

~w

I1

~

h

(19)

e '~

where m

L

o

(14)

m ell" T e

rl[=

7%

W '

v,

(2o)

-l

P /,,=

!2ila~){

=

18.9-rim

(21)

with

Te

(!

_

f I = 2,42.10 e

I~s

(15)

Here too, nl and n w should be understood to be quantum numbers. For the specific resistance Pxx~ we therefore obtain

Super/attices and Microstructures, Vol 7, N o 2, 1990

1 06

--

Pxx~

I

h

npe

e2

(22)

number of collective cells in the given current illa m e n t of length L and we may w r i t e a q u a n t u m number nlpe for it

and thus the f o l l o w i n g relation at nw = I for the resistance of a current filament

nl rl/lle -~

n~ Rp

(n = I)

h

n lie

"~

(23)

2

The essential thrust of (4) is that we should consider ae in a coulomb interaction between t w o charges to be the smallest possible length which can be determined. If we make this statement generally applicable and transfer it to a two-dimensional electron system, then individual electrons can be distinguished only at a density of N -< No, where the critical 2-dimensional density No is given by (24}

This also means that at N _> No the electrons can no l o n g e r be t r e a t e d i n d i v i d u a l l y , b u t m e r g e i n t o groups: collective electron units, probably of coherent phase, are formed. According to our q u a n t u m model, therefore, the value le in (21) becomes a limiting length for a two-dimensional electron-gas system. It determines the limiting number of the current filaments in (12) by way of nw- It also determines the q u a n t u m number nl which we may consider as the m a x i m u m n u m b e r of distinguishable electrons or electron blocks within a current filament of length L. The m o d i f i c a t i o n of nl by nue in (19) essentially amounts to the dissociation of the length L into nl/nue sections each of which corresponds to a single co ective electron state identical to the QHE state (see Fig. I). This means that the fraction nl/npe represents the T I

Drain

l,r.l~ ,

pe

t

2

]

L

n

v,

nf~,~ = 1, 2, 3, ... Normally nl > npe > I, which implies that the trend to collectivization of the electrons predominates in the current filament and is prevented mainly by phonons and ion disturbances. The state of the largest disruption density, ie. at nll~e' = nl, must be expected at a temperature T -,p oo. In this case, the t w o - d i m e n sional resistivity simply becomes p , , ~ = hle2. In conclusion, it should be mentioned that this conductivity model exhibits a certain similarity to Landauer's one-dimensional transport model in which the one-dimensional current path traverses conductive zones delimited by barriers with reflection trans mission properties formulated with the aid of a transmission matrix 15, ~6 c) The Current at N < No As illustrated in Fig. 2 for the case of N < No, the Io calization of the electrons can be countered to some extent by spatial redistribution to create one-dimensional current filaments. The critical principle for the formation of a current filament must be the relation N(nw = I) -> No(nw = I). If we assume that the spacing between the electrons in a current filament is the limiting length Ie given by (21), the application of the boundary conditions for the redistribution defined by

I

[• J

i

(25)

The two-dimensional conductance G, can then be for mulated as follows

e

N<,= (2t~a~} -I = 3,03-1015m 2

-n

i • f~o~',

N = 3.0-I0'5m 2 •

.I.]

ai=o J ®

4~

Drain

i.

" Current electron

•

. • /

®

:: ii : !i i o

--

.

•

•

Drain ]F~--

. . . . .

1

re/ :

®

i.l

(~ 3'

•J

i

le Source

Fig.1 Schematic d i a g r a m of t h e c u r r e n t t r a n s p o r t process flowing by m e a n s of 6 c u r r e n t f i l a m e n t s a t an electron density N = 3 0 x l 0 1 s m 2 le is t h e m i n i m a l w i d t h of t h e current filament, for simplicity, t h e elect r o n s are d e p i c t e d in t h e classical m a n n e r as points. The d a s h e d hnes symbolize t h e b o u n d a r i e s b e t w e e n the colletived electrons.

{

N = 7 5 . 10~4 m 2 ~ Localized "L f Electron • Current Source

Source

Fig.2 Schematic d i a g r a m of t h e possible f o r m a t i o n of a current filament at N < 3x10 Is m-2 The electrons are d e p i c t e d classically as points.

Superlattices and Microstructures, Vol. 7, No. 2, 1990 Nw N = NwNc + NI°(' ~

N--~ N° + Nt°('

(27)

yields the resistivity in the form

PxxE] -

1

Nl

No

Nl

ellN

N W

ep

N(N-Nto,: )

(28)

where NIoc is the two-dimensional density of the remaining, i.e. localized electrons that do not contribute to the formation of the current filaments. NI is the one-dimensional electron density in the current filament and Nw represents the one-dimensional curr e n t - f i l a m e n t - f o r m i n g electron density across the width. The equation for the two-dimensional resistance when N < Noisthus No

R~-

ep

Nl

L

(29)

N(N-Nloct W

Since electrons can be redistributed only in a homogeneous solid in which a nearly constant value is expected for the factor p ("mobility") and in which, to a certain extent, Nloc = 0, in this specific case this model yields the unusual relation RC] ~

(30)

N -2

In the last section it will be shown that the specific type of current transport at N < No is in full agreement with the experimental data. 3. Comparison of the Model with the Experimental Data a) The Quantum Hall Effect The specific property of the integral quantum Hall effect (QHE) in a two-dimensional system is that the Hall resistance is measured in quanta of h/e2 in accordance with

R~

HE -

l

h

i

e2

(31)

where the relation between charge density N per m-2 and the magnetic flux density B, given by N=

e

i--B h

(32)

is satisfied 1. Here i is a quantum number known as the integer filling factor of the Landau levels. An important property of this two-dimensional electron gas in the QHE state is revealed by experiment and signifies that the resistance between source and drain is identical to the Hall resistance R~jQHE as defined by (31) 17. Even more interesting is the fact that a macroscopic area through which the current can

107 f l o w w i t h o u t dissipation is formed in the region between the source and drain, i.e. in this region R = 0 1,2. Another aspect of this two-dimensional electron gas is the fact that this state is independent of the specimen length and width and even of its material, and this state of the electrons is therefore universal in character. In the case discussed below for i = 1, the sourcedrain resistance is defined by hie2. Since this resistance is independent of the material and the dimensions of the specimen, we must assume that in accordance with (19), both = !

n[ = n

(33)

and (34)

npe = 1

must hold. Furthermore, (14) and (17) demand that meff be one of the factors on which the quantum number nue should be dependent. Since the OHE is independent of the material, it follows from (34) that m,

(35) -

I

me|T In their unique way, (33)-(35) express an electron state in which the current-conducting two-dimensional electron gas is completely decoupled from the specimen and its lattice and must form a collective state of macroscopic dimensions because of its independence of charge density. Another reason why (35) is so interesting is that it has already been postulated as an interpretation of how the plateaus in the QHE are produced 18 and because it is also in agreement with the results of the gyromagnetic measurements of superconductive Pb 19 On the basis of experience gained with the integral and fractional QHE 1,2,20, we may conclude that the spatial and temporal order between the electrons must play an important part in the collectivization process. The factor that characterizes this order in the QHE is the externally applied homogeneous magnetic field which can cancel any kind of disturbance up to the outcome of a collective electron state in at least one current filament extending the entire length of the specimen. b) Magnetoresistance in a Two-Dimensional Conductor Bliek et al. studied the decay of the QHE in a narrow constriction of the channel of a (AI,Ga)As/GaAs heterostructure and discovered that pronounced quantum steps in the resistance occurred especially in the specimen with the critical electron density No = 3.0x10 Is m-2 5 In all probability, this concentration seems to be so important because only in this case are the elec trons homogeneously distributed in the collective cells. The quantized behavior has been described by a resistance equation Rs in quantized form as follows:

R = s

n~t

nL

h

no i

e2

n~ = 1,2,3,...

(36)

108

Super/atdces a n d Microstructures, Vo/. 7, No. 2, 1990

where nR iS an empirically determined variable quantum number, nL is the one-dimensional n u m b e r o f electrons along the length L of the constriction, no is the total number of electrons in the relevant constriction and i is the filling factor of the Landau level in the QHE. Since the q u a n t u m number no is defined as no = nL.nw, where nw is the one-dimensional number of electrons across the width, the q u o t i e n t (no.i)/nL must represent the number of current filaments. For one current filament therefore, (36) yields the simple resistance equation w i t h the variable nR

A t t e n t i o n should be drawn to the fact that no material-specific parameters such as meff/mo are present in (19), (23), (26) or (36). Expressed more precisely, the results obtained by Bliek with nR as a sequence of integral numbers show that the number n~e and thus, via (14), the variable time-referred term m o "r/meffT e in (15) can only represent a sequence of integral numbers. We may thus see in this fact an experimental confirmation of the existence of a delimiting measure of time Te assigned to the electron and i n d e p e n d e n t of the material in question, and also of a d e l i m i t i n g measure of length ae by the way of the velocity of light.

R In

=1)=

nR

"

(37)

c) The Independence of the Magnetoresistance from the Specimen Dimensions

If we w a n t to draw a comparison between (23) and (37), we must also bear in mind that (19), (23) and (26) formulate the resistance between source and drain, whereas in s the resistance was measured between the potential probes. It is common k n o w l e d g e that the source-drain delimitation of the current filament alone yields an additional a priori h/ie2 value 17 When we take this circumstance into account, a comparison of (23) with (37) yields the f o l l o w i n g q u a n t u m - n u m ber relation for nR ".

= "~e-

I

(38)

Thus, the variable quantum number nR in 5 represents the number of subdivisions of the original, h o m o geneous collective electron state along the length L of the constriction. Expressed in d i f f e r e n t terms, (nR + 1) = nlue is the number of single collective electron states (collective cells) in a current filament of length L.

The f o l l o w i n g discussion deals with the anomalous connection between the macroscopic dimensions of the specimen and its resistance, which does not correspond to Ohm's law, a property of the 2-dimensio nal electron gas in the magnetic field first observed on Si MOSFETs by Wittmann et al 7 A typical result can be seen in Fig. 3. The diagram shows the source drain resistance RSD of three transistors from the same chip w i t h the length-to-width ratio LAN as a parameter in dependence of the magnetic field. The measurements of the resistance RSD, which was determined by means of the four-terminal method (fourterminal resistance), were carried out at T = 1.5 K and an electron density of 3.0x1016 m-2. For the specific resistance at B = 0 we obtained the values p ,a = 228 ~, P ,b = 228 ~ and p ,c = 250 ~, the maximum mobility being p = 2.8 m2/Vs. What is striking at first glance is the greatly differing modification of RSD by the magnetic field. The extraordinary and unexpected dependence of the Shubnikov-de Haas oscillations on the

I RsD 2O a) L/W = 80 b) L/W= 22.9 c) L/W = 11

(kD)

15

f\

10

i

3

6

(T)

Fig.3 The dependence of the source-drain resistance RSD on the magnetic field B, measured at T = 1.5 K, N = 3x1016 m-2 and ! = 0.S ]JA by the 4-terminal method on 3 Si MOSFETs w i t h the l e n g t h - t o - w i d t h ratio as a p a r a m e t e r , a) L = 3.2 mm, W = 401Jm; b)L = 0.8 ram, W = 35 pm; c)L = 1.76 mm, W = 160 jam 7

9

Superlattices and Microstructures, Vol. 7, No. 2, 1990 specimen dimensions offers a good o p p o r t u n i t y of testing our proposed transport model for its informational value. In accordance with our model, the intensity of the phonon and ion disturbance on the electron transport is so great that up to 8 = 3 T the macroscopically ordering effect of the magnetic field, which is expressed in the reduction of the quantum numbers nw and nlue in (26), is of equall magnitude on both quantum numbers, i.e. Ohm's law remains va d and we can write

109 Here v represents a non-integral number. In analogy to the quantum number i in (36), this number v should be interpreted as the n u m b e r of current levels, formed in parallel to the first surface layer. Starting from (26), and bearing in mind the four-terminal method in the determination of RSD, which analogously to (38) causes a correction of nlpe by 1, we can write 2 I

G[j

=

__e2 h

} n

n:t~e

I

=

e2

I|Vl'

h

n'~r

--(n"e> n/

=

k

-w°~ L

(39)

where L and W are the length and width of the transistor channel, kc~is the specific conductance, which is modified from B = 1 by the ordering as well as disturbing effects of the magnetic field. The brackets < > symbolisethe mean value of npe. As can be seen from the experimental results, an increasing reduction of the size of the maxima of RSD,which is most pronounced for specimen a) with the greatest LAN ratio, occurs from B = 3T. On the basis of our model, this reduction of the amplitude heights can be interpreted to mean that in the process of collectivizing the electrons, the value nw strives toward a certain saturation, in contrast to niveThe determining factor in this process is assigned to the Hall voltage. From the experimental data of Bliek et al.S, we can show that at a constant magnetic field, electron density and current I, the source-drain voltage VSD is proportional to the variable quantum number nR (see (36)). We now characterize this nR as nSD (40) VSI )

ksi )

=

(41) VH = k H n H

kH is the p r o p o r t i o n a l i t y constant, nH a quantum number. By using the Hall voltage equation

v. : (eN) 'JR

(42)

and the relation e2 =

v--

h

B

-

e

(43)

we can formulate a relation between the current I and VH e

I =

,,

2

h

h

e9

n

{ntpe-I)

VSI) -

~,'n H

h

{tlSi)) VSI)

(45)

Using (44) and (45) we obtain the relation o H

VH

(nsi) )

VSl )

(46)

and, in agreement with (40) and (41), the relations n~ k -3,,

(47) =

k nH

=

VH

k ( n ! . e - I) - k(nst) l = Vst )

(48)

Here k is a specimen-specific proportionality constant. nH then specifies the number of the current filaments occuring in each of the v levels and determined by VHThe conductance derived from these considerations now has the form

vn

GU -

H

e

2


(49)

nSl)

Here, ksD is a specimen-specific proportionality constant. As a result of the assumed general validity of (26), at high magnetic fields (40) should also apply to the regions between the QHE states and, by further generalization, also to the Hall voltage VH, i.e.

N

-

v.

(44)

W h e n c o m p a r e d with (39), this e q u a t i o n e x p r e s s e s the independence from the specimen dimensions characteristic of this boundary condition. This becomes evident, when we consider that RSD shows similar values for all three specimens from about B = 8 T. There the equality VSD ~ VH iS approximately given, i.e. nH ~ nSD- This in turn means that nSD mustbe independent of the length of the specimen. Ohm's law has lost its validity for these magnetic fields, only a direct fixed relation obtains between I and VH. Furthermore, the experimental data suggest that a possible relation between I and VSD iS given only on the basis of a special connection between VH and VSD, appearing at high magnetic fields. To study the boundary conditions for nH ~ nSD in detail will be the goal of future investigations. In conclusion we can say that our model allows us to interpret the anomalous transport behavior of the two-dimensional electron gas, which similarly to the OWE state exhibits a specimen resistance independent of dimensions at high magnetic fields. d) The Temperature Dependence o~ the Magnetoresistivity The investigations recently carried out on (AI,Ga)As/ GaAs heterostructures and on Si MOSFETs for determining the temperature dependence of the specific resistance Pxx showed within the region of the QHE

11 0

Super/attices and Microstructures, Vo/. 7, No. 2, 1990 ntp~ h

and at relatively l o w temperatures a behavior which can be formulated by 6,7 I P~xI3

-

h

n2

2

exp ( -

A/3")

:

Dp

(51)

I11

where np represents a quantum number which, refer red to npo = 1, representsa relative measure of the frequent distribution of the nondisturbed to the disturbed electron transport in a filament. The value P I/n

1' =

(52)

P

thus formulates the probability of the appearance of a phonon (or an ion) disturbance during the otherwise non-dissipative electron f l o w b e t w e e n source and drain. If we n o w represent the specific resistance Pxx~j under the size independent boundary conditions by means of (26), we can write

l

~

n i

e 2

P

=

1

--

p

(53)

e2

As will be shown, the transport model described in this paper is also confirmed by the mobility measurements at low temperatures carried out on t w o - d i m e n sional systems such as Si MOSFETs and (AI,Ga)As/GaAs heterostructures 21-30. All known mobility data and the personal communication of G. Weimann 31 un equivocally indicate that the electron density No is a material-independent boundary for achieving a mo~ bility maximum pmax (see Figs. 4 and 5). M e n t i o n

~e

I

/ __

(54}

e) The Mobility and the Critical Electron Density No

1

J

h

This relation can be easily interpreted. The exponen tial expression represents the probability of electronphonon (or electron-ion) collision. Since our considerations relate to the total, specimen-size-independent current, this t e m p e r a t u r e - d e p e n d e n t exponential expression is modified solely by the filling factor number hi, which, in analogy to (36) and (44), we understand as a number of depth levels. This simple and easily understandable interpretation of (54) thus helps us to further underpin the plausibility of our transport model. A more precise represen t a t i o n and analysis of the data o b t a i n e d from Si MOSFETs will be presented elsewhere 7

(m2/Vs)

6

nin

exp I-: A/T)

n~

3

/

1

If we compare (50) and (53), we obtain a relation for the probability P of the appearance of a disturbance

(50)

where ni is a quantum number assigned to the filling factor, T is the temperature and _A represents an activation constant dependent on the magnetic field. This exclusive property of the 2-dimensional electron gas in the magnetic field can also be easily inter~ preted by means of (26). At these l o w temperatures we can assume the f o l l o w i n g for the QHE region: 1) that the q u a n t u m number nw in the sum o f (26) is iVen by the filling factor ni and 2) that the q u a n t u m ctor n~e >> hi, i.e. nil e

-

Pxx5

L

i

i

=

9

12

(10~5m 2)

18

Fig.4 The dependence of the Hall mobility p on the electron density N measured on a Si MOSFET at T = 1.5 K , I = 1 pA, L = 1.76mm, W = 160pro. The diagram shows the data obtained with three different selections of 8 potential probes 22.

Superlattices and Microstructures, Vol. 7, No. 2, 1990

t

~

11 1 II~

j~"

/

(m2/Vs)

"-'~"~

a = 0.8 GaAs[28] GaAs[21.26]

102

~'~ •

101

a = 1.0 GaAs [26]

~/~,~.~,~j/Si

t

I

/

[22]

L

100 i

/

10-It 10TM

r 1015

r 1016 (m-2) N •

Fig.5 The d e p e n d e n c e of p vs. N measured on (AI,Ga)AslGaAs heterostructures. The diagram shows values from Foxon et al. 21, Wittmann 22, Jiang et al. 26 and Harris et al. 28. a is the exponent of the relation p~Na.

should be made of the fact that the measured systems are only quasi two-dimensional and that the roughness of the interface is not n e g l i g i b l e , particularly where Si is concerned; in other words, ~max can be shifted slightly t o w a r d higher values of N. The steep drop of p at N < No t h a t cannot be i n t e r p r e t e d merely by the reduction of the electron shielding effect or by the reduction in the number of Si ions in (AI,Ga)As 21, 26, 32, 33 clearly justifies our model, in which the electrons should be isolated, i.e. localized, at N < No. In other words, we may conclude from these extremly interesting experimental data t h a t the localized electrons w i t h a distance I > Ie cannot make any significant contribution to static current transport and that only those electrons that are in the collectivized state, i.e. for N _> No, are capable of carrying a static current. Moreover, (29) that describes the two-dimensional resistance for N < N o is in complete agreement w i t h the e x p e r i m e n t a l results achieved w i t h (AI,Ga)As/ GaAs heterostructures 21, 26, 28. The studies cited evaluated the experimental data relating to the case of N < No on the basis of a homogeneous electron distribution in the layer, i.e. N = Nw 2 = NI ~ leading to the conclusion that the mobility increases w i t h N as expressed by the relation p ~ N a w i t h a -~0.8~ 1.1 26 (see also Fig.S). It is therefore obvious to us that the experimental data yield the relation

-

N

i~+ll

(55)

w i t h i n the limiting range of N < No. It agrees closely w i t h (30) when assuming Nloc=0, which seems to apply for high-mobility specimens. This view agrees w i t h the experimental result deduced from the temperature relationship in which p h o n o n scattering alone is critical in high-mobility specimens 21 This fact contrasts w i t h transport models which assume the ion scattering to be the significant factor leading to a relation p-N~.s, thus, in this connection also, failing to agree w i t h the experimental findings 2126,34-36 However, as soon as electron localization exceeds the f o r m a t i o n of current filaments and Nloc is no longer negligible, then, in accordance w i t h (29), the static current drops off steeply, a phenomenon observable in Si MOSFETs at N _< 1.5x10~s m-2, and in GaAs heterostructures even b e l o w N <- 3xI014 m 2, depending on the specimen quality (see Figs. 4 and 5) 22,26,37. Extrapolation of the conventional plot betweenpandNshownforaSiMOSFETinFig 4indicates that the mobility reaches p = 0 at N > 0, which seems to be difficult to interpret. In contrast, however, as can be readily appreciated from (29), our localization model renders this fact an absolute necessity. Thus, these experimental findings also constitute a further indication that our model is correct. Finally a further justification of our model, in particular of (26) and (29), is supplied by the transient step-like voltage decrease observed on small-size Si MOSFETs, when the gate bias is reduced at constant current to a voltage close to the threshold value 38. In this case the specific situation N < No is given and (26) yields an excellent basis for the description of all the q u a n t u m phenomena observed here even at room temperature. Finally, it is w o r t h repeating that measurements of mobility in two-dimensional systems furnish i m p o r t a n t experimental confirmation of the existence of a critical density N o in the behavior of electrons. The value N O = 3xi0 Is m-2, applicable to a two-dimenslonal electron gas, has thus become an essential milestone in assessing the correctness of the limiting value Ie in the description of the electron coulomb interaction, including all the associated consequences discussed in this paper. 4. Conclusion In order to provide a more convenient description of the new experimental data in the regime of the 2-dimensional electron gas, a simple q u a n t u m transport model, which represents every specimen resistance as a quantized h/e2 value, was presented for discussion in this paper. The fundamental idea behind this model is a reformulation of the Bohr q u a n t u m model, which is based on a quasi mass me = u2mo assigned to the maximum coulomb energy and specifies the value ae = o-lao = 7.25 nm instead of the f3ohr radius ao as the smallest possible length which can unequivocally occur in the coulomb interaction between t w o elementary charges. The length ae, time Te ~ ae/c and energy Ee = mec2 (i.e. the mass me) are understood here as quantizable basic quantities. It was shown that this idea agrees w i t h the experience of QED, according to which ~, defined in the f o r m a = eo2/4n (in t h e $1 units given by (1 =

11 2

Superlattices and Microstructures, Vol. 7, No. 2, 1990

e2/(4n~o//C) ), is understood to be an electron-photon coupling constant. In the semi-classical Bohr interpretation, in contrast, a is given by the ratio of t w o speeds, a = Vo/C,i.e. by the relation between the velocity of the first Bohr orbit of the hydrogen atom vo and of the velocity of light c, which is w i t h o u t any additional physical justification. The significant feature of our 2-dimensional transport model is its comprehensive applicability: it enables the resistance to be formulated at every temperature with the universal limit value R,, = (h/e2)l./W at T-~ ®, as well as at every magnetic ' f i e l d with the universal limit value R~ = h/(i e2) at the QHE. For an electron in a 2-dimensional electron gas, the density 3.0x101s m-2 becomes a boundary between being able to mediate a static electric current and being a localized electron without this capability. The credibility of this new type of transport model with all its specific statements relating to the coulomb interaction was checked by analyzing numerous experimental results. In addition to the QHE, special attention was given: 1) to the quantum stages in the resistance observed by Bliek et al., measured at the breakdown of the QHE in a narrow channel constriction 5, as it justifies our simple quantum model; 2) to the extraordinarily simple temperature dependence of the magnetoresistivity in the magnetic field region of the QHE, which exhibits a universal character 6,7; and 3) to the independence of the magnetoresistance on the specimen dimensions. This newly recognized property of a 2-dimensional electron gas was observed for the first time on Si-MOSFETs 7 According to our interpretation, this effect should also be universal in character, thus new measurements with GaAs are planned. The essential factor about this transport phenomenon, as can easily be seen from Fig. 3, is the continuous transition of the specimen resistance from ohmic to non-ohmic behavior, the latter's dimensional independence having hitherto been regarded as the characteristic feature of the non-dissipative QHE state alone. Together with Bliek's data s, therefore, this effect is able to provide a significant experimental contribution toward verifying our model, because it consolidates the idea of a continuous transition - na turally regarded w i t h i n the context o f q u a n t i z e d changes - from the ideal OHE transport mechanism up to the usual, i.e. ohmic transport mechanism at B = 0. Depending on its strength, the magnetic field can evidently give rise to a partial or complete cancellation of the disturbances present in the transport mechanism. As a further important experimental result can be considered the characteristic trace of the ]J vs. N curve showing a mobility maximum at or nearby No = 3.0x101s m-2 According to our model, this maximum at No seems to be an experimental support for our proposed modification in the formulation of Bohr's postulate. This modification implies evidently also the cancellation of the classical corpuscular model of an electron rotating around the nucleus in favor of a wave model. An additional, but not insignificant indication of the possible plausibility of our ideas is provided by an analysis of the current magnitude at electron densities N < No. It was shown that also at densities smaller than No an electric current is possible, but only of low magnitude. This low-current state can be

accompanied by characteristic quantum jumps of the voltage as observed at transistors of ultra-small dimensions in the vicinity of the threshold voltage 38 As will be shown in another paper, the analysis of these voltage jumps can be utilized as a fruitful experimental method for critical testing of our simple quantum model. Let us conclude with three remarks: 1) The resulting formulation of the conductance GL expressed in (26) and (39) does not show any explicit material related parameter. But, as can be seen form (14), (17) and (39), the relation of G ~to the given material at B = 0 is automatically included in the value of the average quantum number < n ~ e > . 2) The open question of the compatibility of the results obtained by our model with those of wave mechanics should be, in our opinion, answered positively. It appears to be guaranteed by the fact that the coulomb energy remains an invariant starting point, despite the altered mass me and length ae. 3) The application of the fundamental idea of this transport model is not limited to 2-dimensional systems alone, but can also be transferred to 1- and 3dimensional conductors.

Acknowledgements The a u t h o r is i n d e p t e d to W. Heywang, E. Feldtkeller, D. Schmitt-Landsiedel, W. Haensch, O. Jaentsch, K. Hofmann, E. Gornik, G. Weimann, I. Eisele and W. Brenig for the stimulating conversations and critical comments during the d e v e l o p m e n t of this w o r k , and to C. Weyrich, M. Zerbst and E. Wolfgang for their continuous en couragement. He would also like to thank R. Michell for translating the manuscript and Mrs. U. Kriebitzsch for typing the camera-ready form. References 1 2. 3.

4.

5. 6.

7. 8.

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