Quantum effects in semiconductor components

Physica B 151 (1988) 273-278 North-Holland, Amsterdam

QUANTUM EFFECTS IN SEMICONDUCTOR COMPONENTS G. DORDA Siemens AG, Corporate Research and Development, P.O. Box 830952, D-8000 Miinchen 83, Fed. Rep. Germany

The importance of quantum phenomena in semiconductor components, in particular in Si MOSFETs and GaAIAs/ GaAs heterostructures, is outlined. A short theoretical description of quantization effects in surface potential wells and at high magnetic fields is given. Multiple quantum well structures and modulation-doped heterostructures, as well as their possible applications are described. The discovery of the quantum Hall effect (QHE) is shown to be a result of the development of high-quality components. The features and the importance of the QHE for basic physics are outlined. Recent experimental data are discussed showing that the theoretical description of QHE is still unsatisfactory. A possible analogy of QHE to some features of superconductivity based on the idea of a changed effective mass is considered.

1. Introduction

The fundamental laws of nature are well established in quantum mechanics and electrodynamics, which were elaborated mainly in the first half of this century. The practical usefulness of these laws became evident within the second half of the 20th century when physicists succeeded in describing complex systems developed by new technological processes. Some aspects relating to this development will be shown in this paper. During the scientific attempt to understand and describe the rectification behaviour of metal-semiconductor contacts, the transistor effect was discovered in a germanium crystal by Brattain and Bardeen in 1948. That year marks the beginning of a new technological era characterized by the miniaturization and integration of semiconductor components. The MOSFET (metal-oxide semiconductor field-effect transistor), in particular, developed on the Si crystal in the sixties, became the basic element of integrated circuit systems. The current through the MOSFET and thus the amplification effect of this element is governed by the gate voltage changing the surface electrical potential. The motion of the carriers in the surface potential well is a problem well known from the wave mechanics of crystalline solids. In 1957 Schrieffer first recognized the idea

of surface quantization, proposing that bound states may exist in the inversion layer for motion perpendicular to the surface [ 1 ] . Further theoretical investigation has shown that the electrons in the potential well have to be considered as a two-dimensional electron gas. To describe the electrostatic potential perpendicular to the surface 4ffz) and the energy difference between the quantized levels, the Poisson equation has to be solved, d2~ _ p(z) dz 2

(1)

ee o

where p(z) is the charge density in the inversion layer, given by

p(z) = e ~.~ Ns.il~(z)l i

:z .

(2)

Here Ns.i is the concentration of electrons per unit area in the ith subband and ~b the wave function determined by the Schr6dinger equation. The quantized energy levels are obtained self-consistently. It follows that the energy differences of the levels for Si at the surface field F s = 107 W/m are of the order of 10 meV [2, 3]. When a vertical magnetic field B is applied to the inversion layer, the two-dimensional electron gas is further quantized. The subbands split into a ladder of Landau levels (LL) starting from each subband. Omitting spin splitting, the total

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274

G. Dorda / Quantum effects in semiconductor components

T- ~

energy of the state is now given by

E : E i + ( n + ½ ) f ~ %,

w~

A o ~1

eB m~'

(cm2 dyne-')

60. Xl0 -12

(3)

50"

where E i is the bottom of the ith subband, o)c the cyclotron frequency and rn~ the effective mass parallel to the surface. The energy spectrum is completely discrete and the related densities of states show delta functions separated by the Landau splitting energy ho)c. In real samples the delta function peaks are broadened by the. scattering process. For a characteristic collision time r the level width is of the order of h/,c. Thus Landau level (LL) splitting appears to become significant for ho)c >h/,c and kT, i.e. high magnetic fields and low temperatures are required.

40"

n=0,1,2,...,

2. The MOSFET structure

The first measurements demonstrating the existence of surface quantum phenomena were performed by Fowler et al. [4] on Si M O S F E T s in 1966. The conductance of the transistor exhibits oscillations of constant periodicity at a temperature of 1.34 K and a magnetic field of 9 T. This fact definitely proved the existence of quantized

AI

SiO2 \

Fig. 1. Energy-band diagram indicating quantized energy levels in an n-type Si MOS inversion layer.

30"

Si, n-Channel T - 297 K

~-~N nn (bulk) "

\.

20"

"~'~.~2

10" 0

1.0

0

11001 1'5

2'0 volts 25 Vg ---ib

i

lo' v,lt o.-'

F

.

Fig. 2. Transverse piezoresistance coefficient 7r12for n-type inversion layer on (100) Si vs. gate voltage Vg. A~r/~ro is the relative conductivity change, X is the mechanical stress. (After Dorda [5].) subband levels and LL splitting in the surface inversion layer (fig. 1). The strong effect of quantization on the transport phenomena in surface layers even at room temperature was shown by the author in 1970. In studying the problem of the transformation of accustic energy into electrical energy, the piezoresistance effect of the M O S F E T s was analyzed [5]. A pronounced dependence of the transverse piezoresistance coefficient on the gate field was observed, which clearly showed that the electrons were affected by surface quantization at room temperature even at relatively low surface fields such as F~ = 5 x 106 V/m (see fig. 2). A further indication for the subband level splitting was given by Sch~iffler et al., who showed the presence of discrete absorption peaks between the first two subbands even at room temperature [6]. On the basis of these interesting and surprising results, we can state that every commercially fabricated Si-MOS transistor is governed by quantization laws, thus resulting in a two-dimensional electron gas system, i.e., the splitting of the energy levels appears to be so great, that quantization has to be taken into account even at room temperature.

275

G. Dorda / Quantum effects in semiconductor components

3. The superlattice structure The development of MOS transistors has demonstrated the importance of the crystalline and chemical quality of the interface. Technical progress in this field has made the realization of very thin crystalline layers of semiconductor materials possible. Here again the electrons can be confined in quantum states. Multiple-periodic structures were grown by the molecular-beam epitaxy (MBE) technique, in which the quantum wells are separated by barriers. These barriers come from the use of at least two types of semiconductor materials with different energy gaps (see fig. 3a). Here again the motion along the layer is free, whereas the transverse motion is determined by the rectangular potential well. According to this one-dimensional potential-well model, the energies of the electron and hole are quantized so that we obtain E~ = Ell + t/2

h2T¢2 , 2m.lL2

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

(4)

where Ell is the energy associated with the motion of the carriers along the film, L the film thickness and rn±* the effective mass in the direction perpendicular to the surface. The most frequently studied materials with composition modulations down to single atomic layers have been the GaA1As/GaAs systems,

vs bl

f

t

because of their small lattice mismatch. Negative resistance effects and Bloch oscillations were expected but are hindered by inhomogeneities at the interfaces. A superlattice avalanche photodetector as proposed by Capasso et al. [7] seems to be an interesting application of this system for reducing the multiplication noise. The recently developed resonant-tunneling bipolar transistors contains a GaAs quantum well and two AlAs barriers between emitter and collector (see fig. 4). This device promises to become a new element for applications at high frequencies up to the THz region as well as for high-speed logic systems [8]. The possibility of modifying the splitting energy by the layer thickness L (see eq. 4) offer a wide field of optical applications. Here not only compositional variations but also varied and well-defined dopant profiles are under discussion. An example is represented by n - i - p - i superlattices grown by MBE (molecular-beam epitaxy) or MOCVD (metal-organic chemicalvapor deposition). The advantage of these specific layered systems is the tunability of the carrier concentration and the band gap [9]. Compared with the application of superlattices in which the carriers are transported across barrier layers the modulation-doped superlattices (see figs. 3b, c) with the current flowing along the interfaces are of greater practical and even scientific importance. In this case, the smaller band gap semiconductor is usually undoped, whereas the material with the larger band gap is n-doped. To enhance the mobility, an undoped spacer

~

Ce _

o

--\

ve

gB

c)

~

~

VB EF

- f

VIII

~

n.AIGaAs Fig. 3. Energy-banddiagram: (a) undoped superlattice, (b) superlanice with donor impurities, (c) modulation-dopedsingle interface.

@

~

~

GaAu

n-GaAs

Fig. 4. Energy-banddiagramof the resonanttunnelingbipolar transistor with positive base-emittervoltage.

276

G. Dorda / Quantum effects in semiconductor components

layer is introduced to reduce the Coulomb scattering. With such a structure, high electron mobility transistors (HEMT) have been fabricated for high-frequency and high-speed operation [10].

4. Quantum Hall effect

The high-mobility G a A s / G a A I A s heterostructures achieved major scientific significance in connection with the discovery of the quantum Hall effect (QHE). This effect was first observed in Si MOSFETs by Klaus von Klitzing at liquid helium temperatures and strong transverse magnetic fields in 1980 [11] (see fig. 5). The Q H E is a novel macroscopic quantum phenomenon, relating only to the two-dimensional electron (or hole) gas. As demonstrated by eq. (3), the application of a transverse magnetic field leads to the appearance of Landau quantization of the motion of the carriers parallel to the interface. To describe the Hall effect for a two-dimensional system, a theory was developed by Ando for zero temperatures which included the broadening of the LL system [3]. It has been shown that the Hall conductivity try is a complex function of the scattering process, but for the singular case of completely occupied LLs and in the absence of any scattering, trxy is simply given by

only in extremely homogenous samples. Thus the observation of plateaus in Pxy, i.e., the existence of the quantum state even in the AB or AN~ neighborhood of the singular condition described by eq. (6), was very surprising (see figs. 5 and 6). However, this interesting state of affairs permits this phenomenon to be used as an absolute standard of resistance, i.e. hie 2

The theory of extended and localized states has been developed to describe the plateaus [12]. According to this model the number of free carriers remains constant in a given LL, whereas the Fermi level moves with AB or AN s through localized states, thus resulting in the appearance of plateaus. But it should be emphasized that this interpretation includes a paradox: in agreement with experimental data, the number of localized states should increase strongly with increasing B and decreasing temperature and be accompanied by a distinct reduction of the number of electrons in the extended states. This effect would result in a pronounced reduction of the screening of localized states followed by a UH/mV Upp/mV

f30" '3.0f

I

25 • "2.5

= e (5) Pxy B" Taking the equation of the density of states for /-occupied LLs,

O'xy-

N~ = i --h'- '

(6)

we obtain for the Hall resistivity Pxy the simple form ze

20--2.0

15 - -1.5

eB

h pxy = .-5 ,

(8)

= 25812,806... ~ .

i = 1,2,3 . . . . .

(7)

where h is the Planck constant, e the electric charge and i an integer. Moreover, in this singular case, the current which causes the Hall voltage flows along the sample without any dissipation, i.e. Pxx = 0. It is evident that this specific state should be possible

10 "" 1.0

5 "0.5

OJO 0 -n~O

-~-

n=1

--

2'0

25

-- n = 2

Jl

Vg/V

•

Fig. 5. Hall voltage UH and voltage drop Upp between the potential probes vs. gate voltage V~at T = 1.5 K and B = 18T for a Si MOSFET. (After yon Klitzing et al. [11].)

G. Dorda I Quantum effects in semiconductor components

T'I""

/

t

/ I

//

/ r---J

I

/

_/

1/2"

// // .I

/

Gn~//dGa~ I O. Ebed et sl.

....

/

0

/f'

8 6 l[l

r

4 I

3 i

2

o15

513

4/3

1

'

i

B (ark units)

Fig. 6. Hall resistivity Pxr vs. magnetic field B at T = 50 mK for a GaAs/AIGaAs heterostructure. (After Ebert et al. [26], and Dorda [22].)

strong increase of the scattering. As a result, the widening of the plateaus would be halted with increasing B and decreasing T, which is contrary to the experimental results. Moreover, with increasing sample quality QHE did not disappear, but new plateaus additionally were observed at fractional numbers i in eq. (7), as can be seen in fig. 6. This phenomenon is called the fractional QHE. Whereas a novel type of many-body ground state in the form of an incompressible quantum fluid is suggested to explain the fractional QHE [13], the integral QHE is still believed to be a result of the transport property of single carriers. Some recent experimental data obtained with GaAs samples additionally Be- or Si-doped seemingly support the model of localized states [14], but these data may be just a result of moderate inhomogeneous densities causing the plateau width, as proposed by Woltjer et al. [15]. On the other hand, other recent experimental results such as the quantum Hall potential distribution [16, 17], the breakdown phenomena of this state [18], the substrate bias effect on the QHE [19], the magnetization measurement [20] and others, do not confirm the localization model. Moreover, the radiative recombination measurements performed on Si MOSFETs seem to be of particular importance. They show the existence of a critical temperature for the fractional QHE, which suggests the possibility of a phase transition [21].

277

Thus by considering the non-dissipative current in the QHE regime, a new idea can be developed assuming that the integral QHE is based on a many-body ground state in which the carriers move in a coherent phase induced by the magnetic field B. In this state of phase coherency, the two-dimensional carriers would lose their individual relation to the basic material. In fact, the most important feature of the QHE is its independence on the sample as expressed by eq. (7), which does not make any reference to the material. This circumstance is an extraordinary case in solid-state physics and can be seen in analogy to the Josephson effect. Hence the effective mass of the electrons in coherent phase would be changed, presumably increased up to the value of vacuum m 0. Thus this new type of phase transition is governed by the energy change of the electron state from heB/m* to heB/mo, which results in the appearance of the plateaus and the suppression of the localization effect. The just described energy change also represents the activation energy of the integral QHE. In developing this relationship, a good agreement was achieved with experimental data [22] (see fig. 6). It is evident that this model can be simply proved on materials with a large effective mass, where the QHE should be less pronounced. This idea is supported by the recent data on AlAs quantum wells [23]. The results of the gyromagnetic effect measured on superconducting lead can be seen as a further confirmation of this model [24]. The value of the vacuum m 0 for the electron mass was found by Doll with high precision [24]. It should be noted that electrons in the superconductive state are also in coherent phase, but in that case induced by the electron-phonon coupling. It is also worth mentioning that the gyromagnetic effect is an ideal technique for determining the effective mass, as the state of the electrons is not changed by the method of measurement.

5. Conclusion

Summarizing, we can state that the observation of the QHE on artificially fabricated samples is an impressive demonstration of the im-

278

G. Dorda / Quantum effects in semiconductor components

portance of the quantum laws in matter. This novel phenomenon appears to be the result of a magnetic-field-induced correlated state of the two-dimensional electron gas having no relation to the sample size and the material. The only function of the material of the sample is to transform the three-dimensional state of the electrons (or holes) into a two-dimensional electron gas. Finally, an even more interesting result should also be mentioned: when multiplying the quantum Hall conductance e2/h by half of the permeability of vacuum ½/z0, and the speed of light c, we obtain the fine-structure constant a = ltzoc, e2/h = (137,03599)-1, the magic number which represents the dimensionless measure of the coupling between matter and the electromagnetic field, for which Richard Feynman found the striking remark that "the hand of God wrote that number, and we don't know how He pushed His pencil" [25]. The quantum Hall effect may well be a fantastic telescope allowing us to observe His pencil.

References [1] J.R. Schrieffer, in: Semiconductor Surface Physics, ed. R.H. Kingston (University of Pennsylvania Press, Philadelphia, 1957) p. 55. [2] G. Dorda, in: Festkrrperprobleme 13, ed. H.J. Queisser (Pergamon, Oxford, 1973) p. 215. [3] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [4] A.B. Fowler, F.F. Fang, W.E. Howard and P.J. Stiles, Phys. Rev. Lett. 16 (1966) 901. [5] G. Dorda, Appl. Phys. Lett. 17 (1970) 406. G. Dorda, J. Appl. Phys. 42 (1971) 2053. [6] F. Sch~iffler and K. Koch, Solid State Commun. 37 (1981) 365. [7] F. Capasso, W.T. Tsang, A.L. Hutchinson and G.F. Williams, Appl. Phys. Lett. 40 (1982) 38. ]8] T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwald,

C.D. Parker and D.D. Peck, Appl. Phys. Lett. 43 (1983) 588. F. Capasso, S. Sen, A.C. Gossard, A.L. Hutchinson and J.H. English, 1986 IEDM Digest, p. 282. [9] G.H. Drhler, CRC Rev. in Solid State and Material Sci. 13 (1987) 97. [10] N.T. Link, in: Solid State Devices 1983, ed. E.H. Rhoderick (The Institute of Physics, Bristol and London, 1983) p. 15. [11] K. yon Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [12] H. Aoki and T. Ando, Solid State Commun. 38 (1981) 1079. [13] R.B. Laughlin, in: The Quantum Hall Effect, eds. R.E. Prange and S.M. Girvin (Springer Verlag, New York, 1987) p. 233. [14] R. Haug, R. Gerhardts, K. von Klitzing and K. Ploog, Phys. Rev. Lett. 59 (1987) 1349. [15] R. Woltjer, R. Eppenga, J. Mooren, C.E. Timmering and J.P, Andre, Europhys. Lett. 2 (1986) 149. [16] H.Z. Zheng, D.C. Tsui and A.M. Chang, Phys. Rev. B 32 (1985) 5506. [17] G. Ebert, K. von Klitzing and G. Weimann, J. Phys. C 17 (1985) L257. [18] L. Bliek, G. Hein, V. Kose, J. Niemeyer, G. Weimann and W. Schlapp, in: High Magnetic Fields in Semiconductor Physics, ed. G. Landwehr (Springer Verlag, Berlin, 1987) p. 113. [19] G.M. Gusev, A.N. Dakhin, Z.D. Kvon and V.N. Ovsyuk, Solid State Commun. 56 (1985) 1055. [20] J.P. Eisenstein, H.L. St6rmer, V. Narayanamurti, A.V. Cho, A.C. Gossard and T.W. Tu, Phys. Rev. Lett. 55 (1985) 875. [21] I.V. Kukushkin and V.B. Timofeev, in: Workbook EP2DS-VII Conf., Santa Fe, NM, USA, July 1987, p. 154. [22] G. Dorda, in: Workbook EP2DS-VI Conf., Kyoto, Japan, September 1985, p. 349. [23] T.P. Smith, W.I. Wang, F.F. Fang, L.L. Chang, L.S. Kim, T. Pham and H.D. Drew, in: Workbook EP2DSVII Conf., Santa Fe, NM, USA, July 1987, p. 359. [24] R. Doll, Z. Physik 153 (1958) 207. [25] R.P. Feyman, " Q E D " (Princeton University Press, New York, 1985) p. 129. [26] G. Ebert, K. von Klitzing, J.C. Maan, G. Remenyi, C. Probst, G. Weimann and W. Schlapp, J. Phys. C 17 (1984) L775.

DISCUSSION (Q) G.C. Hegerfeldt: You mentioned your new theory for the QHE based on phase coupling of the electrons. Do you also need impurities or will it work without them? (A) G. Dorda: The model used for the description of the

QHE is based on the assumption that impurities and lattice inhomogeneities destroy or reduce the probability of the QHE state in the region of their influence.