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Electronic states and magnetotransport in δ-layer systems
PHYSICA
Physica B 184 (1993) 298-305 North-Holland
Electronic states and magnetotransport in g-layer systems F. K o c h Physik-Department E16, Technische Universitiit Miinchen, Garching, Germany We review some aspects of magnetotransport in a semiconductor material for which the dopant atoms are incorporated during MBE growth in atomically sharp defined sheets. Hopping transport, in the dilute limit of planar doping, shows interference effects which lead to a distinct resistance variation in a magnetic field. A periodic sequence of B-layers has electronic states with exceptionally high mobility that result from a coherent superposition of wavefunctions throughout the array.
1. Introduction
The quantum properties of a sheet of donor atoms in an epitaxially grown semiconductor were first demonstrated in ref. [1]. For a density N $/>1X1012 Si atoms per cm 2 in GaAs this so-called B-layer is a metallic system of subband states in the self-consistent potential well formed by the positive sheet of Si ions and the electronic screening charge. The soft, V-shaped potential in fig. 1 is a good description of the system when the density is sufficiently high so that the electrons average the granular nature of the Si + charges in the dopant plane. It is known that the levels Eo, El, E 2 , . . . each have a different transport mobility/x. The values of/x0, /£1, ~LL2, are the result of the distinct configurations of the electron densities of the subbands relative to the donor ion sheet. In general/z 0 is much less than /x1,/x 2. . . . . For the system in fig. 1, t% is of order 1000cm2/Vs. For higher subbands the mobility can be an order of magnitude more. Single B-layers in the metallic limit of doping concentration have been studied in many publications. Recent and novel experiments have emphasized the low-density case where the subband description does not apply. Using 8-doping one can fabricate a well-controlled planar system •
"
•
Correspondence to: F. Koch, Physik-Department, Technische Universitfit Miinchen, D-8046 Garching, Germany.
where hopping transport takes place. For a low concentration of dopant atoms unintended background doping and boundary layer depletion must be considered. In this dilute limit it is essential to make use of a multi-layer arrangement with controlled aspect ratio a/b. The average in-plane spacing a is linked to the doping density. The planar separation b is chosen independently to satisfy other criteria. For GaAs samples, where the typical p-type background and surface band-bending reduce the electron density, the b value of 100nm was selected in order to avoi~t excessive depletion of the small number of electrons ir~ the dilutely doped layer. There is also a sacrificial top and bottom sheet to be considered. The latter is totally depleted in 6 - doping
t
E (meV) 12oo
!
Si !1
n
! /
-
-
E
n"
-
-
n
~
~
GaAs
Z =
100
200
Z
Fig. 1. Potential well, subband energies and wavefunctions for a layer of 4.1 × 10 t: Si donor ions per cm 2 in GaAs (see ref. [1]).
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
F. Koch / Electronic states and magnetotransport in 6-layer systems
satisfying the necessary boundary conditions for GaAs. Hopping-transport occurs within each of the many parallel conducting planes separately. The periodic nature of the 8-layers is not significant for hopping as long as a / b is much less than one. More details on GaAs samples are found in [2]. For the formation of minibands the regular array of 8-layers is essential. The use of MOVPE-grown InP with a light, n-type ( - 1 × 1015 cm 3) background is advantageous because it avoids depletion of the miniband charge that exists between the doping planes. For the InP samples there is no reason to expect depletion of the outermost layers. These samples are discussed in ref. [3]. In this paper the intention is to focus on two topics of current interest in magnetotransport studies on 8-layers. We discuss in section 2 the interference effects in the hopping magnetotransport in dilutely doped layers. Section 3 deals with periodic 8-layers and the high-mobility, coherent miniband states that are observed in such systems.
2. Magnetoresistance in hopping transport: interference, resonant tunneling and oscillations The study of electrical transport in dilutely doped semiconductors is a time-honored theme. A large number of publications exists on the subject. Nevertheless, only recently have sheetdoping structures made a mark in this field. Several reports [2-6] highlight general features of hopping magnetoresistance that show up clearly in such planar systems. For ~-layers the hopping sites are deeply embedded inside the epitaxially grown, highly perfect semiconductor layer. Because it is a 2D structure one can contrast the influence of a magnetic field parallel and perpendicular to the layer plane. The potential of each of the donor ions is identical. Only the spacings and positioning of the donors is random. The unintended background doping is adequately known and by adjusting the multilayer spacing it can be controlled deliberately.
299
Other than for MOSFETs, modulation-doped quantum wells or evaporated thin layers, there are no unknown and unspecifiable surface effects. We describe here the basic physical ingredients of the hopping magnetotransport phenomenon and lead up to an intensely debated question- does there exist an oscillatory contribution to the hopping resistance in a magnetic field? The most basic and thoroughly documented ingredient of magnetotransport is the fact that in the limit of high magnetic fields the hopping resistance must increase exponentially with increasing field. Reference [7] describes the B 2dependence that originates from wave function shrinkage. The compression of the hydrogenic impurity ground state reduces the overlap between the hopping sites in a manner such as to exponentially decrease the tunneling probability. The experimental observations on ~-layers (fig. 2) are no exception to the rule. They obey the scaling prediction for between slopes in a log R vs B 2 plot for the two orientations [5,7]. It is expected that in the perpendicular field (B±) the slope has twice the value of the parallel case (BII). While for nearest-neighbor hopping the slope is nearly T-independent, it should with decreasing temperature and for variable-range hopping increase according to a power law. The data in fig. 2 shows the correct tendencies but does not cover a sufficient T-range to uniquely fix the exponent. There is an experimental problem in accurately measuring the exponentially rising resistance at low temperature without heating either the lattice or the electrons. Our measuring limit is in the 109 [). range. While the Bii-data is by and large described adequately in terms of wave-function shrinkage, the perpendicular case in fig. 2 shows a distinct and additional phenomenon- negative magnetoresistance. The resistance decreases linearly at fields of the order of a few tenths of 1T. The reduction saturates and reaches about 50% at the lowest temperatures in fig. 3. The effect is absent in the parallel configuration. Instead of the - 5 0 % decrease at 2 T, the Bit-resistance in fig. 2 has more than doubled. The negative resistance effect is not just a minor correction. It
300
F. Koch / Electronic states and rnagnetotransport in g-layer systems I
0.171(
I
I
I
O. 17K 1.60K
0.60K O.38K
10~
108 I06
D
//
107
f
~. 2K
I0S
I
0
10
20
B~ [T21
0
10 ~
20 IT21
Fig. 2. Variation of the resistance per square of a B-layer with N s = 8 × 101° cm-2 in GaAs with B~ and B~. The theory predicts a straight line. Note the distinct negative magnetoresistance in a perpendicular field.
is the dominant physical p h e n o m e n o n in fields before the wave-function compression takes over. The decreasing resistance is explained by an interference mechanism first postulated in ref. [8] for variable-range hopping. Schirmacher [6] gives a simpler, experimentally more relevant description in terms of the interference of an electron reaching a final site f after tunneling along two alternative paths (fig. 4). In this configuration of hopping sites, an electron starts at site i and tunnels both directly to f, and along a path in which it scatters resonantly off the potential at an intermediate site j. The loss in tunneling amplitude because of the extra distance in the path i ~ j --~ f can be more than compensated by the enhancement effect of the resonance. T h e r e will be destructive interference because of
the phase shift ---~, which the particle incurs during resonant scattering. In a magnetic field the flux through the triangular area in fig. 4 adds an additional phase. On the average, just as in the weak-localization theory for negative magnetoresistance [9], the field acts to decrease the destructive interference. It allows the electron to complete more favorably its hopping process from i to f. One may think of path i---~j---~f in fig. 4 as a coherent, resonant-tunneling process through a quantum well. In that case, the phase shift 7r is the extra half wavelength the particle requires to traverse the potential well at site j. The resonance condition E i ~ Ej for a strong transmission amplitude is evident. It is also obvious that the degree of destructive interference is substantial because the two waves can be of comparable
301
F. Koch / Electronic states and magnetotransport in g-layer systems
0
1
i
O. 17K
-0.2 -0.4
0 &38K
-0.2 -0.4
Ec
0
v,
"7 -e.2
O
m_0.4 0
Ef
Ej
Fig. 4. Interference in the hopping transport along separate paths in a triangular arrangement of impurity sites. Site j provides a resonant-tunneling process for the particle moving from an occupied site i to an unoccupied final state f. The energy scheme is shown below.
-0.2
-0.4
0
-0.2I -0.4
0
I
I
1
2 B IT]
Fig. 3. Normalized variation of the resistance with B± for a GaAs B-layerwith Ns = 8 x 10 cm 2. Note the extra structure at B l - 1.6T. amplitude. The extra decay of intensity for the longer indirect path i--->j---* f is counteracted by the resonance e n h a n c e m e n t factor. There will be totally destructive interference for certain E i Ej and geometrical extra pathlength. The data in fig. 3 poses yet another challenge to the theory. Studying different G a A s samples, and m o r e recently also InP g-layers, leaves little doubt that the nonmonotonic variation of resistance in the negative magnetoresistance range is a real effect. We have included in fig. 5 data for InP. The sample has a somewhat higher carrier sheet density (N s -- 1.5 x 1011 cm -2) and has not yet been measured for T below 1.47 K. Its negative magnetoresistance effect is ~ 2 5 % at the lowest T. Although the exact form of the resistance variation in the 0 - 4 T range is not totally identical with that of the G a A s sample, the
existence of a p e a k in the intermediate field range ( - 2 . 5 T ) may be suspected by a comparison with fig. 3. T h e extra resistance p e a k has caused some excitement and serious concern for our theoretical understanding. The nonmonotonic behavior is not described properly by either the linearinterference negative magnetoresistance, nor by the B e, high-field positive-resistance theory. It is
-0.1 "7
v
I \
~ _o.~I
.
.
.
.
-0.2
-0.3
0
I
2
B[T} Fig. 5. The magnetoresistance variation calculated by Fritzsche and Schirmacher in ref. [11] and a comparison with the experimental result (insert) in fig. 3.
302
F. Koch / Electronic states and magnetotransport in 6-layer systems
a phenomenon reminiscent of the oscillatory resistance variations that occur for g-layer samples in the metallic-density limit. For the hopping transport, on first sight, Landau quantization and Shubnikov-de Haas oscillatory structures are unexpected. In searching for mechanisms to explain the nonmonotonic resistance variation, different approaches have been used. Reference [4] builds on the ideas discussed earlier in ref. [10]. It is argued that the effect appears with the approach to variable-range hopping at the.lowest T in fig. 3. As is well known, Mott's explanation of this hopping mechanism involves the density of impurity states. A high density of states allows for easy hopping at low temperatures because of the correspondingly large number of sites that are then available within the k T range. Reference [10] proves that when the broadening of the state distribution is by random overlap of wavefunctions and irregular placement of the donors, then the wavefunction shrinkage acts to collapse the width, thus increasing the density of states. This extra B2-dependence gives a quadratic decrease of the resistance before the rising magnetoresistance becomes dominant. In ref. [4] it is argued that the addition of the linear resistance decrease (caused by the coherent superposition effect) to the quadratic decrease (incoherent Motthopping) and the B2-increase caused by high fields, will account for the nonmonotonic variation as seen in fig. 3. A different explanation has been argued in ref. [11]. This work quantitatively sums up the random, triangular paths and their respective contributions to the resistance. The authors find that the sum has damped oscillations. A first maximum occurs when a single flux quantum penetrates a certain characteristic area. The area is referred to as the 'interference'-hole. It is related to the triangle surface for which completely destructive interference will occur. By referring to fig. 4 it is possible to understand what is meant by the characteristic area that occurs in the theoretical treatment of ref. [11]. Complete destructive interference requires that the amplitude of the waves along the two paths be equal. Scattering by the potential at site
j causes a resonant enhancement of that wave amplitude. Its magnitude depends on how close the energy E i is t o Ej. It is apparent that for some position of j the extra length of the tunneling path i---~j--->f is just compensated by the resonant enhancement. Destructive interference will occur for a finite area of the triangle. To have a well-defined area, however, requires that averaging over random positions and paths, as well as over the E i - E j , does not broaden the area value too much. According to ref. [11] this is the case when the resonance is sharply defined, or equivalently when the impurity state width is much less than the localization potential V0. It is argued in ref. [11] that a characteristic area exists for the present case and that damped oscillations occur as one, two or more flux quanta thread the area. The argument has a certain similarity with the simple Landau quantization rule for orbital electron motion in metallic transport. The proposed oscillation mechanism and the observed peaks represent a new type of resistance variation in hopping transport. It is reminiscent of the magnetoresistance for artificial quantum-dot lattices where hopping transport takes place between the regularly spaced dots. The case of the randomly spaced impurities of the g-layer is as yet a contested topic. Controversy aside, the fit to experiment that Fritzsche and Schirmacher [11] achieve with their evaluation of the negative magnetoresistance for parameters that describe our samples is very suggestive. We show in fig. 5 their calculated result along with the data (insert).
3. Periodic 8-layers: magnetotransport in high-mobility minibands The quantum arrangement of the electronic screening charges around the sheet of donor ions in fig. 1, has interesting implications for the mobility. Taking for example the n = 1 state, we expect a fairly high mobility because of the wavefunction node at z = 0. For GaAs with a doping of -1012Si ions per c m 2 a value of 10000cm2/Vs applies at low temperature. AI-
303
F. Koch / Electronic states and magnetotransport in 6-layer systems
though this is higher than the mobility of the ground-state electrons, the enhancement is not at all comparable with what has been achieved by modulation-doping of heterojunctions. The single 5-layer does not provide the optimal 'spacer width'. The challenge is to provide a way for electron states of the subbands to spread out more. High conductivity of the 5-doped layer requires that the largest possible number of electrons occupy regions far from the doping sheet. The basic answer to the challenge is provided by properly shaping the wings of the V-shaped, self-consistent potential. The overlap with neighbouring potential wells in a periodic structure can be used to produce flat regions of the potential into which electrons can spread easily. It is also important to use not very high ion densities because this increases the binding strength of the potential. We first noted the existence of high-mobility electrons in periodic 5-layers of GaAs in cyclotron resonance experiments [12]. The carriers were thermally excited into the undoped spaces between the layers. In a Bil-field such electrons give a narrow, high-oJc~" resonance line. Electrons which are tightly bound in the electric subband levels of the layer do not contribute in this configuration. With the field perpendicular to the doping sheets both the carriers in the layers and between them will participate in the resonance. The result is a broad line. This explanation for the work in ref. [12] differs in a subtle way from that in the publication. The effect of thermally excited, high-mobility carriers can also be found in DC magnetotransport experiments. References [3,13] illustrate this by the added contribution to the Hall effect currents. The Hall resistance Rxy is the parallel sum of the resistances for current flowing within the doping layer and current carried by the highly mobile between-layer electrons. The extra component causes the layer system to behave like a two-fluid model in which the bypass current has a substantially bigger mobility. In ref. [3] we have discussed the Rxy(B ) curves for an InP sample with N S = 2.8 × 10 ll cm -2. At the lowest T of 1.6K, the high-mobility carriers practically disappear giving a straight line for Rxy
in this simple case. Raising the density only slightly to a value of 3.6 × 1011 cm -2 per sheet, as for the data in the accompanying fig. 6, we find that the two-fluid effect is retained even at 1.3 K. In this figure, the structure at - 8 T and 1.29 kl-I is the expected i = 2 Hall plateau for the sum of the two carriers. The sample has 10 doping sheets acting in parallel. In principle, the Rxy curves in the figure allow one to determine both the number density and mobility values of the two electronic fluids. In practice, because of the quantum Hall effect structures, it is difficult to achieve an accurate and sensitive fit. Using the classical two-fluid formula we have shown in ref. [13] that a mobility of order 100 000 cmZ/V s is needed to explain the curves. A more straightforward way to determine the parameters of the electrons is cyclotron resonance. The sharp line at 7.2 T in fig. 7 is the resonance of about 6 × 101° cm 2 electrons per sheet with a mobility of 120000 cm2/V s at T=5K. In the spectra one clearly sees the T-dependent increase in the number of carriers that correlates with the changes in Rxy(B ) of the previous figure. The small absorption signal at 3.8 T is the ls---~ 2p impurity magnetoresonance
25 Sample B Ns=3.6x1011cm-e
20
~15 ~
15
as
i0
/
9 .2K
J
T = 6.0K T = 4.1K T = 1.3K
/// 5
~
0 0
J
J
4
8
I'2
BI(T) Fig. 6. Hall Resistance R.y(B±) for an lnP periodic 5-layer system with 10 parallel conducting sheets. The resistance is expressed as a value per sheet to emphasize the value of the Hall plateau at 7.6T. Note the field which represents the Hall current contribution of a high-mobility bypass.
F. Koch I 'Electronic states and magnetotransport in a-layer systems
304
Sample
13
20K
"
o
~
16K
" -~''~
~
12K
-
•~
9K 7K 5K I
0
I
I
I
4
I
i
i
1
8
1
I
I
12
B II (Tesla) Fig. 7. Cyclotron resonance of a periodic b-layer showing high-mobility electrons in miniband states.
for electrons bound to isolated background donors. It is evidence for the expected weak, n-type background doping in M O V P E - g r o w n InP. For the present case, the 1.5 Ixm thick layer has about 1.5 × 1011 c m - 2 of such electrons b o u n d to the impurities. The signal provides a convenient calibration and check on the cyclotron resonance data. T h e r m a l excitation of electrons, with a realspace transfer to high-mobility regions between the doping layers, makes the multi-~-layer syst e m a nonlinearly conducting medium. This effect has been proposed and demonstrated in ref. [14] for a G a A s sample with 10 sheets of N s = 5 X 1011 cm -2 spaced 100 nm apart. For this case, it is found that the ohmic conductivity is a factor two higher than for a randomly doped layer with the same total dopant concentration of 5 x 1012 cm -2 spread over 1 ~m. The differential conductivity dj/dE, m o r e o v e r , rises nearly 50% in a field E - - 1 V/cm. The conductivity of the statistically r a n d o m - d o p e d sample remains nearly constant over the same range of E. T h e most exciting prospects offered by the present investigations are in the design-tailoring
of densities N s and layer-spacing to achieve even at T = 0 K substantial densities of high-mobility electrons. The layer-system can be engineered by p r o p e r placement of impurities to have an optimal electrical conductivity. The electrons of the periodic layer system are miniband states. In particular, the highly mobile electrons of the second and higher miniband states are the result of a coherent superposition of waves each of which avoids the donor ions. The secondminiband (n = 1) electrons of fig. 8. are diffracted waves in the periodic structure that have zero occupation probability at the site of the donor sheets. Scattering in the miniband occurs only via the fluctuation potential in the plane of the d o n o r sheets. The latter is screened by the n = 0 ground state electrons and acts at a distance dictated by the wave-mechanical distribution of charge. Figure 8 is from a calculation for G a A s with N s ~ 5 x 101° cm -2 and spacing of 100 nm. T h e calculation assumes an n-type background doping of 1 x 1015 cm -3. For this case, 0.83N s electrons are predicted to occupy the E 0 level. T h e n = 1 miniband has 0.14N s occupation and is distributed spatially as in the drawing. We note that the flattening of the potential in the wings of the V-shape that results from the superposition of the periodic array leads to an increased average distance of the miniband electrons from the ions. The result is the dramatically improved scattering lifetime that we have demonstrated here in the magnetotransport experiments.
0
E ~
E ,I,
~
'
:
~
~ooo A
~
~
o
V Vl .I
l
Fig. 8. Calculated potential variation, energies and the charge distribution of the n = 1 miniband states for a periodic 6-layer with N~ = 5 × 1011 cm -2 in G a A s . The energy E 1 is 14 meV below E F and the double line is the bandwidth.
F. Koch / Electronic states and magnetotransport in 6-layer systems
4. Conclusions In this p r e s e n t a t i o n we have focused on two subjects of current interest in m a g n e t o t r a n s p o r t in g-layers. T h e discussion on h o p p i n g transport has d e m o n s t r a t e d that the g-layers, because they r e p r e s e n t a particularly clean and well-defined case o f a disordered 2-dimensional system, are a m o d e l system on which to study the m a g n e t o resistance p h e n o m e n a . Next to the negativem a g n e t o r e s i s t a n c e effect, we have emphasized the existence o f a n o n m o n o t o n i c , possibly even oscillatory, variation with B±. F o r a periodic g-layer structure we have discussed thermally excited and m i n i b a n d states of exceptionally high mobility. T h e y result f r o m electron waves moving c o h e r e n t l y in the periodic structure in such a w a y as to avoid scattering by the ions. T h e cyclotron r e s o n a n c e mobility (at B± = 7 . 2 T ) is g r e a t e r t h a n 100 000 cmE/V s.
Acknowledgements In this review I have cited liberally f r o m stud e n t s ' work. J. C z i n g o n has p r o v i d e d the data in figs. 2, 3 and 5. T h e cyclotron data in fig. 7 are f r o m the P h D thesis o f X i a o m e i Feng. T h e y are to be included in a c o m p r e h e n s i v e r e p o r t tog e t h e r with a contribution f r o m J.J. Mares. T h e I n P samples for the experiments have b e e n provided by A . Kohl of the R W T H A a c h e n . T h e G a A s samples stem f r o m an earlier collaboration
305
with K. Ploog. T h e calculation of the minibands in fig. 8 was d o n e by T. I h n and K. Friedland.
References [1] A. Zrenner, H. Reisinger, F. Koch and K. Ploog, in: Proc. 17th Int. Conf. on the Physics of Semiconductors, eds. J.D. Chadi and W.A. Harrison (Springer-Verlag, Berlin, 1985) p. 325. [2] Qiu-yi Ye, B.I. Shklovskii, A. Zrenner, F. Koch and K. Ploog, Phys. Rev. B 41 (1990) 8477. [3] Xiaomei Feng, J.J. Mares, M.E. Raikh, F. Koch, D. Grfitzmacher and A. Kohl, Surf. Sci. 263 (1992) 147. [4] M.E. Raikh, J. Czingon, Qiu-yi Ye, F. Koch, W. Schoepe and K. Ploog, Phys. Rev. B 45 (1992) 6015. [5] Yi-ben Xia, E. Bangert and G. Landwehr, Phys. Stat. Sol. B 144 (1987) 601. [6] W. Schirmacher, Phys. Rev. B 41 (1990) 2461. [7] B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, 1984). [8] B.L. Altshuler, A.G. Aronov and D.E. Khmelnitskii, Pis'ma Zh. Eksp. Teor. Fiz. 36 (1982) 157 [JETP Lett. 36 (1982) 195]. [9] V.L. Nguyen, B.Z. Spivak and B.I. Shklovskii, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 35 [JETP Lett. 41 (1985) 42]. [10] M.E. Raikh, Solid State Commun. 75 (1990) 935. [11] H.-T. Fritzsche and W. Schirmacher, Europhys. Lett., in print. [12] H. Sigg, K. Ploog, Qiu-yi Ye and F. Koch, Phys. Rev. Lett. 65 (1990) 1951. [13] J.J. Mares, Xiaomei Feng, T. Ihn and F. Koch, in: Proc. 21st Conf. on the Physics of Semiconductors (Bejing, 1992), to be published. [14] T. Ihn, H. Kostial, R. Hey, M. Asche, J.J. Mares, Xiaomei Feng and F. Koch, submitted for publication in Japan J. Appl. Phys.
Physica B 184 (1993) 298-305 North-Holland
Electronic states and magnetotransport in g-layer systems F. K o c h Physik-Department E16, Technische Universitiit Miinchen, Garching, Germany We review some aspects of magnetotransport in a semiconductor material for which the dopant atoms are incorporated during MBE growth in atomically sharp defined sheets. Hopping transport, in the dilute limit of planar doping, shows interference effects which lead to a distinct resistance variation in a magnetic field. A periodic sequence of B-layers has electronic states with exceptionally high mobility that result from a coherent superposition of wavefunctions throughout the array.
1. Introduction
The quantum properties of a sheet of donor atoms in an epitaxially grown semiconductor were first demonstrated in ref. [1]. For a density N $/>1X1012 Si atoms per cm 2 in GaAs this so-called B-layer is a metallic system of subband states in the self-consistent potential well formed by the positive sheet of Si ions and the electronic screening charge. The soft, V-shaped potential in fig. 1 is a good description of the system when the density is sufficiently high so that the electrons average the granular nature of the Si + charges in the dopant plane. It is known that the levels Eo, El, E 2 , . . . each have a different transport mobility/x. The values of/x0, /£1, ~LL2, are the result of the distinct configurations of the electron densities of the subbands relative to the donor ion sheet. In general/z 0 is much less than /x1,/x 2. . . . . For the system in fig. 1, t% is of order 1000cm2/Vs. For higher subbands the mobility can be an order of magnitude more. Single B-layers in the metallic limit of doping concentration have been studied in many publications. Recent and novel experiments have emphasized the low-density case where the subband description does not apply. Using 8-doping one can fabricate a well-controlled planar system •
"
•
Correspondence to: F. Koch, Physik-Department, Technische Universitfit Miinchen, D-8046 Garching, Germany.
where hopping transport takes place. For a low concentration of dopant atoms unintended background doping and boundary layer depletion must be considered. In this dilute limit it is essential to make use of a multi-layer arrangement with controlled aspect ratio a/b. The average in-plane spacing a is linked to the doping density. The planar separation b is chosen independently to satisfy other criteria. For GaAs samples, where the typical p-type background and surface band-bending reduce the electron density, the b value of 100nm was selected in order to avoi~t excessive depletion of the small number of electrons ir~ the dilutely doped layer. There is also a sacrificial top and bottom sheet to be considered. The latter is totally depleted in 6 - doping
t
E (meV) 12oo
!
Si !1
n
! /
-
-
E
n"
-
-
n
~
~
GaAs
Z =
100
200
Z
Fig. 1. Potential well, subband energies and wavefunctions for a layer of 4.1 × 10 t: Si donor ions per cm 2 in GaAs (see ref. [1]).
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
F. Koch / Electronic states and magnetotransport in 6-layer systems
satisfying the necessary boundary conditions for GaAs. Hopping-transport occurs within each of the many parallel conducting planes separately. The periodic nature of the 8-layers is not significant for hopping as long as a / b is much less than one. More details on GaAs samples are found in [2]. For the formation of minibands the regular array of 8-layers is essential. The use of MOVPE-grown InP with a light, n-type ( - 1 × 1015 cm 3) background is advantageous because it avoids depletion of the miniband charge that exists between the doping planes. For the InP samples there is no reason to expect depletion of the outermost layers. These samples are discussed in ref. [3]. In this paper the intention is to focus on two topics of current interest in magnetotransport studies on 8-layers. We discuss in section 2 the interference effects in the hopping magnetotransport in dilutely doped layers. Section 3 deals with periodic 8-layers and the high-mobility, coherent miniband states that are observed in such systems.
2. Magnetoresistance in hopping transport: interference, resonant tunneling and oscillations The study of electrical transport in dilutely doped semiconductors is a time-honored theme. A large number of publications exists on the subject. Nevertheless, only recently have sheetdoping structures made a mark in this field. Several reports [2-6] highlight general features of hopping magnetoresistance that show up clearly in such planar systems. For ~-layers the hopping sites are deeply embedded inside the epitaxially grown, highly perfect semiconductor layer. Because it is a 2D structure one can contrast the influence of a magnetic field parallel and perpendicular to the layer plane. The potential of each of the donor ions is identical. Only the spacings and positioning of the donors is random. The unintended background doping is adequately known and by adjusting the multilayer spacing it can be controlled deliberately.
299
Other than for MOSFETs, modulation-doped quantum wells or evaporated thin layers, there are no unknown and unspecifiable surface effects. We describe here the basic physical ingredients of the hopping magnetotransport phenomenon and lead up to an intensely debated question- does there exist an oscillatory contribution to the hopping resistance in a magnetic field? The most basic and thoroughly documented ingredient of magnetotransport is the fact that in the limit of high magnetic fields the hopping resistance must increase exponentially with increasing field. Reference [7] describes the B 2dependence that originates from wave function shrinkage. The compression of the hydrogenic impurity ground state reduces the overlap between the hopping sites in a manner such as to exponentially decrease the tunneling probability. The experimental observations on ~-layers (fig. 2) are no exception to the rule. They obey the scaling prediction for between slopes in a log R vs B 2 plot for the two orientations [5,7]. It is expected that in the perpendicular field (B±) the slope has twice the value of the parallel case (BII). While for nearest-neighbor hopping the slope is nearly T-independent, it should with decreasing temperature and for variable-range hopping increase according to a power law. The data in fig. 2 shows the correct tendencies but does not cover a sufficient T-range to uniquely fix the exponent. There is an experimental problem in accurately measuring the exponentially rising resistance at low temperature without heating either the lattice or the electrons. Our measuring limit is in the 109 [). range. While the Bii-data is by and large described adequately in terms of wave-function shrinkage, the perpendicular case in fig. 2 shows a distinct and additional phenomenon- negative magnetoresistance. The resistance decreases linearly at fields of the order of a few tenths of 1T. The reduction saturates and reaches about 50% at the lowest temperatures in fig. 3. The effect is absent in the parallel configuration. Instead of the - 5 0 % decrease at 2 T, the Bit-resistance in fig. 2 has more than doubled. The negative resistance effect is not just a minor correction. It
300
F. Koch / Electronic states and rnagnetotransport in g-layer systems I
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Fig. 2. Variation of the resistance per square of a B-layer with N s = 8 × 101° cm-2 in GaAs with B~ and B~. The theory predicts a straight line. Note the distinct negative magnetoresistance in a perpendicular field.
is the dominant physical p h e n o m e n o n in fields before the wave-function compression takes over. The decreasing resistance is explained by an interference mechanism first postulated in ref. [8] for variable-range hopping. Schirmacher [6] gives a simpler, experimentally more relevant description in terms of the interference of an electron reaching a final site f after tunneling along two alternative paths (fig. 4). In this configuration of hopping sites, an electron starts at site i and tunnels both directly to f, and along a path in which it scatters resonantly off the potential at an intermediate site j. The loss in tunneling amplitude because of the extra distance in the path i ~ j --~ f can be more than compensated by the enhancement effect of the resonance. T h e r e will be destructive interference because of
the phase shift ---~, which the particle incurs during resonant scattering. In a magnetic field the flux through the triangular area in fig. 4 adds an additional phase. On the average, just as in the weak-localization theory for negative magnetoresistance [9], the field acts to decrease the destructive interference. It allows the electron to complete more favorably its hopping process from i to f. One may think of path i---~j---~f in fig. 4 as a coherent, resonant-tunneling process through a quantum well. In that case, the phase shift 7r is the extra half wavelength the particle requires to traverse the potential well at site j. The resonance condition E i ~ Ej for a strong transmission amplitude is evident. It is also obvious that the degree of destructive interference is substantial because the two waves can be of comparable
301
F. Koch / Electronic states and magnetotransport in g-layer systems
0
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Fig. 4. Interference in the hopping transport along separate paths in a triangular arrangement of impurity sites. Site j provides a resonant-tunneling process for the particle moving from an occupied site i to an unoccupied final state f. The energy scheme is shown below.
-0.2
-0.4
0
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0
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Fig. 3. Normalized variation of the resistance with B± for a GaAs B-layerwith Ns = 8 x 10 cm 2. Note the extra structure at B l - 1.6T. amplitude. The extra decay of intensity for the longer indirect path i--->j---* f is counteracted by the resonance e n h a n c e m e n t factor. There will be totally destructive interference for certain E i Ej and geometrical extra pathlength. The data in fig. 3 poses yet another challenge to the theory. Studying different G a A s samples, and m o r e recently also InP g-layers, leaves little doubt that the nonmonotonic variation of resistance in the negative magnetoresistance range is a real effect. We have included in fig. 5 data for InP. The sample has a somewhat higher carrier sheet density (N s -- 1.5 x 1011 cm -2) and has not yet been measured for T below 1.47 K. Its negative magnetoresistance effect is ~ 2 5 % at the lowest T. Although the exact form of the resistance variation in the 0 - 4 T range is not totally identical with that of the G a A s sample, the
existence of a p e a k in the intermediate field range ( - 2 . 5 T ) may be suspected by a comparison with fig. 3. T h e extra resistance p e a k has caused some excitement and serious concern for our theoretical understanding. The nonmonotonic behavior is not described properly by either the linearinterference negative magnetoresistance, nor by the B e, high-field positive-resistance theory. It is
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302
F. Koch / Electronic states and magnetotransport in 6-layer systems
a phenomenon reminiscent of the oscillatory resistance variations that occur for g-layer samples in the metallic-density limit. For the hopping transport, on first sight, Landau quantization and Shubnikov-de Haas oscillatory structures are unexpected. In searching for mechanisms to explain the nonmonotonic resistance variation, different approaches have been used. Reference [4] builds on the ideas discussed earlier in ref. [10]. It is argued that the effect appears with the approach to variable-range hopping at the.lowest T in fig. 3. As is well known, Mott's explanation of this hopping mechanism involves the density of impurity states. A high density of states allows for easy hopping at low temperatures because of the correspondingly large number of sites that are then available within the k T range. Reference [10] proves that when the broadening of the state distribution is by random overlap of wavefunctions and irregular placement of the donors, then the wavefunction shrinkage acts to collapse the width, thus increasing the density of states. This extra B2-dependence gives a quadratic decrease of the resistance before the rising magnetoresistance becomes dominant. In ref. [4] it is argued that the addition of the linear resistance decrease (caused by the coherent superposition effect) to the quadratic decrease (incoherent Motthopping) and the B2-increase caused by high fields, will account for the nonmonotonic variation as seen in fig. 3. A different explanation has been argued in ref. [11]. This work quantitatively sums up the random, triangular paths and their respective contributions to the resistance. The authors find that the sum has damped oscillations. A first maximum occurs when a single flux quantum penetrates a certain characteristic area. The area is referred to as the 'interference'-hole. It is related to the triangle surface for which completely destructive interference will occur. By referring to fig. 4 it is possible to understand what is meant by the characteristic area that occurs in the theoretical treatment of ref. [11]. Complete destructive interference requires that the amplitude of the waves along the two paths be equal. Scattering by the potential at site
j causes a resonant enhancement of that wave amplitude. Its magnitude depends on how close the energy E i is t o Ej. It is apparent that for some position of j the extra length of the tunneling path i---~j--->f is just compensated by the resonant enhancement. Destructive interference will occur for a finite area of the triangle. To have a well-defined area, however, requires that averaging over random positions and paths, as well as over the E i - E j , does not broaden the area value too much. According to ref. [11] this is the case when the resonance is sharply defined, or equivalently when the impurity state width is much less than the localization potential V0. It is argued in ref. [11] that a characteristic area exists for the present case and that damped oscillations occur as one, two or more flux quanta thread the area. The argument has a certain similarity with the simple Landau quantization rule for orbital electron motion in metallic transport. The proposed oscillation mechanism and the observed peaks represent a new type of resistance variation in hopping transport. It is reminiscent of the magnetoresistance for artificial quantum-dot lattices where hopping transport takes place between the regularly spaced dots. The case of the randomly spaced impurities of the g-layer is as yet a contested topic. Controversy aside, the fit to experiment that Fritzsche and Schirmacher [11] achieve with their evaluation of the negative magnetoresistance for parameters that describe our samples is very suggestive. We show in fig. 5 their calculated result along with the data (insert).
3. Periodic 8-layers: magnetotransport in high-mobility minibands The quantum arrangement of the electronic screening charges around the sheet of donor ions in fig. 1, has interesting implications for the mobility. Taking for example the n = 1 state, we expect a fairly high mobility because of the wavefunction node at z = 0. For GaAs with a doping of -1012Si ions per c m 2 a value of 10000cm2/Vs applies at low temperature. AI-
303
F. Koch / Electronic states and magnetotransport in 6-layer systems
though this is higher than the mobility of the ground-state electrons, the enhancement is not at all comparable with what has been achieved by modulation-doping of heterojunctions. The single 5-layer does not provide the optimal 'spacer width'. The challenge is to provide a way for electron states of the subbands to spread out more. High conductivity of the 5-doped layer requires that the largest possible number of electrons occupy regions far from the doping sheet. The basic answer to the challenge is provided by properly shaping the wings of the V-shaped, self-consistent potential. The overlap with neighbouring potential wells in a periodic structure can be used to produce flat regions of the potential into which electrons can spread easily. It is also important to use not very high ion densities because this increases the binding strength of the potential. We first noted the existence of high-mobility electrons in periodic 5-layers of GaAs in cyclotron resonance experiments [12]. The carriers were thermally excited into the undoped spaces between the layers. In a Bil-field such electrons give a narrow, high-oJc~" resonance line. Electrons which are tightly bound in the electric subband levels of the layer do not contribute in this configuration. With the field perpendicular to the doping sheets both the carriers in the layers and between them will participate in the resonance. The result is a broad line. This explanation for the work in ref. [12] differs in a subtle way from that in the publication. The effect of thermally excited, high-mobility carriers can also be found in DC magnetotransport experiments. References [3,13] illustrate this by the added contribution to the Hall effect currents. The Hall resistance Rxy is the parallel sum of the resistances for current flowing within the doping layer and current carried by the highly mobile between-layer electrons. The extra component causes the layer system to behave like a two-fluid model in which the bypass current has a substantially bigger mobility. In ref. [3] we have discussed the Rxy(B ) curves for an InP sample with N S = 2.8 × 10 ll cm -2. At the lowest T of 1.6K, the high-mobility carriers practically disappear giving a straight line for Rxy
in this simple case. Raising the density only slightly to a value of 3.6 × 1011 cm -2 per sheet, as for the data in the accompanying fig. 6, we find that the two-fluid effect is retained even at 1.3 K. In this figure, the structure at - 8 T and 1.29 kl-I is the expected i = 2 Hall plateau for the sum of the two carriers. The sample has 10 doping sheets acting in parallel. In principle, the Rxy curves in the figure allow one to determine both the number density and mobility values of the two electronic fluids. In practice, because of the quantum Hall effect structures, it is difficult to achieve an accurate and sensitive fit. Using the classical two-fluid formula we have shown in ref. [13] that a mobility of order 100 000 cmZ/V s is needed to explain the curves. A more straightforward way to determine the parameters of the electrons is cyclotron resonance. The sharp line at 7.2 T in fig. 7 is the resonance of about 6 × 101° cm 2 electrons per sheet with a mobility of 120000 cm2/V s at T=5K. In the spectra one clearly sees the T-dependent increase in the number of carriers that correlates with the changes in Rxy(B ) of the previous figure. The small absorption signal at 3.8 T is the ls---~ 2p impurity magnetoresonance
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F. Koch I 'Electronic states and magnetotransport in a-layer systems
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for electrons bound to isolated background donors. It is evidence for the expected weak, n-type background doping in M O V P E - g r o w n InP. For the present case, the 1.5 Ixm thick layer has about 1.5 × 1011 c m - 2 of such electrons b o u n d to the impurities. The signal provides a convenient calibration and check on the cyclotron resonance data. T h e r m a l excitation of electrons, with a realspace transfer to high-mobility regions between the doping layers, makes the multi-~-layer syst e m a nonlinearly conducting medium. This effect has been proposed and demonstrated in ref. [14] for a G a A s sample with 10 sheets of N s = 5 X 1011 cm -2 spaced 100 nm apart. For this case, it is found that the ohmic conductivity is a factor two higher than for a randomly doped layer with the same total dopant concentration of 5 x 1012 cm -2 spread over 1 ~m. The differential conductivity dj/dE, m o r e o v e r , rises nearly 50% in a field E - - 1 V/cm. The conductivity of the statistically r a n d o m - d o p e d sample remains nearly constant over the same range of E. T h e most exciting prospects offered by the present investigations are in the design-tailoring
of densities N s and layer-spacing to achieve even at T = 0 K substantial densities of high-mobility electrons. The layer-system can be engineered by p r o p e r placement of impurities to have an optimal electrical conductivity. The electrons of the periodic layer system are miniband states. In particular, the highly mobile electrons of the second and higher miniband states are the result of a coherent superposition of waves each of which avoids the donor ions. The secondminiband (n = 1) electrons of fig. 8. are diffracted waves in the periodic structure that have zero occupation probability at the site of the donor sheets. Scattering in the miniband occurs only via the fluctuation potential in the plane of the d o n o r sheets. The latter is screened by the n = 0 ground state electrons and acts at a distance dictated by the wave-mechanical distribution of charge. Figure 8 is from a calculation for G a A s with N s ~ 5 x 101° cm -2 and spacing of 100 nm. T h e calculation assumes an n-type background doping of 1 x 1015 cm -3. For this case, 0.83N s electrons are predicted to occupy the E 0 level. T h e n = 1 miniband has 0.14N s occupation and is distributed spatially as in the drawing. We note that the flattening of the potential in the wings of the V-shape that results from the superposition of the periodic array leads to an increased average distance of the miniband electrons from the ions. The result is the dramatically improved scattering lifetime that we have demonstrated here in the magnetotransport experiments.
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Fig. 8. Calculated potential variation, energies and the charge distribution of the n = 1 miniband states for a periodic 6-layer with N~ = 5 × 1011 cm -2 in G a A s . The energy E 1 is 14 meV below E F and the double line is the bandwidth.
F. Koch / Electronic states and magnetotransport in 6-layer systems
4. Conclusions In this p r e s e n t a t i o n we have focused on two subjects of current interest in m a g n e t o t r a n s p o r t in g-layers. T h e discussion on h o p p i n g transport has d e m o n s t r a t e d that the g-layers, because they r e p r e s e n t a particularly clean and well-defined case o f a disordered 2-dimensional system, are a m o d e l system on which to study the m a g n e t o resistance p h e n o m e n a . Next to the negativem a g n e t o r e s i s t a n c e effect, we have emphasized the existence o f a n o n m o n o t o n i c , possibly even oscillatory, variation with B±. F o r a periodic g-layer structure we have discussed thermally excited and m i n i b a n d states of exceptionally high mobility. T h e y result f r o m electron waves moving c o h e r e n t l y in the periodic structure in such a w a y as to avoid scattering by the ions. T h e cyclotron r e s o n a n c e mobility (at B± = 7 . 2 T ) is g r e a t e r t h a n 100 000 cmE/V s.
Acknowledgements In this review I have cited liberally f r o m stud e n t s ' work. J. C z i n g o n has p r o v i d e d the data in figs. 2, 3 and 5. T h e cyclotron data in fig. 7 are f r o m the P h D thesis o f X i a o m e i Feng. T h e y are to be included in a c o m p r e h e n s i v e r e p o r t tog e t h e r with a contribution f r o m J.J. Mares. T h e I n P samples for the experiments have b e e n provided by A . Kohl of the R W T H A a c h e n . T h e G a A s samples stem f r o m an earlier collaboration
305
with K. Ploog. T h e calculation of the minibands in fig. 8 was d o n e by T. I h n and K. Friedland.
References [1] A. Zrenner, H. Reisinger, F. Koch and K. Ploog, in: Proc. 17th Int. Conf. on the Physics of Semiconductors, eds. J.D. Chadi and W.A. Harrison (Springer-Verlag, Berlin, 1985) p. 325. [2] Qiu-yi Ye, B.I. Shklovskii, A. Zrenner, F. Koch and K. Ploog, Phys. Rev. B 41 (1990) 8477. [3] Xiaomei Feng, J.J. Mares, M.E. Raikh, F. Koch, D. Grfitzmacher and A. Kohl, Surf. Sci. 263 (1992) 147. [4] M.E. Raikh, J. Czingon, Qiu-yi Ye, F. Koch, W. Schoepe and K. Ploog, Phys. Rev. B 45 (1992) 6015. [5] Yi-ben Xia, E. Bangert and G. Landwehr, Phys. Stat. Sol. B 144 (1987) 601. [6] W. Schirmacher, Phys. Rev. B 41 (1990) 2461. [7] B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, 1984). [8] B.L. Altshuler, A.G. Aronov and D.E. Khmelnitskii, Pis'ma Zh. Eksp. Teor. Fiz. 36 (1982) 157 [JETP Lett. 36 (1982) 195]. [9] V.L. Nguyen, B.Z. Spivak and B.I. Shklovskii, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 35 [JETP Lett. 41 (1985) 42]. [10] M.E. Raikh, Solid State Commun. 75 (1990) 935. [11] H.-T. Fritzsche and W. Schirmacher, Europhys. Lett., in print. [12] H. Sigg, K. Ploog, Qiu-yi Ye and F. Koch, Phys. Rev. Lett. 65 (1990) 1951. [13] J.J. Mares, Xiaomei Feng, T. Ihn and F. Koch, in: Proc. 21st Conf. on the Physics of Semiconductors (Bejing, 1992), to be published. [14] T. Ihn, H. Kostial, R. Hey, M. Asche, J.J. Mares, Xiaomei Feng and F. Koch, submitted for publication in Japan J. Appl. Phys.