Application of High Magnetic Fields in Semiconductor Physics: Proceeding of the International Conference Held in Grenoble, France, September 13–17, 1982

548 Pages·1983·29.669 MB·English

by  K. v. Klitzing, B. Tausendfreund, H. Obloh (auth.), G. Landwehr (eds.)

PRECISION DETERMINATION OF h/e 2- AND THE FINE-STRUCTURE
CONSTANT FROM MAGNETO-TRANSPORT MEASUREMENTS ON
2D ELECTRONIC SYSTEMS
K. v.Klitzing, B. Tausendfreund, H. Obloh, and T. Herzog
Physik-Department, Technische Universit~t M~nchen,
D-8046 Garching, FRG
Introduction
High magnetic fields are necessary for the observation of the quantized
Hall resistance - a quantum phenomenon which leads finally to a new type
of resistor R H with a resistance value depending only on the Planck
constant h and the elementary charge e I :
R H = h/e2i ~ 25812.8 ~/i i = 1,2,3,... )I(
Such a resistor allows a direct determination of h/e 2 and therefore the
fine-structure constant ,z( because ~ is given by the equation (SI units)
= (e2/h) (~oC/2) )2(
with the velocity of light c = 2.99792458 x IoSm/s and
~o-- 49 x 10 -7 H/m.
A simple experimental arrangement for the determination of ~ is shown
in Fig. .I The current I is flowing through both the resistor R H and
a standard resistor R N (resistors calibrated in SI units with an uncer-
tainty of about 10 -7 are maintained at the national laboratories).
From the ratio of the voltage drops across R H and R N (the absolute
values of the voltages need not be known) a determination of h/e 2 and
therefore v, is possible. The uncertainty in h/e 2 from such an experi-
ment is at least as large as the uncertainty of about 10 -7 in the value
of the reference resistor R N. If the experimental uncertainty is smaller
than 10 -7 , then the quantized Hall resistance R H can be used as a
resistance standard. Different national laboratories are investigating
such an application. Up to now, experimental problems limit the accur-
acy of the data, and the main purpose of this article is to discuss some
Uo
Fig. I: Experimental
set-up for measurements
/ of the fine-structure
constant using the
quantum Hall effect.
R H is the quantized
Hall resistance, R N the
standard resistance,
and S is a two-way
selector switch.
of the problems related to high-precision measurements of h/e 2 with
uncertainties of less than 10 -6 .
Systems used for h/e2-measurements
The realization of a resistor with resistance values according to Eq. I
is based on Hall effect measurements on a two-dimensional electron gas
(2DEG) 2. Up to now, all experimental results are obtained on MOS
devices or heterostructures, where a 2DEG can be created at the inter-
face between a semiconductor and an insulator or at the interface bet-
ween two semiconductors. The two-dimensional character originates from
a quantization of the energy for the motion perpendicular to the inter-
face into well separated electric subbands Eo, El, ... due to the con-
finement of the electrons within a narrow channel. Fig. 2 shows cross-
sections and top-views of typical devices used in our experiments.
................. ~-~., Gate
/ aG As (undoped) 20 nm
Source Gate Drain / . . . .
-,, / / / b~-aopea 061 nm
IA aG As
/\ /~/////,////~///////~ ............... .............L ......... / /~ /5 nm
\ SiOz / ................... undo~Jed .....................
-2 GED
n /+ / p- Si/ GaAs (undoped) lpm
n- inversion layer
(2 )GED /Ga As (semi-insulating)
/
Fi~. 2: Top view and cross-section of typical devices used
in our experiments. Left side: Silicon MOSFET. The
InSb MISFET is built up of lacquer as an insulator
on top of p-InSb. Right side: GaAs/AlxGa1_xAS
heterostructure with Schottky gate.
The carrier density of the 2DEG can be changed by applying a gate vol-
tage Vg across the metal-insulator (SiO 2 or AlxGal-xAS)-semiconductor
(Si, InSb, GaAs) capacitor. The voltage drops between the potential
probes for a constant source-drain current are proportional to the
components Pxx and Pxy = RH of the resistivity tensor.
A strong magnetic field B perpendicular to the plane of the 2DEG quan-
tizes the energy for the motion in the plane leading schematically to
a discrete energy spectrum
Ej, n = Ej + (n + ~)~c n,j = O,1,2 .... )3(
with the cyclotron energy ~c = ~eB/m.
The energy splitting between adjacent Landau levels (~ n = )I increases
with increasing magnetic field, so that well separated energy levels
are expected, if this splitting is much larger than the broadening of
the energy levels. Since the degeneracy factor N of each Landau level
Ej, n is equal to 3
N = eB/h )4(
a Hall resistance R H corresponding to
R H = h/e2i )5(
is expected, if exactly an integer number i of energy levels is occupied
i( is identical with g(n + I), where g is the degeneracy factor due to
spin- and valley-splitting and n is the Landau quantum number).
The condition of a fully occupied Landau level is identical with no
elastic scattering of an electron from an occupied to an empty state
and therefore to a vanishing energy loss. Theoretical calculations
based on the self-consistent Born approximation at T = 0 K predict that
the resistivity Pxx is basically proportional to the square of the den-
sity of states at the Fermi energy 4, so that the condition Pxx = O
characterizes the situation that all (mobile) states of a Landau level
are occupied, whereas the states of higher energetic levels are empty.
Exactly at these carrier densities n s (Pxx = O) a quantized Hall resis-
tance R H = h/ie 2 is expected.
In a simple approximation, the Hall resistance R H should decrease
monotonically with increasing carrier density n s and therefore with
increasing gate voltage: R H -- 1/n s -- I/Vg. This means that the carrier
density should be adjusted and stabilized at the value n s = iN = ieB/h
with an accuracy corresponding to the desired accuracy of better than
10 6 for the quantized Hall resistance. This is not possible due to
instabilities in the correlation between gate voltage and carrier den-
sity (threshold shifts) and a not well defined gate voltage for which
the resistivity Pxx becomes zero.
Fortunately, localized states lead to a stabilization of R H at the
value of R H = h/ie 2 (Hall plateaus). Ando 5 has shown that the tail
states in each Landau level are localized. Different authors demon-
strated that, if the occupation of localized states is changed, the
resistivity components Pxx and Pxv remain constant at a value corres-
ponding to fully occupied Landau ~evels 6,7. All these theories are
based on the assumption of zero temperature, so that the resistivity
Pxx should be exactly zero as long as the Fermi energy lies in the
mobility gap between two Landau levels.
Practically, thermal excitations across the energy gap 8 or variable
range hopping between "localized" states 9 lead to a finite scatter-
ing probability and therefore to a finite resistivity Pxx- One important
question is whether this finite scattering rate leads to a correction
in high-precision measurements of the quantized Hall resistance.
Experimental results
Results for the magneto-resistivity and the Hall resistance obtained
for different two-dimensional systems (Si, InSb, GaAs) as a function
of the gate voltage are summarized in the figures 3-5. The overlap bet-
ween adjacent energy levels for the InSb MISFET is relatively strong
even at B = 12 T, visible as a finite value of the minima in the Pxx-
oscillation. The reason for the unresolved Landau levels is that the
quality of the InSb-insulator interface is not very high, which leads
to a reduced mobility ~ and therefore to an increased broadening 1 ~ of
the Landau levels (F -- I/~ in the self-consistent Born approximation
I-LRN:EBORf
= . K 8=14.5 T I=20E-6 g x
00001 0OOZ
2500 0051
t
0005 0001
/
2500 005
~ 0
4 S 8 10 12 14 16 B1 20 22 24 26 28 30 32 34
,vo Iv
Fig. 3: Gate voltage dependence of the Hall resistance R H = Pxy and
the resistivity Pxx at B = 14.5 T for a silicon MOSFET.
52
12 InSb-MISFET
0 I(E T = 1.5 K
1.0 20
B=12T
~)xx/kQ T L 0f(E~) Qxy/k~
0.8
6.0
E(IO o )
01
lf(Eo) 1;~ °) 0~E I} 2f(E o)
I
;,Z.O -
02
0
I I I I I I I
-5o o so oo1 o5~ 200
~vozv
Fig. 4: Gate voltage dependence of the Hall resistance R H = Pxy and
the resistivity Pxx at B = 12 T for an InSb-MISFET.
TEF-sAmC/SAT'%G'°A I eborP G-D,~B0-4 =vRl
I T=I,5K
OK
x R 1
4-- T 54
Kn
2T jT
1
T5,1
T5,1
l
vGv %v
Fi 9. 5: Hall resistance and resistivity at different magnetic fields
as a function of the gate voltage for a GaAs-AIo.3Gao. 7As
heterostructure.
4). In addition, the occupation of a second electric subband E I for
this material at a relatively low carrier density leads to a super-
position of different Landau levels and probably to an increased level
broadening due to intersubband scattering. This means that well separat.
ed energy levels are not observed for the InSb system, although the
small effective mass for electrons in this material leads to a cyclot-
ron energy larger than oberserved for Si or GaAs.
The most pronounced Hall steps are visible for the GaAs/AlxGa1_xAS
heterostructure. Compared to silicon MOSFETs, the mobility of the
electrons in this heterostructure is typically one order of magnitude
higher and the effective mass a factor of three smaller, so that much
better resolved Landau levels are expected. However, the highly doped
AlxGa1_xAS layer is not a good insulator, which may lead to leakage
currents from the gate to the interface channel and therefore to errors
in high-precision measurements of the quantized Hall resistance. There-
fore measurements at zero gate voltage are necessary with a fixed car-
rier density n s determined mainly by the ionized donors in the
AlxGa1_xAS depletion layer. The quantum condition n s = ieB/h can be
obtained by changing the magnetic field. A typical result for such a
measurement is shown in Fig. 6. High resolution measurements of the
flatness of the Hall steps on a similar sample show that the value of
the quantized Hall resistance remains constant within the experimental
uncertainty of ± 0.06 ppm even if the magnetic field is changed by
11% (Fig. 7). However, the absolute value of R H is not known with such
an accuracy, because the reference resistors of 6453.2 ~or 12906.4
in our ac bridge shown in Fig. 8 are not calibrated. Tsui et al. 10
HU h
T = e 2 .... --- /
3 f 7
002- Vm ? , " / / \ -Vm001
/tsnoc=A~101=I~ / X
I/\
21U. = Up 32U = U H / / / U P •
HU h
T 9
-- T# .......... -7-
/
001- Vm ~ U H~/.// ~ ~ Vm05
A"
/u_ :L___ J
01 , alseT
5 cp itengam dleif B 51
Fig. 6: Hall effect and resistivity measurements on a GaAs-AI O 3Gao.7As
heterostructure as a function of the magnetic field B."
&gxy/gxy ppm OaAs i=2
B = 8,2 Tesla
T=1,5 K
+/ ISD = 1 3 I~A
2
i
0 -
-2 --
.4/- --
t I I I I I I ~B %
B
-10 0 01
Fi@. 7: High-resolution measurements of the flatness of the Hall
plateau R H = h/2e 2 for a GaAs-AIo. 3Gao. 7As heterostructure
(similar sample as shown in Fig. 6).
1.0000 _33M I
Fig. :8
Sample Experimental set-
up for high-resol-
ution measurements
of the flatness of
the Hall ~lateau
R H = h/4e z.
-' 1.0000 Lf.M
reported a value of h/4e 2 = (6453.2004 ± O.OO11)~. From this result
the following value for the fine-structure constant is obtained,
which may be compared with G determined by other methods 11:
-1
= 137.035968(23) )?( (quantum Hall effect)
-I
= 137.O35963(15) )?( (7~-method)
-I
= 137.035993(5) )9( (anomalous moment) )6(
-I
= 137.035969(21) (46) (muonium hfs)
-I
= 137.O36040(110) (recommended 1973)
The first errors listed are experimental and the second theoretical.
The question mark in the condensed-matter determinations indicate that
the theoretical uncertainties are unknown; they are possibly very small
in comparison to the experimental ones.
Systematic errors in high-precision measurements of the quantized Hall
resistance using heterostructures may be due to a leakage current
through the highly doped AlxGal_xAS layer. The conductivity of this
layer can be increased with infrared illumination, and the result in
Fig. 9 shows that after illumination the resistivity minima Pxx in-
creases and a finite slope for the Hall steps is observed. (Results
for the same device without illumination are shown in Fig. 6~) High-
precision measurements of h/e 2 on such a device (Fig. 9) are not
possible.
Conductive layers parallel to the 2DEG of a silicon MOSFET are prac-
tically not present, because the SiO 2 and the depletion layer between
the 2DEG and the weakly doped substrate material are nearly ideal in-
sulators at helium temperatures.
Therefore, the silicon MOSFET seems to be the most reliable system for
high-precision measurements of h/e 2 , but the experimental conditions
are more complicated, because magnetic fields higher than used for the
GaAs-AlxGa1_xAS heterostructure are necessary in order to separate the
25 -5
20 ¸ T=1.5 K
ns~6 x1011cm -2
jRx
(after IR iUuminQtion)
15 ¸
T
10-
5-
5 0I 51
BIT
Fig. 9 : Hall resistance and magneto-resistivity measurements
on a GaAs-AIo. 3Gao. 7As heterostructure after illumination
with infrared light. The IR illumination leads to an
increased carrier density of the 2DEG and to bypass
conductivity of the silicon-doped AIo. 3Gao.7As layer.
The experimental result without illumination is shown
in Fig. 6.
Landau levels. Even at B = 14.5 T the Hall plateau R H = h/4e 2 in Fig. 3
remains constant (within ± 0.2 ppm) only in a relatively small gate
voltage region of Vq = 12.3 ± 0.2 V. At lower magnetic fields B( <
14.5 T) or higher temperatures (T > 1.5 K) the gate voltage region
where Pxv stays constant becomes very small and usually a finite slope
dpxy/dVg~can be resolved at the gate voltage where a fully occupied
Landau level is expected. This slope is proportional to the resistivity
Pxx as shown in Fig. 10, which means that for a typical value of Pxx
= 0.01 Ohm and an uncertainty of I% for the gate voltage a deviation of
Pxy/Pxy ~ 3 x 10 -7 is expected. However, this uncertainty can be re-
duce~ by increasing the magnetic field or reducing the temperature,
which leads to smaller Pxx-values. The thermally activated behaviour
Pxx ~ exp(-~EI/kT) observed in the plateau region 8 is also visible
in the deviation of the Hall conductivity ~ ~xy relative to the quan-
tized value ~qu (Fig. I I). The activation energy ~ E is the same for
both Pxx and ~ ~xy, as long as the activation to one mobility edge
dominates (Fermi energy not exactly in the center between the two
mobility edges of adjacent Landau levels). The correction to the Hall
conductivity at very low temperature, where variable range hopping
dominates 12,9, is so small that no quantitative data are available.
1 Si i=4 e z
9xy %=4- 6
01 :- • HPg/1;11T
• H96 ;13T
"HPg/2 ;12T
lo-'
x e
,Fo
• x
X 0
Ox
0
01 ~4
I I I I 1/T J'K
52,0 5,0 57,0
1,0 1 01 001 9×× Q
Fi 9. 10: Slope of the Hall resis- Fi~. 11: Deviation of the
tance dPxy/dVg -- dpmy/dNs at Hall conductivity
the center of the Hall plateaus ~xv relative to
as a function of the minimal the ~uantized value
resistivity Pxx" The magnetic ~xy = 4e2/h as a
field is varied between 11T < function of the in-
B < 13 T, the temperature bet- verse temperature
ween 1.5 K < T < 4.2 K, and I/T.
the substrate bias voltage
between - 9 V < VSB < 0 V.
In summary, high-precision measurements of h/e 2 from magneto-transport
measurements - this is not only a question of high-resolution measure-
ments of voltages (the problems related to thermoelectric phenomena or
high-frequency pick-up have not been discussed in this article), but
also a question of a microscopic interpretation of the transport proper-
ties of a 2DEG in a strong magnetic field. Corrections to the value of
the quantized Hall resistance are always present because the resistivity
Pxx is never exactly zero. Such a finite resistivity leads on the one
hand to a geometrical correction in Hall effect measurements (the Hall
angle is not 90 ° , and therefore the finite aspect ratio of the device
leads to a reduction in the Hall voltage 13), and on the other hand
to corrections due to the scattering process. A quantitative theory of
the Hall effect in the localized regime at finite temperatures is not
available. Since all thinkable corrections to the value of the quan-
tized Hall resistance should decrease with increasing magnetic field,
measurements at different magnetic fields are the best check of the
reliability of the quantum Hall effect for the determination of the
fine-structure constant.
At the present level of uncertainty of the order 10 -6 , magnetic fields
of 14 T seem to be high enough for a quantitative determination of
h/e 2 from Hall effect measurements on silicon MOSFETs (at T = I .5 )K ,
but still higher magnetic fields are necessary, if a higher precision
is desired.
0~
Acknow led@ements :
The experiments were performed on a large number of devices, and we
would like to thank K. Ploog (Max-Planck-Institut fHr Festk6rperfor-
schung, Stuttgart), M. Pepper (Cavendish Laboratory, Cambridge),
R.J. Wagner (Naval Research Laboratory, Washington), and G. Dorda
(Siemens Forschungslaboratorien, MHnchen) for providing us with samples
This work was supported by the Deutsche Forschungsgemeinschaft via
SFB 128.
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