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Observation of quantized hall effect and vanishing resistance at fractional Landau level occupation
Physica 117B & 118B (1983) 688-690 North.Holland Publishing Company
688
OBSERVATION OF QUANTIZED HALL EFFECT AND VANISHING RESISTANCE AT FRACTIONAL LANDAU LEVEL OCCUPATION H. L. St6rmer a*, D. C. Tsui b*, A. C. Gossard a, and J. C. M. Hwang a a) Bell Laboratories, Murray Hill, NJ 07974 b) Department of Electrical Engineering and Computer Science Princeton University, Princeton, NJ 08544 *) Visiting Scientists at the National Magnet Lab., Cambridge, MA
02138
Quantization of the Hall resistance Pxy and the approach of a zero-resistance state in Pxx are observed at fractional filling of Landau levels in the magneto-transport of the two-dimensional electrons in GaAs-(AIGa)As heterostructures. At the lowest temperatures (T~0.5K), the Hall resistance is quantized to values Pxy = h/(i/3 e 2) and Pxy = h/(2/3 e2). This observation, unexpected from current theoretical models for the quantized Hall effect, suggests the formation of a new electronic state at fractional level occupation.
Quantized Hall resistance and zero-resistance state of two-dimenslonal electrons are now well documented phenomena [1-4]. At low temperatures (T) and high magnetic fields (B) the Hall resistance Px- of these systems is quantized to values Oxy = ~/ie2, where h is Planck's constant, e is the electron charge and i is an integer. At regions where Px- assumes its quantized values, the d i a g o n a ~ resistivity 0xx extrapolates to Pxx = 0 as T approaches T = 0. The observation of the quantized Hall effect and the zero-resistance state is attributed to the existence of localized states within the gap between Landau levels and spin levels which pin the Fermi energy, EF, for finite regions of B or electron density n[5-10]. The high accuracy to which Px~ is quantized to Pxy = h/ie2 is a consequence el gauge invariance and the existence of these mobility gaps [7]. The integer i represents the number of completely filled Landau and spin levels below E F [8]. Accordingly, the quantized Hall resistance is limited to intesral values of i, with Pxy = h/e2 being its largest quantized value. We have been studying the magneto-transport of the two-dimensional electrons in GaAs-(AIGa)As heterojunctions in the extreme quantum limit Ill, 12] when all electrons occupy the lowestenergy, spin-polarized Landau level, i.e. the Landau level filling factor, defined by = nh/eB, is ~ < I. We observed that, in our high mobility samples, plateaus and minima are developed in Pxy and Oxx, respectively, at low temperatures at ~ = 1/3 and 2/3. At our lowest temperature (T~0.5K) the Hall plateaus are quantized to Pxy = h/ie2 with i = 1/3 and 2/3 at = 1/3 and 2/3, respectively. These results are unexpected from the current theoretical models for the quantized Hall effect. The low temperature (T ~ 0.SK) high magnetic field (B ~ 200kG) experiments were performed at the Francis Bitter National Magnet Lab on lowdensity, high mobility, modulation doped GaAs-(AIGa)As heterojunctions [12], prepared by molecular beam epitaxy. The parameters of the two samples discussed in this paper are given
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
in Table I. Standard Hall Bridges were used for the experiments. Figure i shows 0x v and px x of sample i as a function of B at fou~ different temperatures. The scale on top of Fig. i shows the Landau level filling factor ~. At integer values, ~ counts the number of occupied levels, with ~ = i representing the beginning of the extreme quantum limit where only the lower spin state of the lowest Landau level is occupied. Fractional values of ~ indicate fractional occupation. At all integral values of ~ the data reproduce well the characteristic features of the quantized Hall resistance and zero-resistance state. As observed earlier, the plateaus in 0x~ as well as the minima in Pxx become increasingly pronounced when lowering the temperatures. The development of the regions of odd ~, where E F is pinned between two spin levels of the same Landau level, lag behind regions of even ~, where E F if pinned between two Landau levels, due to a smaller spin splitting as compared to Landau splitting [13]. In the extreme quantum limit at high temperatures, Pxy follows to a good approximation a linear relation with the magnetic field, 0xy ~ B/he, as one would expect from a classical independent electron model. However, on lowering the temperature, a feature occurs at ~ = 1/3 which approaches a plateau at the lowest temperature employed. It is accompanied by a strong reduction in the diagonal resistance Oxx, evolving into a minimum at the position of the plateau. The development of these features is completely analogous to the development of the plateaus in Oxy and the minima in Pxx at integral values of W at higher temperatures. Determining the value of the resistance at the position of the plateau for a number of traces, we find Oxy to be quantized to Pxy = h/(i/3 e 2) to an accuracy of 1%. Closer inspection of the data reveals additional structure in Pxx near ~ = 2/3, accompanied by a slight change in the slope of pxy" These features are more clearly seen in the data from sample 2, shown in Fig. 2. In this sample, due
11.L. Stfrmer et al. / Observation o f quantized Hall effect
432
FILLING FACTOR 2/3 112 1/3
t
12
t/4
3
_
689
I
I
G)
o,
I
• o
I
%•
,
'0 I0 4.15K
"g6
• 1.235tIOIICm-2(AT//• 1/3) o 3,06 X"IO"cm'2(AT~,-2/3
oo
%4 2
I
i o
o
b)
i
u
c •
•
uJ 0 8
,
,
,//
,
• 1.23~t I0" Cm'alAT b, • b' 3 | ~ 0.4
~ I /
..
I//
T,o, o,o, o ~
/
4
.
1
5
I00 150 MAGNETIC FIELD O (kG)
Figure i. Pxy and Ox x different temperatures Notice zero-offsets in shows sample geometry. factor is defined by ~
z
o• o•
0.2
o;
e ~Oe•
I
I
i
i
I
2
3
4
TEMPERATURF T{K)
K
. . . . . . . . . .
50
/t.GSK
o 3 ~ ~ iOllcm- 2( AT~,.2/3 |
o %
,,,,
200
o f sample 1 vs. B for between 0.48K and 4.15K. both graphs. Insert The Landau level filling = nh/eB.
FILLING ~ C T O R v
Figure 3. T dependence o f (a) the minimum in Pxx and (b) slope o f Px at v = 1/3 and v = 2/3 in sample 1 and sample ~ respectively. Notice two different scales in (a).
to its higher density, the V = 2/3 features are at similar B-fields as the V = 1/3 features of sample i. Once shifted to comparable B, the strength and T-dependence of the V =2/3 and v = 1/3 features are very similar. Apart from these prominent effects, sample 2 as well as sample i exhibit additional structures in the vicinity of v =3/2, most noticeable in p of sample 2. However, the strength of thes~Xdips was strongly current dependent and did not show a similar dependence on T. For filling factors V < 1/3, Pxx increases strongly (see Fig. i) with decreasing T though does not undergo a similarly strong v a r i a t i o ~ y Figure 3 illustrates the development of Px- and Pxx at two selected points. Figure 3b sho~s the slope of Pxv at the plateau position ~ = 1/3 of sample 1 an~ V = 2/3 of sample 2 normalized to the high temperature slope. The slope is clearly approaching zero for the lowest temperatures employed. Extrapolating the data on a logarithmic slope vs. inverse T plot to a slope of i, results in a temperature T o % 5K for the onset of the effect. Figure 3a comprises the temperature dependence of 0xx for both samples.
I(30
I~
MA~E~CFIELD
ZOO
Summarizing the experimental results we find that, similar to the development of the quantized plateaus in O and the vanishing of 0 x x . . x at integral values o~ the f1111ng factor v, quantized Hall plateaus of 0x = h/ie 2 with i = 1/3 and 2/3, and vanishing 0 y are developed at lower temperatures in the extreme quantum limit at fractional values of v = 1/3 and v = 2/3.
B(kG)
Figure 2. PXU and Pxx o f sample 2 vs. B for different temperatures between 0.5K and 4.5K. Notice zero-offsets in both graphs. B-axis is truncated for clarity.
Our experimental results cannot be understood within the framework of current explanations of the quantized Hall effect [5-10]. At this time no satisfactory theoretical model is available.
14..L. StOrmer et al. / Observation o f quantized Hall effect
690
TABLE I.
SAMPLE
i. Layer GaAs undoped
2. Layer (AlGa)As undoped
3. Layer (AlGa)As Si-doped
SAMPLE PARAMETERS
4. Layer GaAs Si-doped
A1 Concentr. [%]
Si Concentr [cm -3 ]
n(4.2K) [cm-2]
p(4.2K) [cm2/Vsec]
1
~Ip
500~
600~
200~
32
~6 1017
1.23 I0 II
90 000
2
~I~
180~
500~
o
30
~2 1018
3.06 i0 II
400 000
However, since the quantized Hall effect is generally accepted to be a consequence of the existence of gaps in the energy spectrum of the system and subsequent pinning of E F within the gap region, one might conclude on the appearance of additional gaps at fractional filling factors of ~ = 1/3 and v = 2/3. Their nature has to be dissimilar from the Landau-level and spin-level gaps since those appear exclusively at integral filling factors. Speculations about the origin of the gap concentrate on the formation of a new electron-liquid. In order to discern among various possible explanations, we performed preliminary measurements on the I-V characteristic of our samples in the field region where these features appear. We found that at ~ = 1/3 and ~ =2/3, P behaves strictly ohmic over a field range fromX~0 V/cm down to 10 -5 V/cm. For fields higher than i0 V/cm, a deviation from an ohmic behavior occurs, which we attribute to heating. However, no deviation from a linear I-V characteristic is found at the lowest voltages employed. Tentatively, this observation suggests the absence of pinning of the ground state, since pinning would result in a finite threshold voltage for electrical conduction.
REFERENCES i. Klitzing, D. v., Dorda, G., and Pepper, Phys. Rev. Lett. 45 (1980) 494. 2. Tsui, D. C., and Gossard, A. C., Appl. Phys. Lett. 37 (1981) 550. 3. Tsui, D. C., StSrmer, H. L., and Gossard, A. C., Phys. Rev. B25 (1982) 1405. 4. Paalanen, M. A., Tsui, D. C., and Gossard, A. C., Phys. Rev. B25 (1982) 5566. 5.
Prange, R. E., Phys. Rev. B23 (1981) 4802.
6. Aoki, H., and Ando, T., Solid State Comm. 38 (1981) 1079. 7.
Laughlin,
R. B., Phys. Rev. B25 (1982) 2185.
8.
Halperin,
B. I., Phys. Rev. B25 (1982) 2185.
9. Baraff, G., and Tsui, D. C., Phys. Rev. B24 (1981) 2274. i0. Kazarinov, R., and Luryi, S., to be published in Phys. Rev. B (1982).
ACKNOWLEDGMENT
ii. Tsui, D. C., St~rmer, H. L., Gossard, A. C., and Wiegmann, W., Phys Rev. 21 (1980) 1589.
We thank P. M. Tedrow for the loan of the He 3 refrigerator and K. Baldwin, T. M. Brennan, G. Kaminsky, and W. Wiegmann for technical assistance.
12. Tsui, D. C., St~rmer, H. L., and Gossard, A. C., Phys. Rev. Lett. 48 (1982) 1562. 13. Englert, Th., Tsui, D. C., and Gossard, A. C., Surf. Sci. 113 (1982) 295.
688
OBSERVATION OF QUANTIZED HALL EFFECT AND VANISHING RESISTANCE AT FRACTIONAL LANDAU LEVEL OCCUPATION H. L. St6rmer a*, D. C. Tsui b*, A. C. Gossard a, and J. C. M. Hwang a a) Bell Laboratories, Murray Hill, NJ 07974 b) Department of Electrical Engineering and Computer Science Princeton University, Princeton, NJ 08544 *) Visiting Scientists at the National Magnet Lab., Cambridge, MA
02138
Quantization of the Hall resistance Pxy and the approach of a zero-resistance state in Pxx are observed at fractional filling of Landau levels in the magneto-transport of the two-dimensional electrons in GaAs-(AIGa)As heterostructures. At the lowest temperatures (T~0.5K), the Hall resistance is quantized to values Pxy = h/(i/3 e 2) and Pxy = h/(2/3 e2). This observation, unexpected from current theoretical models for the quantized Hall effect, suggests the formation of a new electronic state at fractional level occupation.
Quantized Hall resistance and zero-resistance state of two-dimenslonal electrons are now well documented phenomena [1-4]. At low temperatures (T) and high magnetic fields (B) the Hall resistance Px- of these systems is quantized to values Oxy = ~/ie2, where h is Planck's constant, e is the electron charge and i is an integer. At regions where Px- assumes its quantized values, the d i a g o n a ~ resistivity 0xx extrapolates to Pxx = 0 as T approaches T = 0. The observation of the quantized Hall effect and the zero-resistance state is attributed to the existence of localized states within the gap between Landau levels and spin levels which pin the Fermi energy, EF, for finite regions of B or electron density n[5-10]. The high accuracy to which Px~ is quantized to Pxy = h/ie2 is a consequence el gauge invariance and the existence of these mobility gaps [7]. The integer i represents the number of completely filled Landau and spin levels below E F [8]. Accordingly, the quantized Hall resistance is limited to intesral values of i, with Pxy = h/e2 being its largest quantized value. We have been studying the magneto-transport of the two-dimensional electrons in GaAs-(AIGa)As heterojunctions in the extreme quantum limit Ill, 12] when all electrons occupy the lowestenergy, spin-polarized Landau level, i.e. the Landau level filling factor, defined by = nh/eB, is ~ < I. We observed that, in our high mobility samples, plateaus and minima are developed in Pxy and Oxx, respectively, at low temperatures at ~ = 1/3 and 2/3. At our lowest temperature (T~0.5K) the Hall plateaus are quantized to Pxy = h/ie2 with i = 1/3 and 2/3 at = 1/3 and 2/3, respectively. These results are unexpected from the current theoretical models for the quantized Hall effect. The low temperature (T ~ 0.SK) high magnetic field (B ~ 200kG) experiments were performed at the Francis Bitter National Magnet Lab on lowdensity, high mobility, modulation doped GaAs-(AIGa)As heterojunctions [12], prepared by molecular beam epitaxy. The parameters of the two samples discussed in this paper are given
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
in Table I. Standard Hall Bridges were used for the experiments. Figure i shows 0x v and px x of sample i as a function of B at fou~ different temperatures. The scale on top of Fig. i shows the Landau level filling factor ~. At integer values, ~ counts the number of occupied levels, with ~ = i representing the beginning of the extreme quantum limit where only the lower spin state of the lowest Landau level is occupied. Fractional values of ~ indicate fractional occupation. At all integral values of ~ the data reproduce well the characteristic features of the quantized Hall resistance and zero-resistance state. As observed earlier, the plateaus in 0x~ as well as the minima in Pxx become increasingly pronounced when lowering the temperatures. The development of the regions of odd ~, where E F is pinned between two spin levels of the same Landau level, lag behind regions of even ~, where E F if pinned between two Landau levels, due to a smaller spin splitting as compared to Landau splitting [13]. In the extreme quantum limit at high temperatures, Pxy follows to a good approximation a linear relation with the magnetic field, 0xy ~ B/he, as one would expect from a classical independent electron model. However, on lowering the temperature, a feature occurs at ~ = 1/3 which approaches a plateau at the lowest temperature employed. It is accompanied by a strong reduction in the diagonal resistance Oxx, evolving into a minimum at the position of the plateau. The development of these features is completely analogous to the development of the plateaus in Oxy and the minima in Pxx at integral values of W at higher temperatures. Determining the value of the resistance at the position of the plateau for a number of traces, we find Oxy to be quantized to Pxy = h/(i/3 e 2) to an accuracy of 1%. Closer inspection of the data reveals additional structure in Pxx near ~ = 2/3, accompanied by a slight change in the slope of pxy" These features are more clearly seen in the data from sample 2, shown in Fig. 2. In this sample, due
11.L. Stfrmer et al. / Observation o f quantized Hall effect
432
FILLING FACTOR 2/3 112 1/3
t
12
t/4
3
_
689
I
I
G)
o,
I
• o
I
%•
,
'0 I0 4.15K
"g6
• 1.235tIOIICm-2(AT//• 1/3) o 3,06 X"IO"cm'2(AT~,-2/3
oo
%4 2
I
i o
o
b)
i
u
c •
•
uJ 0 8
,
,
,//
,
• 1.23~t I0" Cm'alAT b, • b' 3 | ~ 0.4
~ I /
..
I//
T,o, o,o, o ~
/
4
.
1
5
I00 150 MAGNETIC FIELD O (kG)
Figure i. Pxy and Ox x different temperatures Notice zero-offsets in shows sample geometry. factor is defined by ~
z
o• o•
0.2
o;
e ~Oe•
I
I
i
i
I
2
3
4
TEMPERATURF T{K)
K
. . . . . . . . . .
50
/t.GSK
o 3 ~ ~ iOllcm- 2( AT~,.2/3 |
o %
,,,,
200
o f sample 1 vs. B for between 0.48K and 4.15K. both graphs. Insert The Landau level filling = nh/eB.
FILLING ~ C T O R v
Figure 3. T dependence o f (a) the minimum in Pxx and (b) slope o f Px at v = 1/3 and v = 2/3 in sample 1 and sample ~ respectively. Notice two different scales in (a).
to its higher density, the V = 2/3 features are at similar B-fields as the V = 1/3 features of sample i. Once shifted to comparable B, the strength and T-dependence of the V =2/3 and v = 1/3 features are very similar. Apart from these prominent effects, sample 2 as well as sample i exhibit additional structures in the vicinity of v =3/2, most noticeable in p of sample 2. However, the strength of thes~Xdips was strongly current dependent and did not show a similar dependence on T. For filling factors V < 1/3, Pxx increases strongly (see Fig. i) with decreasing T though does not undergo a similarly strong v a r i a t i o ~ y Figure 3 illustrates the development of Px- and Pxx at two selected points. Figure 3b sho~s the slope of Pxv at the plateau position ~ = 1/3 of sample 1 an~ V = 2/3 of sample 2 normalized to the high temperature slope. The slope is clearly approaching zero for the lowest temperatures employed. Extrapolating the data on a logarithmic slope vs. inverse T plot to a slope of i, results in a temperature T o % 5K for the onset of the effect. Figure 3a comprises the temperature dependence of 0xx for both samples.
I(30
I~
MA~E~CFIELD
ZOO
Summarizing the experimental results we find that, similar to the development of the quantized plateaus in O and the vanishing of 0 x x . . x at integral values o~ the f1111ng factor v, quantized Hall plateaus of 0x = h/ie 2 with i = 1/3 and 2/3, and vanishing 0 y are developed at lower temperatures in the extreme quantum limit at fractional values of v = 1/3 and v = 2/3.
B(kG)
Figure 2. PXU and Pxx o f sample 2 vs. B for different temperatures between 0.5K and 4.5K. Notice zero-offsets in both graphs. B-axis is truncated for clarity.
Our experimental results cannot be understood within the framework of current explanations of the quantized Hall effect [5-10]. At this time no satisfactory theoretical model is available.
14..L. StOrmer et al. / Observation o f quantized Hall effect
690
TABLE I.
SAMPLE
i. Layer GaAs undoped
2. Layer (AlGa)As undoped
3. Layer (AlGa)As Si-doped
SAMPLE PARAMETERS
4. Layer GaAs Si-doped
A1 Concentr. [%]
Si Concentr [cm -3 ]
n(4.2K) [cm-2]
p(4.2K) [cm2/Vsec]
1
~Ip
500~
600~
200~
32
~6 1017
1.23 I0 II
90 000
2
~I~
180~
500~
o
30
~2 1018
3.06 i0 II
400 000
However, since the quantized Hall effect is generally accepted to be a consequence of the existence of gaps in the energy spectrum of the system and subsequent pinning of E F within the gap region, one might conclude on the appearance of additional gaps at fractional filling factors of ~ = 1/3 and v = 2/3. Their nature has to be dissimilar from the Landau-level and spin-level gaps since those appear exclusively at integral filling factors. Speculations about the origin of the gap concentrate on the formation of a new electron-liquid. In order to discern among various possible explanations, we performed preliminary measurements on the I-V characteristic of our samples in the field region where these features appear. We found that at ~ = 1/3 and ~ =2/3, P behaves strictly ohmic over a field range fromX~0 V/cm down to 10 -5 V/cm. For fields higher than i0 V/cm, a deviation from an ohmic behavior occurs, which we attribute to heating. However, no deviation from a linear I-V characteristic is found at the lowest voltages employed. Tentatively, this observation suggests the absence of pinning of the ground state, since pinning would result in a finite threshold voltage for electrical conduction.
REFERENCES i. Klitzing, D. v., Dorda, G., and Pepper, Phys. Rev. Lett. 45 (1980) 494. 2. Tsui, D. C., and Gossard, A. C., Appl. Phys. Lett. 37 (1981) 550. 3. Tsui, D. C., StSrmer, H. L., and Gossard, A. C., Phys. Rev. B25 (1982) 1405. 4. Paalanen, M. A., Tsui, D. C., and Gossard, A. C., Phys. Rev. B25 (1982) 5566. 5.
Prange, R. E., Phys. Rev. B23 (1981) 4802.
6. Aoki, H., and Ando, T., Solid State Comm. 38 (1981) 1079. 7.
Laughlin,
R. B., Phys. Rev. B25 (1982) 2185.
8.
Halperin,
B. I., Phys. Rev. B25 (1982) 2185.
9. Baraff, G., and Tsui, D. C., Phys. Rev. B24 (1981) 2274. i0. Kazarinov, R., and Luryi, S., to be published in Phys. Rev. B (1982).
ACKNOWLEDGMENT
ii. Tsui, D. C., St~rmer, H. L., Gossard, A. C., and Wiegmann, W., Phys Rev. 21 (1980) 1589.
We thank P. M. Tedrow for the loan of the He 3 refrigerator and K. Baldwin, T. M. Brennan, G. Kaminsky, and W. Wiegmann for technical assistance.
12. Tsui, D. C., St~rmer, H. L., and Gossard, A. C., Phys. Rev. Lett. 48 (1982) 1562. 13. Englert, Th., Tsui, D. C., and Gossard, A. C., Surf. Sci. 113 (1982) 295.