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Fractional quantum hall effect in tilted magnetic fields
252
Surface Science 196 (1988) 252-256 North-Holland, Amsterdam
FRACTIONAL Q U A N T U M H A L L EFFECT IN T I L T E D 1VL~GNETIC FIELlrlS D.A. SYPHERS* Physics Department, Bowdoin COllege, Brunswick, ME 04011, USA
and J.E. FURNEAUX* Naval Research Laboratory, Code 6874, Washington, DC 203 75, USA
Received 2 June 1987; accepted for publication 10 August 1987
The effects of a parallel magnetic field on the fractional quantum Hall effect state of a quasi-twodimensional syst.zm with finite layer thickness are investigated. The presence of this parallel field decreases the layer thickness and changes the effective mass. It is shown that this decreased layer thickness causes an increasein the experimentallydetermined activation energy.The effect of changes in the effective mass is discussed. Data from high/1 (up to 400 mr/V-s) GaAs/AIGaAs samples tilted in-situ are presented, and compared to the theory.
The most complete published measurements of the activation energies [ 1 ] J/2, of the fractional quantum Hall effects, FQHE, obtained assuming the diagonal resistivity, p , , ~ exp( - d / 2 T) at the appropriate p.,..,,m i n i m u m , fall a great deal below the theoretical predictions [ 2,3 ] for an ideal two-dimensional electron gas, 2DEG. Corrections due to the finite thickness of the quasi-2D layer bring the theoretical predictions closer to the experimental results [4,5 ], although there is still substantial disagreement between the two sets of values. Conventional w i s d o m would attribute this disparity to the finite mobility kt, of the samples used. Even though the quality of the samples was quite high, their/l is still far below the theoretical limit. The effect of tilted magnetic fields on the F Q H E state should provide a critical test of our understanding of this state. For quasi-2D systems the L a n d a u level fillin~ factor is determined by the perpendicu19,, c o m p o n e n t of the magnetic field. To first order, a tilted magnetic field would be expected to have no effect on the FQHE state of an ideal 2DEG. The existence of a parallel magnetic field, B , would be expected to have an effect, however, on systems like the GaAs/A1GaAs heterostructures where the half-width (za,) of the quasi-2D layer is about 70 ,/k [ 6 ], which * visiting scientisl Francis Bitter Nationa! Magnet Laboratory, MIT. 003%6028/88/$ 03.50 © Elsevier Scienc,- Publishers B.V. (North-Holland Physics Publishing Divis{on)
D.A. S),phers, .I.E. Furneaux/FQHE in tKted magnetic fields
253
is comparable to the magnetic length l = (c~/eB)':2-81 A at 10 T. As this parallel field increases, Zav decreases [ 7 ]. For the highest # 2DEG systems the dominant effect should be a change in these finite layer thickness corrections, resulting in an increase in the experimentally determined 2 as B, is increased. For systems well below the theoretical limits of # we expect to see competing effects. Because increasing B, leads to a decreasing Zav, there will be increased scattering due to ionized impurities and surface roughness. Therefore the measured d should also decrease [ 1 ]. In order to obtain more quantitative results on the effect of B, we follow Stern [ 7 ]. In the ~bsence of B~, the carriers move in a potential well V(z) near the interface, which is a combination of electrostatic, image, interface grading and exchange-correlation potentials [6]. In the presence of B, the Hamiltonian for the system becomes /'/2o = [Px +ezB (tan 8)/c]2+ [py +eBx/c]2 + V(z),
(1)
where B= B~ is the perpendicular magnetic field value, and 0 the angle of tilt measured from the perpendicular, for the gauge .4= (Bztan 0, Bx, 0). This reduces to the Hamiltonian in the Landau gauge, HLt., for a 2D system in a perpendicular magnetic field with two addition 0 dependent terms [ 7,8 ] 2 2 H2D=HLL +htoc(zkx) tan O+ 1zmttocz tan20,
(2)
where t.oc= eBImtc is the cyclotron frequency and m~ the effective mass. For these studies the important term is the z 2 term. This term affects E, the subband energies, zav, and mt [7,8]. These changes in zav and in rn, both may have important effects on A. We use perturbation theory to arrive at the first order expressions for zav and mt as a function of 8
z,,, ~z~,,o - m , ~ . o [ ( z ~ ) - (z 2 ) ~z) ] tan:8, [
mt,~,mt.o
(3)
iz..ol 2 ]-,,2 1-2m~'°°gcz ,>o ~ E,,-Eo tan20 '
(4)
where ¢O~.o,Za~.O,and mt.o are ~o~,za~, and m, for 8 = 0 . The change in m~ in eq. (4) is due to the change in mx resulting from B , coupled with the fact that m:, is unchanged. The effects of the decrease in Za~ have been discussed above. The effect of an increased as well as anisotropic m, on the FQHE state is not as clear. Work on the FQHE A's in Si MOSFETs [9] could be interpreted as indicating that ae increased ~,~ m~It~,c n h c ~ , ~ t ; n n
~f th~ I~AHI~ o~nc
o~ci~r
C ~ n t h e nt|'~or h ~ n d
an [nPren~ed
m, would decrease ~ . For a system whose scattering time z is unrelated to m, this increased m, would decrease the value of ~ r . Since ~ z is directly related to ff.e ratio P.JP.,.x, an increased m, would result in an observed increase in Px.,-experimentally. Thus, it is not clear what effect an increased mt would have on the thermal activation of the p,..,, minima. However, the Si MOSFET argument is weaker
D.A. Syphers, J.E. Furneaux/FQHE in tilted magnetic fields
254 Table 1 Tilt angle 0 (dog)
34 40 49.5 53.5
Za~
-_.,(0=0)
0.895 0.84 0.74 0.69
Predicted increase in measured A due to finite size corrections
Observed increases in A
Percentage correction (%)
Absolute correction ( K )
Percentage change (%)
Absolute change ( K )
5 Il i3 !6
1.0 2.2 2.6 3.2
30 40 50 47
0.2 0.3 0.4 0.4
since it deals with an entirely different system. We therefore would exvect large increases in mt alone to decrease the experimentally determined A. For large values ¢fB, this effect competes with the increases in A due to decreasing zav o f the 2DEG. Resistivity data were taken on samples tilted in situ, in order to measure 3 of FQHE states as a function o f 0. Detailed m e a s u r e m e n t s on the 2/3 state were taken on a sample with a p of 30 m2/V-s at a perpendicular field of 15.3 T. T h e measured d was found to increase with increasing 0 for small 0. For larger 0, A reached a m a x i m u m at about 50 °, and showed no further change at 53.5 ° within experimental error. T h e results on this sample and the predictions from the theory [6] are summarized in table 1. The discrepancy between theory and experiment can probably be attributed to the fact that samples with this g have an onset magnetic field for observing the FQHE, whereas finite thickness corrections do not [ 5 ]. t h e saturation at the highest tilt angles is probably due to a decrease in oJ~r. We are investigating whether the source of this decrease is primarily an increased m, or a decreased z. At this writing we have some partial, but facinating, data on a sample with a .u of 400 m2/V.s. A typical trace of the resistivity data at 0.5 K is shown in fig. 1. Note that there is fractional structure at about 2 T, indicating an onset field below 1.7 T. The p o f this sample approaches the theoretical limit for a heterostructure, and the onset field is very close to the theoretical onset at zero B. Our present data are limited to different tilt angles at a constant t e m p e r a t u r e of 0.5 K. The inset of fig. 1 shows the effect of tilt en the 2/3 m i n i m u m . At 0 = 13.4 ° the 2/3 m i n i m u m decreased to 64% and the 3,5 m i n i m u m decreased to about 67% of their 0 = 0 values. At 0 - 2 2 ° the 2/3 m i n i m u m decreased to 23% of its zero tilt value. If ":,'e assume p,, is exponentially activated with no 0 - d e p e n d e m prefactor (a reasonable assumption for a sample of this quality), we can estimate the za, value o f A, Ao, ~ sing me ratle of °~'" " u m c , c m O, mc va~u~ u~ "a,' , , u , , cq. ( ~ X "~"~ * ~ finite size corrections from ref. [5]. For the 2/3 state, the t3.4 ° tilt gives A~,= 15 K, and the 22 ° till gives Ao= 25 K. For the 3/5 state, ~he 13.4 ~ till gives Au ~ 12 Ko These derived v~!ues compare ¢~a,'u~ . . . . auly '- with the theoretical value of Ao= 19 K for the 2/3 state [2-41. For the 3/5 state [2] do is somewhat higher ~han the 5 K pre-
D.A. Syphers, .I.E. Ft~rneaux/FQHE in tilted magnetic fields
,!
t~l
3. . . .
^
0 025
• fo
140
,],
0
255
5
,
10
,
,
B (T
Fig. !. Resistivity in units o f h/e 2 as a function o f magnetic field for the 400 m2/V.s sample at T = O. 5 K. The upper curve is p , . and the lower curve is p , , expanded by a factor o f 15. Inset: p, • near the F Q H E 2/3 state expanded at constant B: to show the effect of tipped fields. The solid line is 0 = 0 °; the long dashed line is 0 = 13.4 °; and the short dashed line is 0 = 22 =.
dicted theoretically [ 2 ], but the ratio Of Ao for the 3/5 state to that of the 2/3 state is consistent with previous experimental results [ l ]. Too much reliance should not be placed on these comparisons with previous work because there is probably a factor of 2 uncertainty in these estimates. These results are intriguing and show the necessity for continuing this study on these high quality samples. We would like to acknowledge the support of the National Science Foundation, the Office of Naval Research, and the staff of the Francis Bitter National Magnet Laboratory. The sample used for detailed studies was provided by S. Palmateer of GE. The high/z samples were provided by W. Wang of IBM and Columbia University with the processing done by W. Tseng of NRL. Their contributions are gratefully acknowledged.
References ........... ~+,,~,~, ~+L. S~i3~m~r, L~.L. ~sm, J.C.M. ttwang, A. ~ " Weimann, Surface Sci. 170 (1986) 129, G.S. Boebinger, PhD Thesis, MIT (1986). [2] B.I. Ha|perin, Surface Sci 170 (1986) 1 t5.
~" Tu and G.
256
D.A. Syphers, ,I.E. Furneaux/FQHE in tilted magnetic fields
[ 3 ] ED.M. Haldane and E.H. Rezayi, Phys. Rev. Letters 54 (1985) 237; S. M. Girvin, A.H. Mac[~anald and P.M. Platzman, Phys. Rev. Letters 54 (1985) 581. [ 4 | A.H. MacDonald and G.C. Aers, Phys. Rev. B29 (1984 ) 5976; D. Yoshioka, ,i. Phys. Sac. Japan 55 (1986) 85. [5] F.C. Zhang and S. Das Sarma, Phys. Rev. B33 (1986) 2903. [6] F. Stern and S. Das Sarma, Phys. Rev. B30 (1984) 840. [7] F. Stern, Phys. Rev. Letters 21 (1968) 1687. [8] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [ 9 ] I.V. Kukushkin and V.B. Timofeev, Surface Sci. ! 70 (1986) 148; .I.E. Furneaux, D.A. Syphers, J.S. Brooks, G.M. Schmiedeshoff, R.G. Wheele¢ and P.J. Stiles, Surface Sci. 170 (1986) 154; and unpublished data on MOSFETs showing FQHE features at mobilities as low as 1.5 m:/V.s.
Surface Science 196 (1988) 252-256 North-Holland, Amsterdam
FRACTIONAL Q U A N T U M H A L L EFFECT IN T I L T E D 1VL~GNETIC FIELlrlS D.A. SYPHERS* Physics Department, Bowdoin COllege, Brunswick, ME 04011, USA
and J.E. FURNEAUX* Naval Research Laboratory, Code 6874, Washington, DC 203 75, USA
Received 2 June 1987; accepted for publication 10 August 1987
The effects of a parallel magnetic field on the fractional quantum Hall effect state of a quasi-twodimensional syst.zm with finite layer thickness are investigated. The presence of this parallel field decreases the layer thickness and changes the effective mass. It is shown that this decreased layer thickness causes an increasein the experimentallydetermined activation energy.The effect of changes in the effective mass is discussed. Data from high/1 (up to 400 mr/V-s) GaAs/AIGaAs samples tilted in-situ are presented, and compared to the theory.
The most complete published measurements of the activation energies [ 1 ] J/2, of the fractional quantum Hall effects, FQHE, obtained assuming the diagonal resistivity, p , , ~ exp( - d / 2 T) at the appropriate p.,..,,m i n i m u m , fall a great deal below the theoretical predictions [ 2,3 ] for an ideal two-dimensional electron gas, 2DEG. Corrections due to the finite thickness of the quasi-2D layer bring the theoretical predictions closer to the experimental results [4,5 ], although there is still substantial disagreement between the two sets of values. Conventional w i s d o m would attribute this disparity to the finite mobility kt, of the samples used. Even though the quality of the samples was quite high, their/l is still far below the theoretical limit. The effect of tilted magnetic fields on the F Q H E state should provide a critical test of our understanding of this state. For quasi-2D systems the L a n d a u level fillin~ factor is determined by the perpendicu19,, c o m p o n e n t of the magnetic field. To first order, a tilted magnetic field would be expected to have no effect on the FQHE state of an ideal 2DEG. The existence of a parallel magnetic field, B , would be expected to have an effect, however, on systems like the GaAs/A1GaAs heterostructures where the half-width (za,) of the quasi-2D layer is about 70 ,/k [ 6 ], which * visiting scientisl Francis Bitter Nationa! Magnet Laboratory, MIT. 003%6028/88/$ 03.50 © Elsevier Scienc,- Publishers B.V. (North-Holland Physics Publishing Divis{on)
D.A. S),phers, .I.E. Furneaux/FQHE in tKted magnetic fields
253
is comparable to the magnetic length l = (c~/eB)':2-81 A at 10 T. As this parallel field increases, Zav decreases [ 7 ]. For the highest # 2DEG systems the dominant effect should be a change in these finite layer thickness corrections, resulting in an increase in the experimentally determined 2 as B, is increased. For systems well below the theoretical limits of # we expect to see competing effects. Because increasing B, leads to a decreasing Zav, there will be increased scattering due to ionized impurities and surface roughness. Therefore the measured d should also decrease [ 1 ]. In order to obtain more quantitative results on the effect of B, we follow Stern [ 7 ]. In the ~bsence of B~, the carriers move in a potential well V(z) near the interface, which is a combination of electrostatic, image, interface grading and exchange-correlation potentials [6]. In the presence of B, the Hamiltonian for the system becomes /'/2o = [Px +ezB (tan 8)/c]2+ [py +eBx/c]2 + V(z),
(1)
where B= B~ is the perpendicular magnetic field value, and 0 the angle of tilt measured from the perpendicular, for the gauge .4= (Bztan 0, Bx, 0). This reduces to the Hamiltonian in the Landau gauge, HLt., for a 2D system in a perpendicular magnetic field with two addition 0 dependent terms [ 7,8 ] 2 2 H2D=HLL +htoc(zkx) tan O+ 1zmttocz tan20,
(2)
where t.oc= eBImtc is the cyclotron frequency and m~ the effective mass. For these studies the important term is the z 2 term. This term affects E, the subband energies, zav, and mt [7,8]. These changes in zav and in rn, both may have important effects on A. We use perturbation theory to arrive at the first order expressions for zav and mt as a function of 8
z,,, ~z~,,o - m , ~ . o [ ( z ~ ) - (z 2 ) ~z) ] tan:8, [
mt,~,mt.o
(3)
iz..ol 2 ]-,,2 1-2m~'°°gcz ,>o ~ E,,-Eo tan20 '
(4)
where ¢O~.o,Za~.O,and mt.o are ~o~,za~, and m, for 8 = 0 . The change in m~ in eq. (4) is due to the change in mx resulting from B , coupled with the fact that m:, is unchanged. The effects of the decrease in Za~ have been discussed above. The effect of an increased as well as anisotropic m, on the FQHE state is not as clear. Work on the FQHE A's in Si MOSFETs [9] could be interpreted as indicating that ae increased ~,~ m~It~,c n h c ~ , ~ t ; n n
~f th~ I~AHI~ o~nc
o~ci~r
C ~ n t h e nt|'~or h ~ n d
an [nPren~ed
m, would decrease ~ . For a system whose scattering time z is unrelated to m, this increased m, would decrease the value of ~ r . Since ~ z is directly related to ff.e ratio P.JP.,.x, an increased m, would result in an observed increase in Px.,-experimentally. Thus, it is not clear what effect an increased mt would have on the thermal activation of the p,..,, minima. However, the Si MOSFET argument is weaker
D.A. Syphers, J.E. Furneaux/FQHE in tilted magnetic fields
254 Table 1 Tilt angle 0 (dog)
34 40 49.5 53.5
Za~
-_.,(0=0)
0.895 0.84 0.74 0.69
Predicted increase in measured A due to finite size corrections
Observed increases in A
Percentage correction (%)
Absolute correction ( K )
Percentage change (%)
Absolute change ( K )
5 Il i3 !6
1.0 2.2 2.6 3.2
30 40 50 47
0.2 0.3 0.4 0.4
since it deals with an entirely different system. We therefore would exvect large increases in mt alone to decrease the experimentally determined A. For large values ¢fB, this effect competes with the increases in A due to decreasing zav o f the 2DEG. Resistivity data were taken on samples tilted in situ, in order to measure 3 of FQHE states as a function o f 0. Detailed m e a s u r e m e n t s on the 2/3 state were taken on a sample with a p of 30 m2/V-s at a perpendicular field of 15.3 T. T h e measured d was found to increase with increasing 0 for small 0. For larger 0, A reached a m a x i m u m at about 50 °, and showed no further change at 53.5 ° within experimental error. T h e results on this sample and the predictions from the theory [6] are summarized in table 1. The discrepancy between theory and experiment can probably be attributed to the fact that samples with this g have an onset magnetic field for observing the FQHE, whereas finite thickness corrections do not [ 5 ]. t h e saturation at the highest tilt angles is probably due to a decrease in oJ~r. We are investigating whether the source of this decrease is primarily an increased m, or a decreased z. At this writing we have some partial, but facinating, data on a sample with a .u of 400 m2/V.s. A typical trace of the resistivity data at 0.5 K is shown in fig. 1. Note that there is fractional structure at about 2 T, indicating an onset field below 1.7 T. The p o f this sample approaches the theoretical limit for a heterostructure, and the onset field is very close to the theoretical onset at zero B. Our present data are limited to different tilt angles at a constant t e m p e r a t u r e of 0.5 K. The inset of fig. 1 shows the effect of tilt en the 2/3 m i n i m u m . At 0 = 13.4 ° the 2/3 m i n i m u m decreased to 64% and the 3,5 m i n i m u m decreased to about 67% of their 0 = 0 values. At 0 - 2 2 ° the 2/3 m i n i m u m decreased to 23% of its zero tilt value. If ":,'e assume p,, is exponentially activated with no 0 - d e p e n d e m prefactor (a reasonable assumption for a sample of this quality), we can estimate the za, value o f A, Ao, ~ sing me ratle of °~'" " u m c , c m O, mc va~u~ u~ "a,' , , u , , cq. ( ~ X "~"~ * ~ finite size corrections from ref. [5]. For the 2/3 state, the t3.4 ° tilt gives A~,= 15 K, and the 22 ° till gives Ao= 25 K. For the 3/5 state, ~he 13.4 ~ till gives Au ~ 12 Ko These derived v~!ues compare ¢~a,'u~ . . . . auly '- with the theoretical value of Ao= 19 K for the 2/3 state [2-41. For the 3/5 state [2] do is somewhat higher ~han the 5 K pre-
D.A. Syphers, .I.E. Ft~rneaux/FQHE in tilted magnetic fields
,!
t~l
3. . . .
^
0 025
• fo
140
,],
0
255
5
,
10
,
,
B (T
Fig. !. Resistivity in units o f h/e 2 as a function o f magnetic field for the 400 m2/V.s sample at T = O. 5 K. The upper curve is p , . and the lower curve is p , , expanded by a factor o f 15. Inset: p, • near the F Q H E 2/3 state expanded at constant B: to show the effect of tipped fields. The solid line is 0 = 0 °; the long dashed line is 0 = 13.4 °; and the short dashed line is 0 = 22 =.
dicted theoretically [ 2 ], but the ratio Of Ao for the 3/5 state to that of the 2/3 state is consistent with previous experimental results [ l ]. Too much reliance should not be placed on these comparisons with previous work because there is probably a factor of 2 uncertainty in these estimates. These results are intriguing and show the necessity for continuing this study on these high quality samples. We would like to acknowledge the support of the National Science Foundation, the Office of Naval Research, and the staff of the Francis Bitter National Magnet Laboratory. The sample used for detailed studies was provided by S. Palmateer of GE. The high/z samples were provided by W. Wang of IBM and Columbia University with the processing done by W. Tseng of NRL. Their contributions are gratefully acknowledged.
References ........... ~+,,~,~, ~+L. S~i3~m~r, L~.L. ~sm, J.C.M. ttwang, A. ~ " Weimann, Surface Sci. 170 (1986) 129, G.S. Boebinger, PhD Thesis, MIT (1986). [2] B.I. Ha|perin, Surface Sci 170 (1986) 1 t5.
~" Tu and G.
256
D.A. Syphers, ,I.E. Furneaux/FQHE in tilted magnetic fields
[ 3 ] ED.M. Haldane and E.H. Rezayi, Phys. Rev. Letters 54 (1985) 237; S. M. Girvin, A.H. Mac[~anald and P.M. Platzman, Phys. Rev. Letters 54 (1985) 581. [ 4 | A.H. MacDonald and G.C. Aers, Phys. Rev. B29 (1984 ) 5976; D. Yoshioka, ,i. Phys. Sac. Japan 55 (1986) 85. [5] F.C. Zhang and S. Das Sarma, Phys. Rev. B33 (1986) 2903. [6] F. Stern and S. Das Sarma, Phys. Rev. B30 (1984) 840. [7] F. Stern, Phys. Rev. Letters 21 (1968) 1687. [8] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [ 9 ] I.V. Kukushkin and V.B. Timofeev, Surface Sci. ! 70 (1986) 148; .I.E. Furneaux, D.A. Syphers, J.S. Brooks, G.M. Schmiedeshoff, R.G. Wheele¢ and P.J. Stiles, Surface Sci. 170 (1986) 154; and unpublished data on MOSFETs showing FQHE features at mobilities as low as 1.5 m:/V.s.