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Oscillations in the low-field diffusion thermopower of a 2D electron gas
ii,
ii
ii
surface science ELSEVIER
Surface Science 361/362 (1996) 529-532
Oscillations in the low-field diffusion thermopower of a 2D electron gas R. Fletcher a.,, P.T. Coleridge b, y. Feng b =PhysicsDepartment, Queen's University, Kingston, Ontario K7L 3N6, Canada bMicrostructaral Sciences, National Research Council, Ottaw.a, Ontario KI A OR6, Canada Received 5 June 1995; accepted for publication 10 August 1995
Alamact Oscillations in the diffusion thermopower of a GaAs/Gat_xAl~,s heterojunction have been dearly observed for the first time. Phonon drag has been strongly reduced by growing the 2DEG on a heavily-doped conducting substrate and by working below 1 K. The low-field oscillations in the Nemst-Etlingshausen coefficient have been investigated and are found to show the unique thermal damping factor of diffusion.
Keywords: Thermomagneti~ Thermopower
1. Introduction
Quantum oscillations in the diffusion thermopower of 2DEGs at high magnetic fields were theoretically investigated soon after the discovery of the quantum Hall effect. A number of experimental groups published results which were interpreted in terms of these theories and there appeared to be reasonable agreement, at least initially. Ref. [ 1] has a comprehensive review of the early work on the thermopower of 2DEC-s, and many references may be found therein. A theoretical study [2] of the low-field diffusion case also used then-current experimental data to compare with predictions. However, it was later shown both experimentally and theoretically that phonon drag overwhelms the diffusion component over the range of temperatures where the experiments were made, usually * Corresponding author. Faya + 1 613 54564630.
the 4He range. Meanwhile the mobility of samples had steadily increased and there remained some uncertainty as to the extent of how far the mobility of the 2DEG affects phonon drag. It seemed possible that the early low-mobility samples might have a much smaller phonon drag component than later high-mobility samples, but recent work [3] has shown that phonon drag is essentially independent of mobility down to at least 0.1 m2/V-s. There remains little doubt that the early experiments actually observed oscillations in phonon drag, and not diffusion. The aim of the present work is to reduce phonon drag to a level where diffusion dominates and to look for the diffusion oscillations. Under high-field conditions, diffusion and phonon drag oscillations are expected to be rather similar in form and difficult to separate reliably. High-field conditions correspond to h ~ >>F, where ms is the cyclotron frequency and F is the halfwidth of the Landau levels. In the opposite extreme
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved PI1 S0039-6028 (96) 00462-1
R. Fletcher et aL /Surface Science 361/362 (1996) .529-532
530
of low-field conditions, the amplitude of diffusion oscillations can be calculated with no unknowns (except for F, which can be obtained from an analysis of resistivity oscillations) and has a unique thermal damping factor. This is the main reason we have chosen to work at low fields. A full account of the low-field theory and experimental details will be published elsewhere [4].
the thermal energy ksT involved in each scattering event. What happens with Sy~ at low fields is unclear, but the fact that the monotonic phonon drag contribution is quite small and in fact is zero as a first approximation [6], is perhaps an advantage.
3. Results and analysis 2. Theory The thermopower S is defined by E = S V T, where E is the electric field developed in the sample under the applied temperature gradient V T. If we take the magnetic field B to be perpendicular to the plane of the 2DEG and along $, and the temperature gradient to be along ~, then the transverse thermopower (oRen called the NernstEttingshausen coefficient) is defined by Syx= Ey/V T~. The oscillatory diffusion component gy~ for elastic scattering of the electrons at low fields is found to be given by [2,4] ~y=
4c°o~t
×D'(X) exp ( -
sin
(1)
Here, ev is the Fermi energy and zt and Zq are the transport and quantum lifetimes, respectively, the latter being related to F by F - - h/2~q. The quantity D'(X) is the thermal damping factor and is the derivative of D(X) with respect to X, where D(X) = X/sinh X and X f 2x2knT/hmo. The oscillations in the longitudinal component S~= are predicted to be very similar but smaller by a factor 1/)xocxt. Note that there is a phase difference of n/2 between the oscillations in p= and Sy~ predicted by Eq. (1). We also expect oscillations to occur in the phonon drag thermopower, but no theory yet exists for the low-field case. Experimentally it is known [5] that phonon drag oscillations in S ~ behave roughly like those in the resistivity and increase monotonically with field, though the amplitude is much more rapidly reduced at low fields, i.e. the effective impurity damping factor is much stronger. This presumably arises because of
The sample had an electron density of ~ 1.9 x 10is m - 2 and a mobility of ~40 m2/V.s. By growing the sample on a strongly doped substrate the lattice thermal conductivity was reduced; this in turn decreases the flux of phonon momentuna in the applied temperature gradient so there was less phonon momentum available to transfer to the electrons as compared to the undoped case, thereby reducing phonon drag. Substrate doping has no effect on the diffusion component which at low temperatures depends only on the scattering of electrons by impurities close to the 2DEG. To further accentuate diffusion it is an advantage to reduce the temperature below 1 K where phonon drag decreases much more rapidly with temperature than diffusion. Fig. 1 gives experimental data on ~yx at a temperature of 0.55 IC The two experimental traces are from two different pairs of transverse probes on the sample, and the upper data are continued up to 8 T. These traces are derived from data taken with + B by eliminating the part even in field, which is about 20% of the total Near 1 T there is a dear change in the appearance of the oscillations. At high fields the oscillations are found to resemble the derivative of p ~ with respect to field. This is the form expected for high-field diffusion oscillations [7] but phonon drag seems to have a similar form, so the origin of these oscillations is not yet unambiguous. The lowest trace is calculated according to Eq. (1) using parameters appropriate to the sample and m*--0.07m, but is multiplied by a factor of two. There is substantial agreement between experiment and theory, especially in the appearance of a maximum in the amplitude near 0.5 T. This maximum is basically due to if(X) in Eq. (1), though modified by the term in zt, and is believed to be unique to diffusion thermopower. It has
531
R. Fletcher et aL /Surface Science 361/362 (1996) 529-532
2ot{ i
i
•
T=O.55K
.-.>~
::.
looo
o •
o
',
v O9
,L
-20
~m "~ ~©.=2 Eo -40
" vvvvvVVVVVVVVVVVVVVV o,o°,.0,=
~ ~'~'Q~I
.mL
o 0 c-
--60
,
0
I
,
2
I
(I/T)
Fig. 1. The osdllatiom in St. as a function of inverse magnetic field I/B. The upper two curves are experimental data from different voltage probes, the lowest curve is calculated using parameters appropriate to the present sample but the magnitude is multipfied by a factor of two. The bottom two traces axe offset by 20 and 40 #V K -1, respectively. (The calcuhted curve contains the monotonic component but this is essentially negligible in this range of field).
previously been seen in bulk metal systems (see, for example, Ref. [8]). In comparison, the amplitude of ~ = monotonically increases with B in this range. Observation of the predicted phase difference between the oscillations in p = and Syx would provide a further test. However, as can be seen in Fig. 1, the experimental data on ff,~, from two independent pairs of sample probes show relative phase shifts. This might be due to inhomogeneities in the sample. As a final test, we use both/~= and ffyxto extract Zq, which should be the same for both coefficients. The amplitude data for g,~ are plotted in Fig. 2 in the form log[~x(1 +o92~'~)/ogc~tD'(X)]a s a function of lIB over the temperature range 0.35-0.72 K for two pairs of sample probes. The scaled data follow a straight line within experimental error, and the fine through the data has a slope corresponding to Xq--1.8 ps which is identical to that obtained from a similar analysis [9] of p = oscillations. If phonon drag was important, and ff the phonon drag oscillations in S,x behave similarly to those
o
•
v
• 0.55K
=
• 0.45K " 0.39K * 0.35K ' ,3
A
,
o
4 1/B
~r~
100
10
0
' 1
' 2
0.72K
uqrl
'
4
' 5
6
B-~ (T-~) Fig. l The amplitude of ~rx multiplied by (1 + ~/(c%ztD'(X)) as a function of lIB. The open and closed symbols are for two independent pairs of sample probes, respectively. Only the part odd in field b included in , ~ .
in S=, then we would expect a much smaller effective Zq [5]. The intercept at l / B = 0 is 2 . 3 + I . 0 m V K -z, which is about a factor of two larger than the expected value Of 41tie~e= 1.08 mV K - 1; this factor of two was also indicated from the comparison of theory with experiment in Fig. 1.
4. Conclusions The observed thermopower oscillations exhibit the expected thermal damping factor for diffusion. They also yield the same impurity damping term as the resistivity oscillations, as is expected because both are determined by the same impurity scattering of electrons. In contrast, previous experiments have shown that phonon drag oscillations behave as if they have a much stronger effective impurity damping term. The only significant discrepancy with calculation is that the absolute amplitude of the thermoelectric oscillations is about a factor of two higher than predicted.
532
R. Fletcher et al./ Surface Science 361/362 (1996) 529-532
Acknowledgement This work was partially supported by the Natural Sciences and Engineering Research Council of Canada.
References 1"1] B.L. Gallagher and P.N. Butcher, in: Handbook on Se~conductors, VoL 1, Ed. P.T. Landsberg (Elsevier, Amsterdam~ 1992) p. 817.
[2] H. Havlov~ and L. Smr~ka, Phys. Status Solidi B 137 (1988) 331. [3] R. Fletcher, JJ. Harris and C.T. Foxon, M. Tsaousidou and P.N. Butcher, Phys. Rev. B 50 (1994) 14991. [4] 1~ Fletcher, P.T. Coleridge and Y. Feng, Phys. Rev. B 52 (1995) 2823. 1"5-1 M. D'Iorio, R. Stoner and R. Fletcher, Solid State Commun. 65 (1988) 697. 1"6"] X. Zjanni; P.N. Butcher and MJ. Kearney, Phys. Rev. B 49 (1994) 7520. [7] tL Oji, J. Phys. C: Solid State Phy~ 17 (1984) 3059. 1"8] I~ Fletcher, Phys. Rev. 28 (1983) 1721. Phys. Rev. 28 (1983) 6670. 1"9] P.T. Coleridge, R. Stoner and R. Fletcher, Phys. Rev. B 39 (1989) 1120.
ii
ii
surface science ELSEVIER
Surface Science 361/362 (1996) 529-532
Oscillations in the low-field diffusion thermopower of a 2D electron gas R. Fletcher a.,, P.T. Coleridge b, y. Feng b =PhysicsDepartment, Queen's University, Kingston, Ontario K7L 3N6, Canada bMicrostructaral Sciences, National Research Council, Ottaw.a, Ontario KI A OR6, Canada Received 5 June 1995; accepted for publication 10 August 1995
Alamact Oscillations in the diffusion thermopower of a GaAs/Gat_xAl~,s heterojunction have been dearly observed for the first time. Phonon drag has been strongly reduced by growing the 2DEG on a heavily-doped conducting substrate and by working below 1 K. The low-field oscillations in the Nemst-Etlingshausen coefficient have been investigated and are found to show the unique thermal damping factor of diffusion.
Keywords: Thermomagneti~ Thermopower
1. Introduction
Quantum oscillations in the diffusion thermopower of 2DEGs at high magnetic fields were theoretically investigated soon after the discovery of the quantum Hall effect. A number of experimental groups published results which were interpreted in terms of these theories and there appeared to be reasonable agreement, at least initially. Ref. [ 1] has a comprehensive review of the early work on the thermopower of 2DEC-s, and many references may be found therein. A theoretical study [2] of the low-field diffusion case also used then-current experimental data to compare with predictions. However, it was later shown both experimentally and theoretically that phonon drag overwhelms the diffusion component over the range of temperatures where the experiments were made, usually * Corresponding author. Faya + 1 613 54564630.
the 4He range. Meanwhile the mobility of samples had steadily increased and there remained some uncertainty as to the extent of how far the mobility of the 2DEG affects phonon drag. It seemed possible that the early low-mobility samples might have a much smaller phonon drag component than later high-mobility samples, but recent work [3] has shown that phonon drag is essentially independent of mobility down to at least 0.1 m2/V-s. There remains little doubt that the early experiments actually observed oscillations in phonon drag, and not diffusion. The aim of the present work is to reduce phonon drag to a level where diffusion dominates and to look for the diffusion oscillations. Under high-field conditions, diffusion and phonon drag oscillations are expected to be rather similar in form and difficult to separate reliably. High-field conditions correspond to h ~ >>F, where ms is the cyclotron frequency and F is the halfwidth of the Landau levels. In the opposite extreme
0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved PI1 S0039-6028 (96) 00462-1
R. Fletcher et aL /Surface Science 361/362 (1996) .529-532
530
of low-field conditions, the amplitude of diffusion oscillations can be calculated with no unknowns (except for F, which can be obtained from an analysis of resistivity oscillations) and has a unique thermal damping factor. This is the main reason we have chosen to work at low fields. A full account of the low-field theory and experimental details will be published elsewhere [4].
the thermal energy ksT involved in each scattering event. What happens with Sy~ at low fields is unclear, but the fact that the monotonic phonon drag contribution is quite small and in fact is zero as a first approximation [6], is perhaps an advantage.
3. Results and analysis 2. Theory The thermopower S is defined by E = S V T, where E is the electric field developed in the sample under the applied temperature gradient V T. If we take the magnetic field B to be perpendicular to the plane of the 2DEG and along $, and the temperature gradient to be along ~, then the transverse thermopower (oRen called the NernstEttingshausen coefficient) is defined by Syx= Ey/V T~. The oscillatory diffusion component gy~ for elastic scattering of the electrons at low fields is found to be given by [2,4] ~y=
4c°o~t
×D'(X) exp ( -
sin
(1)
Here, ev is the Fermi energy and zt and Zq are the transport and quantum lifetimes, respectively, the latter being related to F by F - - h/2~q. The quantity D'(X) is the thermal damping factor and is the derivative of D(X) with respect to X, where D(X) = X/sinh X and X f 2x2knT/hmo. The oscillations in the longitudinal component S~= are predicted to be very similar but smaller by a factor 1/)xocxt. Note that there is a phase difference of n/2 between the oscillations in p= and Sy~ predicted by Eq. (1). We also expect oscillations to occur in the phonon drag thermopower, but no theory yet exists for the low-field case. Experimentally it is known [5] that phonon drag oscillations in S ~ behave roughly like those in the resistivity and increase monotonically with field, though the amplitude is much more rapidly reduced at low fields, i.e. the effective impurity damping factor is much stronger. This presumably arises because of
The sample had an electron density of ~ 1.9 x 10is m - 2 and a mobility of ~40 m2/V.s. By growing the sample on a strongly doped substrate the lattice thermal conductivity was reduced; this in turn decreases the flux of phonon momentuna in the applied temperature gradient so there was less phonon momentum available to transfer to the electrons as compared to the undoped case, thereby reducing phonon drag. Substrate doping has no effect on the diffusion component which at low temperatures depends only on the scattering of electrons by impurities close to the 2DEG. To further accentuate diffusion it is an advantage to reduce the temperature below 1 K where phonon drag decreases much more rapidly with temperature than diffusion. Fig. 1 gives experimental data on ~yx at a temperature of 0.55 IC The two experimental traces are from two different pairs of transverse probes on the sample, and the upper data are continued up to 8 T. These traces are derived from data taken with + B by eliminating the part even in field, which is about 20% of the total Near 1 T there is a dear change in the appearance of the oscillations. At high fields the oscillations are found to resemble the derivative of p ~ with respect to field. This is the form expected for high-field diffusion oscillations [7] but phonon drag seems to have a similar form, so the origin of these oscillations is not yet unambiguous. The lowest trace is calculated according to Eq. (1) using parameters appropriate to the sample and m*--0.07m, but is multiplied by a factor of two. There is substantial agreement between experiment and theory, especially in the appearance of a maximum in the amplitude near 0.5 T. This maximum is basically due to if(X) in Eq. (1), though modified by the term in zt, and is believed to be unique to diffusion thermopower. It has
531
R. Fletcher et aL /Surface Science 361/362 (1996) 529-532
2ot{ i
i
•
T=O.55K
.-.>~
::.
looo
o •
o
',
v O9
,L
-20
~m "~ ~©.=2 Eo -40
" vvvvvVVVVVVVVVVVVVVV o,o°,.0,=
~ ~'~'Q~I
.mL
o 0 c-
--60
,
0
I
,
2
I
(I/T)
Fig. 1. The osdllatiom in St. as a function of inverse magnetic field I/B. The upper two curves are experimental data from different voltage probes, the lowest curve is calculated using parameters appropriate to the present sample but the magnitude is multipfied by a factor of two. The bottom two traces axe offset by 20 and 40 #V K -1, respectively. (The calcuhted curve contains the monotonic component but this is essentially negligible in this range of field).
previously been seen in bulk metal systems (see, for example, Ref. [8]). In comparison, the amplitude of ~ = monotonically increases with B in this range. Observation of the predicted phase difference between the oscillations in p = and Syx would provide a further test. However, as can be seen in Fig. 1, the experimental data on ff,~, from two independent pairs of sample probes show relative phase shifts. This might be due to inhomogeneities in the sample. As a final test, we use both/~= and ffyxto extract Zq, which should be the same for both coefficients. The amplitude data for g,~ are plotted in Fig. 2 in the form log[~x(1 +o92~'~)/ogc~tD'(X)]a s a function of lIB over the temperature range 0.35-0.72 K for two pairs of sample probes. The scaled data follow a straight line within experimental error, and the fine through the data has a slope corresponding to Xq--1.8 ps which is identical to that obtained from a similar analysis [9] of p = oscillations. If phonon drag was important, and ff the phonon drag oscillations in S,x behave similarly to those
o
•
v
• 0.55K
=
• 0.45K " 0.39K * 0.35K ' ,3
A
,
o
4 1/B
~r~
100
10
0
' 1
' 2
0.72K
uqrl
'
4
' 5
6
B-~ (T-~) Fig. l The amplitude of ~rx multiplied by (1 + ~/(c%ztD'(X)) as a function of lIB. The open and closed symbols are for two independent pairs of sample probes, respectively. Only the part odd in field b included in , ~ .
in S=, then we would expect a much smaller effective Zq [5]. The intercept at l / B = 0 is 2 . 3 + I . 0 m V K -z, which is about a factor of two larger than the expected value Of 41tie~e= 1.08 mV K - 1; this factor of two was also indicated from the comparison of theory with experiment in Fig. 1.
4. Conclusions The observed thermopower oscillations exhibit the expected thermal damping factor for diffusion. They also yield the same impurity damping term as the resistivity oscillations, as is expected because both are determined by the same impurity scattering of electrons. In contrast, previous experiments have shown that phonon drag oscillations behave as if they have a much stronger effective impurity damping term. The only significant discrepancy with calculation is that the absolute amplitude of the thermoelectric oscillations is about a factor of two higher than predicted.
532
R. Fletcher et al./ Surface Science 361/362 (1996) 529-532
Acknowledgement This work was partially supported by the Natural Sciences and Engineering Research Council of Canada.
References 1"1] B.L. Gallagher and P.N. Butcher, in: Handbook on Se~conductors, VoL 1, Ed. P.T. Landsberg (Elsevier, Amsterdam~ 1992) p. 817.
[2] H. Havlov~ and L. Smr~ka, Phys. Status Solidi B 137 (1988) 331. [3] R. Fletcher, JJ. Harris and C.T. Foxon, M. Tsaousidou and P.N. Butcher, Phys. Rev. B 50 (1994) 14991. [4] 1~ Fletcher, P.T. Coleridge and Y. Feng, Phys. Rev. B 52 (1995) 2823. 1"5-1 M. D'Iorio, R. Stoner and R. Fletcher, Solid State Commun. 65 (1988) 697. 1"6"] X. Zjanni; P.N. Butcher and MJ. Kearney, Phys. Rev. B 49 (1994) 7520. [7] tL Oji, J. Phys. C: Solid State Phy~ 17 (1984) 3059. 1"8] I~ Fletcher, Phys. Rev. 28 (1983) 1721. Phys. Rev. 28 (1983) 6670. 1"9] P.T. Coleridge, R. Stoner and R. Fletcher, Phys. Rev. B 39 (1989) 1120.