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Structure in the rectifying behavior of mesoscopic Si MOSFETs
Surface Science 196 ( 1988 ) 93-100 North-Holland, Amsterdam
93
STRUCTURE IN THE RECTIFYING BEHAVIOR OF MESOSCOPIC Si MOSFETs S.B. K A P L A N IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA
Reveived 1 June 1987; accepted for publication 10 Augusl 1987
The conductance of Si MOSFETs of submicrometer dimensions was studied as a function of source-drain voltage. Aperiodic structure was observed in the conductance as the gate voltage and magnetic field were varied. This structure was found to depend on the polarity of the source-drain vol ~age.The structure in the rectified portion (antisymmetric part) of the conductance was observed to nave a magnetic field scale similar to that of the symmetric part. The magnitude of this structure (relati ve to the symmetric conductance structure) increased linearly at first, then sublinearly, with increasing source-dJain voltage, as is predicted by the existing theories of conductance fluctuations due to quantum interference.
1. Introduction The quantum-mechanical transmission coefficient for electronic transport through a disordered conductor depends on the magnitude and sign of the electrochemical potential differences that exist within it. In mac :oscopic conductors, many nonlinear effects are self-averaged. This paper will concern itself with mesoscopic conductors, which are small enough that a great degree of electronic phase correlation exists. The r a n d o m placement ofelast ic scattering centers in disordered samples results in a potential which lacks in' ersion symmetry. In the mesoscopic regime, the asymmetry imposed by the pot ential should begin to become apparent. In particular, structure in the rectifying behavior of small conductors should be observed due to q u a n t u m interference of electronic waves, as predicted by Al'tshuler and Khmel'nitskii [ 1 ]. I have observed fluctuations in the rectifying b e h a v i o r of Si MOSFETs with submicrometer dimensions which varied aperiodically with gate voltage and magnetic field, and which increased with the magnitude of the source-drain voltage VsD. This is believed to be the first report of rectification in aperiodic m~gnetostructure due to quantum interference in weakly localized semiconductor systems. The asymmetric aperiodic structure discussed in this paper is intimately related to the magnetostructure reported by" U m b a c h et at. [ 2 ]. These so-called "universal conductance fluctuations" were first explained (for the case T = 0) by two theorists 0039-6028/88/$ 03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
94
S.B. Kaplan/Structure in tt,'e rectifi,ing behavior of Si MOSFETs
working independently. Al'tshu!er [ 311 used a weak-scattering diagrammatic technique to predict that sample-to-sam,;fie conductance fluctuations may arise because of sample-specific interference conditions. Stone [4] used the results of simulations to argue that a change in an external parameter results in precisely the same effect in any given sample. In a Si MOSFET, for example, a change in gate voltage (VG) changes electronic wa'~,elengths; a change in the magnetic flux perpendicular to the inversion layer changes electronic phases. A variation in the source~,'ain voltage (Vso) changes the scattering matrix elements, which results in changes in the interference conditions. Because of the random nature of the potential, the conductance fluctuations depend on the sign as well as the magnitude of VsD. Structure is observed in the antisymmetric COmponent of the conductance as a result of quantum interference. In order to discuss the magnitude of this effect, the theory of conductance fluctuations must be discussed in further detail. A number of workers have made use of diagrammatic methods to study the fluctuations [ 5 ]. It was found that electrons are correlated within length and energy ranges. The phase correlation length L~ = ~/D%, where D is the diffusion constant and % is the electronic lifietime for phase-breaking events (such as inelastic scattering). Let us consider a sample with maximum dimension L w, Lo. As long as the energy difference between two electronic waves is small enough, the phase difference between them should remain small enough to allow phase coherence to exist. Therefore, electrons should be correlated over an energy scale
17.o.~ h i % ,
(1)
where Zd = L :/D is the time for an electron to diffuse across the sample. According 1o the zero-temperature theo~,, the conductance autocorrelation function has a width at half maximum equal to E~ and a maximum value of order (e2/h) 2. If the voltage across the device is varied, the scattering patterns are changed, resulting in additional random corrections to the conductance. Since an energy change of E~ results in a phase change of 2n, it is expected that for small VSD,an antisymmetric component of the fluctuations appears, which is gi'ren by
6 G A - G ( V S D ) - G ( - VsD) ~go(eVsD/Eo),
(2)
where go-- e2/h, as predicted in ref. [ 1 ]. K.,hmel'nitskii and Larkin [ 6 ] have argued that when VSD>Eo, the energy range is divided into eVsD/Eo coherent levels. They predict that ~.he total rms fluctuation amplitude is the sum of random contributions from each level: 6 G~ ~ go(eVs~IE~) ~<2
(3)
For samples larger than Lo, the theory discussed above needs to be amended [ 5 ]. Let us consider a two-dimensional sample with length and width both greater than L~ in a magnetic field B strong enough that correction to the conductance due to weak localizatien is quenched. The width of the conductance autocorrelation func-
S.B. Kaplan/Structure in the rectifying behavior of Si ~IOSFETs
95
tion is given by E o from eq. (1), but with L = L~. The (sym;netric) rms fluctuation amplitude per square is given by
5GD = O . 6 1 g o / ~ ,
(4)
where M and N are the respective wic~th and length of the sample in units of L o. In other words, the amplitude is the ste,chastic average of MN samples. It was also predicted [ 5 ] that at finite temperatu'res, the thermal smearin~ of energy limits the maximum correlation length (irrespective of scattering) to L~ ~ h D ~ . When LT < Lo, the width of the correlation function is kT, and one must average coherent regions of size LT. It is expected that similar corrections reduce GAo It will be argued that the observed asymmetric structure is due to quantum interference. The antisymmetric part of the co~ductance will be comp~ red to the symmetric part to show that the relative magnitude, magnetic field scale arid the source-drain voltage scale compare favorably with theoretical r,~ :dictions.
2. Experimental methods The samples used in these experiments are MOSFETs made on p-type epitaxially grown Si layers. The gate oxide ~s 12.5 rtm thick, and the gate lengths and widths are ~ 0.9 and 0.5/zm, respectiw:ly. All measurements were made at a bath temperature of T = 0.45 K. Magnetoconductance measurements were made using an AC lock-in technique at 375 Hz. The AC component of the source-drain voltage was kept low enough to avoid electron heating. The DC component was added by a Keithley 230 programmable voltage source. The AC current through the channel was monitored with an Ithaco 1201 curren -sensitive weamplifier and a PAR 5301 lock-in amplifier. Data were logged using ,~ mu|fich~ reel analyzer program running on an IBM personal computer. In order t.o minimize measurement errors, Vst, was toggled between positive and negative valaes before sweeping to the next value of gate voltage or magnetic field. The resultant current noise amplitude was usually less than 5 × 10- ~3 A.
3. Results and discussion In order to compare results with theory, L,, must be known. To this end, the conductance of a larger adjacent Si MOSFET was measured. The presence of fluctuations in this 3. t gm wide by 2.4 #m long device prevented a determination e f Lo by negative magnetoresistance measuremer is, so the fluctuation amplitude of ~ 3.6× 10 - 6 S was used along with eq. (4) to ,nfer a value of Lo,".0.32 Izm. The width of the field correlation function was found to be Bc ~, 0.12 .-+_0.02 T. A similar value was found for the small MOSFET. Since Lo is directly related to Be, we as-
96
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B(T) Fig. 1. The fluctuating component of the antisymmetric conductance for VG=3 V is plotted versus the applied field for various values of VsD.All curves but the bottom one have been arbitrarily placed on the vertical scale for the sake of clarity. same that the inelastic diffusion length is the same in the small M O S F E T at the same n u m b e r density. For VG < 4 V, the data were very reproducible as long as the sample was kept at cryogenic temperatures. Above this value, occasional changes were seen over time. The fluctuation amplitude 5Gs at low VsD was found to be ~ 2 × 10 -6 S, which is a factor of three less than expected. This may be due to sample inhomogeneities, or to the statistics of small samples. However, the absolute m a g n i t u d e does not affect the conclusions of this study. In general, the antisymmetric component had a constant background which varied with gate voltage; this will be discussed later. The fluctuating component o f the antisymmetric conductance for VG = 3 V is plotted versus the applied field in fig. 1 for various values of VsD, which are corrected for contact resistance. ( T h e contact resistance Rc was estimated to be ,-, 2.9 k ~ . This was done by measuring neighboring gates o f differing sizes to obtain the contact resistance of the larger device. Since Bc is approximately the same for both the small and the large devices described above, a similar conductance per square is implied. ) We note that the structure in the cur~,es increases in magnitude with VsD, and begins to change shape on a scale of ~'sD~ 0.3 inV. This behavior is expected when e s , - LEo/Lo ~ 0.45 meV (when ~be potential drop across a coherent region is E~, and where the electric field has been assumed to be approximately constant). The asymmetric conductance and its fluctuation amplitude did not exhibit a systematic magnetic field d~.pend~.n~. We expect that the fluctt~ation amplitude at either polarity should increase with
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gsD(mV) Fig. 2. (a) The rms amNiludes of the antisymmetric (6GA) and the symmetric t~Gs) components of the conducla~ce are plotted versus Vso. (b) The ratio g G a / 6 G s is plotted versus VsD.
VSD as described above. However, there is also a decrease with increasing VsD as the electrons gain energy from the electric field. This increases both electron-electron and electron-phonon scattering, and thus decreases L o, which results in a decrease in the fluctuation amplitude through eq. (4). In fig. 2a, we plot the antisymmetric m s fluctuation amplitude 5GA, as well as the symmetric component given by"~ G~,-=0.5[G(VsD) + G( - VsD)]. We notice that 5Gs begins to drop with increasing This provides a crude estimator of eiecnon ..... :. . . . . ~ presumably also affects ~SGa. The ratio ,~fthese two quantities is shown in fig. 2b° There is a linear iacrease in this quantity until VsD~0.3 mV, which compares favorably with eq. (2). There is a change to a sublinear slope as VSD increases further. One would expect this to occur when the potential drop across a coherent
98
S.B. Kaplan/Structure in the rectifvingbehaviorof Si MOSFETs
region equals Eo, or eVso..~0.4 meV. The actual value is in fair agreement with theory. The data shown in fig. 2 are suggestive of this type of effect. Similar data were obtained at higher gate ",oRage. The field scale of the fluctuations in Gs and GA was determined by calculating the widths of the respective autocorrelation functions. Bc was found to start at ~ 0 . 1 2 _ + 0.02 T for small VsD, and increases with VSD.This indicates a decrease in the correlation length, as expected. A more detailed discussion of this effect is beyond the scope of this paper, and will be given elsewhere. The conductance has also been measured under the opposite polarity of magnetic field. Despite the fact that strong nonequilibrium conditions occur in the MOSFET, the Onsager relations are found to hold: i.e. G( Vso, B) = G( VSD, - B ) and G ( - Vso, B ) = G ( - VsD, - B ) . This symmetry, and the lack of a magnetic field dependence of the fluctuation amplitude suggests that Hall effects do not substantially affect the data (However, we do not note that in a mesoscopic system, the current and electric field vectors are not locally collinear.) The question arises whether the conductance of the contacts affects the magnetostructure in this two-point measurement. A number of gate voltage sweeps were made to answer this question. In general, structure in 8 GA was found to increase with VSD,and changes shape when list) varied more than ~ 0.2 mV. However, there was a large background with a larger voltage scale and an amplitude of several times that of the magnetostructure which tended not to change shape. In fig. 3a we plot data representing 8Gs taken vSth I VsDI----0.2 mV and for various values of substrate bias. We see large dips in the background at VG~ 2.6 V and 2.9 V. We would expect that if these dips were associated with the inversion layer, they would line up 'better when plotted versus the gate voltage in excess of threshold, as shown in fig. 3b. The lack of a lineup suggests that these features arise at the contacts. The background rectification persists up to temperatures of > 15 K, where the interference effects have already become too small to measure because of thermal smearing. During our low-temperature measurements, which are made at a constant gate voltage, the background rectification is constant and does not affect our results.
4. Conclusions The magnetoconductance of submicron Si MOSFETs was measured at various values of source-drain voltage. Aperiodic fluctuations were obsewed in the antisymmetric component of the magnetoconductance. The magnitude of these fluctuations varied approximately linearly with source-drain voltage up to a value which was of the same order as the electron cowelation energy; at higher voltages, sublincar dependence was seen, The magnetic field scale of both symmetric and antisymmetric components of conductance increased in a similar manner with PsD. It is believed that this is the first report of the modulation of rectifying behavior in metallic inversion layers. The results are in fair agreement with theory.
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(v) Fig. 3. (a) A gate voltage sweep of amisymmetric conductance data taken with [I~Di =0.2 m V and for various values of substrate bias• NB: These are raw data, including the contact resistance. (b) The same data plotted againstthe gate voltage in excess of threshold. All curves but the bottom one in each plot have been arbli~arilydisplaced along the vertica!scalefor sake of clarity.
The existence o f asymmetric structure due ,~ ~," Oec'~ric ~eM in a mesoscopic conductor implies the possibility of observing negative dynamic resistance [ 6,7 ]. However, in the present work, the fluctuations are far too small to observe this effect in the metallic regime. It would be of interest to study MOSFETs or semiconductor heterostructures which are short enough that inelastic scattering is not a limiting factor.
100
S.B. Kaplan/Structure in the rectifying behavior of Si MOSFETs
Acknowledgement The author would like to thank J. Sun for providing the samples, and M. Biittiker, A. Williams, A. Hartstein, S. Washburn, P.A. Lee and Y. lmry for interesting discussions. The technical assistance of N. Albert was well appreciated.
References [ 1] B.L. AFtshuler and D,E. Khmel'nitskii, JETP Letters 42 (1985) 361. [2] C.P. Umbach, S. Washburn, R.B. Laibowitz and R.A. Webb, Phys. Rev. B30 (1984) 4048. [3] B.L. Al'tshuler, Pis'ma Zh. Eskp. Teor. Fiz. 41 (1985) 530 [JETP Letters 41 (1985) 648]. [4] A,D. Stone, Phys. Rev. Letters 54 (1985) 2693. [ 5 ] See for a review: P.A. Lee, A.D. Stone and H. Fukuyama, Phys. Rev. B35 (! 987) 1039. [6] A,I. Larkin and D.E. Khmel'nitskii, Zh. Eksp. Teor. Fiz. 91 (1986) 1815. [7 ] G. Timp, A.M. Chang, J.E. Cunning~ham,T.Y. Chang, P. Mankiewich, R. Behringer and R.E Howard, Phys. Rev. Letters 58 (1987) 2814.
93
STRUCTURE IN THE RECTIFYING BEHAVIOR OF MESOSCOPIC Si MOSFETs S.B. K A P L A N IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA
Reveived 1 June 1987; accepted for publication 10 Augusl 1987
The conductance of Si MOSFETs of submicrometer dimensions was studied as a function of source-drain voltage. Aperiodic structure was observed in the conductance as the gate voltage and magnetic field were varied. This structure was found to depend on the polarity of the source-drain vol ~age.The structure in the rectified portion (antisymmetric part) of the conductance was observed to nave a magnetic field scale similar to that of the symmetric part. The magnitude of this structure (relati ve to the symmetric conductance structure) increased linearly at first, then sublinearly, with increasing source-dJain voltage, as is predicted by the existing theories of conductance fluctuations due to quantum interference.
1. Introduction The quantum-mechanical transmission coefficient for electronic transport through a disordered conductor depends on the magnitude and sign of the electrochemical potential differences that exist within it. In mac :oscopic conductors, many nonlinear effects are self-averaged. This paper will concern itself with mesoscopic conductors, which are small enough that a great degree of electronic phase correlation exists. The r a n d o m placement ofelast ic scattering centers in disordered samples results in a potential which lacks in' ersion symmetry. In the mesoscopic regime, the asymmetry imposed by the pot ential should begin to become apparent. In particular, structure in the rectifying behavior of small conductors should be observed due to q u a n t u m interference of electronic waves, as predicted by Al'tshuler and Khmel'nitskii [ 1 ]. I have observed fluctuations in the rectifying b e h a v i o r of Si MOSFETs with submicrometer dimensions which varied aperiodically with gate voltage and magnetic field, and which increased with the magnitude of the source-drain voltage VsD. This is believed to be the first report of rectification in aperiodic m~gnetostructure due to quantum interference in weakly localized semiconductor systems. The asymmetric aperiodic structure discussed in this paper is intimately related to the magnetostructure reported by" U m b a c h et at. [ 2 ]. These so-called "universal conductance fluctuations" were first explained (for the case T = 0) by two theorists 0039-6028/88/$ 03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
94
S.B. Kaplan/Structure in tt,'e rectifi,ing behavior of Si MOSFETs
working independently. Al'tshu!er [ 311 used a weak-scattering diagrammatic technique to predict that sample-to-sam,;fie conductance fluctuations may arise because of sample-specific interference conditions. Stone [4] used the results of simulations to argue that a change in an external parameter results in precisely the same effect in any given sample. In a Si MOSFET, for example, a change in gate voltage (VG) changes electronic wa'~,elengths; a change in the magnetic flux perpendicular to the inversion layer changes electronic phases. A variation in the source~,'ain voltage (Vso) changes the scattering matrix elements, which results in changes in the interference conditions. Because of the random nature of the potential, the conductance fluctuations depend on the sign as well as the magnitude of VsD. Structure is observed in the antisymmetric COmponent of the conductance as a result of quantum interference. In order to discuss the magnitude of this effect, the theory of conductance fluctuations must be discussed in further detail. A number of workers have made use of diagrammatic methods to study the fluctuations [ 5 ]. It was found that electrons are correlated within length and energy ranges. The phase correlation length L~ = ~/D%, where D is the diffusion constant and % is the electronic lifietime for phase-breaking events (such as inelastic scattering). Let us consider a sample with maximum dimension L w, Lo. As long as the energy difference between two electronic waves is small enough, the phase difference between them should remain small enough to allow phase coherence to exist. Therefore, electrons should be correlated over an energy scale
17.o.~ h i % ,
(1)
where Zd = L :/D is the time for an electron to diffuse across the sample. According 1o the zero-temperature theo~,, the conductance autocorrelation function has a width at half maximum equal to E~ and a maximum value of order (e2/h) 2. If the voltage across the device is varied, the scattering patterns are changed, resulting in additional random corrections to the conductance. Since an energy change of E~ results in a phase change of 2n, it is expected that for small VSD,an antisymmetric component of the fluctuations appears, which is gi'ren by
6 G A - G ( V S D ) - G ( - VsD) ~go(eVsD/Eo),
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where go-- e2/h, as predicted in ref. [ 1 ]. K.,hmel'nitskii and Larkin [ 6 ] have argued that when VSD>Eo, the energy range is divided into eVsD/Eo coherent levels. They predict that ~.he total rms fluctuation amplitude is the sum of random contributions from each level: 6 G~ ~ go(eVs~IE~) ~<2
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For samples larger than Lo, the theory discussed above needs to be amended [ 5 ]. Let us consider a two-dimensional sample with length and width both greater than L~ in a magnetic field B strong enough that correction to the conductance due to weak localizatien is quenched. The width of the conductance autocorrelation func-
S.B. Kaplan/Structure in the rectifying behavior of Si ~IOSFETs
95
tion is given by E o from eq. (1), but with L = L~. The (sym;netric) rms fluctuation amplitude per square is given by
5GD = O . 6 1 g o / ~ ,
(4)
where M and N are the respective wic~th and length of the sample in units of L o. In other words, the amplitude is the ste,chastic average of MN samples. It was also predicted [ 5 ] that at finite temperatu'res, the thermal smearin~ of energy limits the maximum correlation length (irrespective of scattering) to L~ ~ h D ~ . When LT < Lo, the width of the correlation function is kT, and one must average coherent regions of size LT. It is expected that similar corrections reduce GAo It will be argued that the observed asymmetric structure is due to quantum interference. The antisymmetric part of the co~ductance will be comp~ red to the symmetric part to show that the relative magnitude, magnetic field scale arid the source-drain voltage scale compare favorably with theoretical r,~ :dictions.
2. Experimental methods The samples used in these experiments are MOSFETs made on p-type epitaxially grown Si layers. The gate oxide ~s 12.5 rtm thick, and the gate lengths and widths are ~ 0.9 and 0.5/zm, respectiw:ly. All measurements were made at a bath temperature of T = 0.45 K. Magnetoconductance measurements were made using an AC lock-in technique at 375 Hz. The AC component of the source-drain voltage was kept low enough to avoid electron heating. The DC component was added by a Keithley 230 programmable voltage source. The AC current through the channel was monitored with an Ithaco 1201 curren -sensitive weamplifier and a PAR 5301 lock-in amplifier. Data were logged using ,~ mu|fich~ reel analyzer program running on an IBM personal computer. In order t.o minimize measurement errors, Vst, was toggled between positive and negative valaes before sweeping to the next value of gate voltage or magnetic field. The resultant current noise amplitude was usually less than 5 × 10- ~3 A.
3. Results and discussion In order to compare results with theory, L,, must be known. To this end, the conductance of a larger adjacent Si MOSFET was measured. The presence of fluctuations in this 3. t gm wide by 2.4 #m long device prevented a determination e f Lo by negative magnetoresistance measuremer is, so the fluctuation amplitude of ~ 3.6× 10 - 6 S was used along with eq. (4) to ,nfer a value of Lo,".0.32 Izm. The width of the field correlation function was found to be Bc ~, 0.12 .-+_0.02 T. A similar value was found for the small MOSFET. Since Lo is directly related to Be, we as-
96
S.B. Kaplan/Structure in the rectifj,ing behavior of Si MOSFETs
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B(T) Fig. 1. The fluctuating component of the antisymmetric conductance for VG=3 V is plotted versus the applied field for various values of VsD.All curves but the bottom one have been arbitrarily placed on the vertical scale for the sake of clarity. same that the inelastic diffusion length is the same in the small M O S F E T at the same n u m b e r density. For VG < 4 V, the data were very reproducible as long as the sample was kept at cryogenic temperatures. Above this value, occasional changes were seen over time. The fluctuation amplitude 5Gs at low VsD was found to be ~ 2 × 10 -6 S, which is a factor of three less than expected. This may be due to sample inhomogeneities, or to the statistics of small samples. However, the absolute m a g n i t u d e does not affect the conclusions of this study. In general, the antisymmetric component had a constant background which varied with gate voltage; this will be discussed later. The fluctuating component o f the antisymmetric conductance for VG = 3 V is plotted versus the applied field in fig. 1 for various values of VsD, which are corrected for contact resistance. ( T h e contact resistance Rc was estimated to be ,-, 2.9 k ~ . This was done by measuring neighboring gates o f differing sizes to obtain the contact resistance of the larger device. Since Bc is approximately the same for both the small and the large devices described above, a similar conductance per square is implied. ) We note that the structure in the cur~,es increases in magnitude with VsD, and begins to change shape on a scale of ~'sD~ 0.3 inV. This behavior is expected when e s , - LEo/Lo ~ 0.45 meV (when ~be potential drop across a coherent region is E~, and where the electric field has been assumed to be approximately constant). The asymmetric conductance and its fluctuation amplitude did not exhibit a systematic magnetic field d~.pend~.n~. We expect that the fluctt~ation amplitude at either polarity should increase with
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VSD as described above. However, there is also a decrease with increasing VsD as the electrons gain energy from the electric field. This increases both electron-electron and electron-phonon scattering, and thus decreases L o, which results in a decrease in the fluctuation amplitude through eq. (4). In fig. 2a, we plot the antisymmetric m s fluctuation amplitude 5GA, as well as the symmetric component given by"~ G~,-=0.5[G(VsD) + G( - VsD)]. We notice that 5Gs begins to drop with increasing This provides a crude estimator of eiecnon ..... :. . . . . ~ presumably also affects ~SGa. The ratio ,~fthese two quantities is shown in fig. 2b° There is a linear iacrease in this quantity until VsD~0.3 mV, which compares favorably with eq. (2). There is a change to a sublinear slope as VSD increases further. One would expect this to occur when the potential drop across a coherent
98
S.B. Kaplan/Structure in the rectifvingbehaviorof Si MOSFETs
region equals Eo, or eVso..~0.4 meV. The actual value is in fair agreement with theory. The data shown in fig. 2 are suggestive of this type of effect. Similar data were obtained at higher gate ",oRage. The field scale of the fluctuations in Gs and GA was determined by calculating the widths of the respective autocorrelation functions. Bc was found to start at ~ 0 . 1 2 _ + 0.02 T for small VsD, and increases with VSD.This indicates a decrease in the correlation length, as expected. A more detailed discussion of this effect is beyond the scope of this paper, and will be given elsewhere. The conductance has also been measured under the opposite polarity of magnetic field. Despite the fact that strong nonequilibrium conditions occur in the MOSFET, the Onsager relations are found to hold: i.e. G( Vso, B) = G( VSD, - B ) and G ( - Vso, B ) = G ( - VsD, - B ) . This symmetry, and the lack of a magnetic field dependence of the fluctuation amplitude suggests that Hall effects do not substantially affect the data (However, we do not note that in a mesoscopic system, the current and electric field vectors are not locally collinear.) The question arises whether the conductance of the contacts affects the magnetostructure in this two-point measurement. A number of gate voltage sweeps were made to answer this question. In general, structure in 8 GA was found to increase with VSD,and changes shape when list) varied more than ~ 0.2 mV. However, there was a large background with a larger voltage scale and an amplitude of several times that of the magnetostructure which tended not to change shape. In fig. 3a we plot data representing 8Gs taken vSth I VsDI----0.2 mV and for various values of substrate bias. We see large dips in the background at VG~ 2.6 V and 2.9 V. We would expect that if these dips were associated with the inversion layer, they would line up 'better when plotted versus the gate voltage in excess of threshold, as shown in fig. 3b. The lack of a lineup suggests that these features arise at the contacts. The background rectification persists up to temperatures of > 15 K, where the interference effects have already become too small to measure because of thermal smearing. During our low-temperature measurements, which are made at a constant gate voltage, the background rectification is constant and does not affect our results.
4. Conclusions The magnetoconductance of submicron Si MOSFETs was measured at various values of source-drain voltage. Aperiodic fluctuations were obsewed in the antisymmetric component of the magnetoconductance. The magnitude of these fluctuations varied approximately linearly with source-drain voltage up to a value which was of the same order as the electron cowelation energy; at higher voltages, sublincar dependence was seen, The magnetic field scale of both symmetric and antisymmetric components of conductance increased in a similar manner with PsD. It is believed that this is the first report of the modulation of rectifying behavior in metallic inversion layers. The results are in fair agreement with theory.
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The existence o f asymmetric structure due ,~ ~," Oec'~ric ~eM in a mesoscopic conductor implies the possibility of observing negative dynamic resistance [ 6,7 ]. However, in the present work, the fluctuations are far too small to observe this effect in the metallic regime. It would be of interest to study MOSFETs or semiconductor heterostructures which are short enough that inelastic scattering is not a limiting factor.
100
S.B. Kaplan/Structure in the rectifying behavior of Si MOSFETs
Acknowledgement The author would like to thank J. Sun for providing the samples, and M. Biittiker, A. Williams, A. Hartstein, S. Washburn, P.A. Lee and Y. lmry for interesting discussions. The technical assistance of N. Albert was well appreciated.
References [ 1] B.L. AFtshuler and D,E. Khmel'nitskii, JETP Letters 42 (1985) 361. [2] C.P. Umbach, S. Washburn, R.B. Laibowitz and R.A. Webb, Phys. Rev. B30 (1984) 4048. [3] B.L. Al'tshuler, Pis'ma Zh. Eskp. Teor. Fiz. 41 (1985) 530 [JETP Letters 41 (1985) 648]. [4] A,D. Stone, Phys. Rev. Letters 54 (1985) 2693. [ 5 ] See for a review: P.A. Lee, A.D. Stone and H. Fukuyama, Phys. Rev. B35 (! 987) 1039. [6] A,I. Larkin and D.E. Khmel'nitskii, Zh. Eksp. Teor. Fiz. 91 (1986) 1815. [7 ] G. Timp, A.M. Chang, J.E. Cunning~ham,T.Y. Chang, P. Mankiewich, R. Behringer and R.E Howard, Phys. Rev. Letters 58 (1987) 2814.