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in the present paper we examine the experimental setup well known for routine low - t transport measurements . let a 2d electron gas ( 2deg ) sample be placed ( fig.1 ) in a sample chamber kept at liquid - helium temperature .
the current leads are attached to the sample , and then connected to external measuring terminal kept at room temperature .
the measuring circuit is connected to the terminal . without external excitation ,
the dc voltmeter connected to arbitrary 2deg sample contacts demonstrates [ 1]-[5 ] a puzzling nonzero voltage(nv ) on the order of @xmath1v .
the value and the sign of the dc potential depends on a actual contact pair .    in the presence of a magnetic field , the measured dc potential demonstrates strong ( @xmath2mv ) oscillations named `` zero '' oscillations zo , which exhibit a @xmath3-periodicity similar to the well known shubnikov - de haas(sdh ) oscillations .
the zo period allows one to extract the carrier density , whereas the temperature dependence of the zo amplitude is similar to that for sdh oscillations and gives the correct value of the carrier effective mass .
in contrast to sdh oscillations , zo are skew - symmetric . the amplitude and the phase shift of the zero oscillations depend on a chosen contact pair .
moreover , for a certain contact pair , the zo shape is strongly affected when other sample leads are connected to(disconnected from ) the measuring circuit[6 ] .
we emphasize that the effects observed are , in general , universal and are observed in various 2deg systems and for arbitrary sample configuration .
the basic idea put forward in refs.[3,5 ] in order to explain the nv and zo consists in the possible rectification of the input noise by the 2d-3d schottky diodes formed at the sample contacts . as demonstrated in refs.[3,5 ] , a careful screening of the circuit diminishes the amplitude of the rectified voltage .
then , shunting of the sample contacts by a capacitance suppresses [ 5 ] the dc potential as well . in order to quantitatively
examine the influence of the noise , both the voltmeter and the ac generator playing the role of the noise source were attached to same sample contacts(see fig.1 , contacts 1,2 ) .
the experimental data demonstrate that the amplitude of the rectified voltage is @xmath4 to the ac voltage applied by the generator .
this finding is rather puzzling as regards the rectification consept proposed in refs.[3,5 ] . in the present paper , we suggest a phenomenological analysis of the experimental findings and then explain the important specific features of the effect .
setup configuration .
the noise is simulated by the ac generator . ]
we further use the simplest model of a current - voltage characteristic of the 3d-2d schottky contact[7,8 ] . in the thermionic diode approximation at finite temperatures ,
the current is given by @xmath5 where @xmath6 is the backward saturation current ; @xmath7 , equilibrium contact potential ; and , @xmath8 , voltage drop across the contact .
then , @xmath9 is the richardson constant for the thermionic emission , and @xmath10 is the quantum well width .
we emphasize that the schottky diodes at the left ( fig.1 , index 1 ) and right ( index 2 ) contacts have the opposite polarity and , in general , are different from each other .
therefore , the relationship between the total voltage drop across the sample , @xmath11 , and the current , @xmath12 , reads : @xmath13 where @xmath14 is the dimensionless voltage ; @xmath15 , current scaled with respect to reverse saturation current(@xmath16 ) of the left - contact diode ; and , @xmath17 , asymmetry parameter of the contacts .
then , @xmath18 is the dimensionless resistance of the 2d electron gas , and @xmath19 is the 2deg resistance .
current - voltage characteristics(panel a ) specified by eq.(2 ) for a contact - asymmetry parameter @xmath20 and a 2deg resistance @xmath21 .
the dashed line represents the low - field ohmic dependence .
the low - voltage(bold line ) and high - voltage(thin line ) ac input signals are represented in panel b. the respective responses are shown in panel c. ]    we are primarily interested in the low - current case @xmath22 and , therefore , linearize eq.(2 ) with respect to the current as @xmath23 as expected , the current - voltage characteristic exhibits the ohmic behavior @xmath24 or @xmath25 , where @xmath26 is the zero - field total resistance of the sample , and @xmath27 are the schottky resistances of contacts 1 and 2 , respectively . in the opposite case of a high applied voltage ,
the current - voltage characteristic is strongly nonlinear . indeed , in this case , the forward and reverse currents are limited ( see fig.2,a ) by the schottky diode saturation currents @xmath28 and @xmath16 , respectively .
let us first seek the response of the 3d/2deg/3d system to an applied ac voltage @xmath29 .
the ac voltage is provided by the generator shown in fig.1 . at low voltages
@xmath30 , eq.(3 ) allows one to extract the current as @xmath31 where @xmath32 . according to eq.(4 ) , the in - phase response of the system to the applied ac voltage consists of the ohmic contribution and an , additional part , because @xmath33 .
thus , we conclude that the widely used lock - in ac measurement method give [ 3 ] a sample resistance that is somewhat different from zero - field sample resistance @xmath34 .    we now intend to resolve the primary problem formulated in the present paper .
let us investigate the dc response of the circuit(see fig.1 ) to an applied ac voltage .
we emphasize that the second - order in voltage term in eq.(4 ) describes the rectification properties of the 2deg sample at @xmath35 .
equation 4 yields the time - averaged current @xmath36 and , then , the voltage drop measured by dc voltmeter is given by : @xmath37 the polarity of the measured dc voltage is determined by the contact asymmetry . as expected , the transmission characteristic @xmath38 is quadratic law at @xmath39 . in the opposite case of a strong ac excitation @xmath40 , the dc response can be found qualitatively with the help of fig.2,c .
indeed , the rectified current can be regarded as a rectangular meander sequence with linear fronts .
the higher the applied ac voltage , the sharper the front of the current pulse .
after simple averaging , we obtain the dc current and , finally , the rectified voltage as @xmath41 .
\label{average - high}\ ] ] at a high input ac signal @xmath42 , the measured voltage saturates @xmath43 .
one could expect that , at a moderate ac signal level @xmath44 , the low and high ac input cases merge , and , consequently , there could exist a certain part of the transmission characteristic which could be associated with a linear dependence[3,5 ] .    to confirm our qualitative predictions , we present in fig.3 the result of our numerical calculations .
we use current - voltage characteristic specified by eq.(2 ) . reversing this equation with respect to current ,
we find numerically the dependence @xmath45 .
the successive averaging of the current caused by the input ac signal gives the related dc voltage drop across the sample , and , hence the transmission characteristic .
as expected , the transmission characteristic follows the asymptote given by eq.(5 ) at low excitations . at a high - level ac input @xmath40 ,
the dependence @xmath38 can be approximated with eq.(6 ) , and then levels - off . for intermediate voltages @xmath46 ,
the transmission characteristic exhibits a nearly linear behavior @xmath47 in accordance with the experimental findings[5 ] .
main panel : the transmission characteristic for contact asymmetry @xmath20 and 2deg resistance @xmath21(upper curve ) . dotted lines a and b represent the low and high - voltage approximations specified by eq.(5 ) and eq.(6 ) , respectively .
the dashed line demonstrates a nearly linear dependence reported in [ 5 ] .
the lower curve corresponds to the case of voltmeter connected to 1 - 3 contacts ( see fig.1 ) when @xmath48 .
inset : the observed [ 5 ] transmission characteristics correspond to those in the main panel . ]
it is instructive to note that , in ref.[5 ] , the dc voltmeter was connected to the 1st(basic ) and intermediate probe contacts ( see contact 3 in fig.1 ) as well . in this case
, the transmission characteristic must be multiplied by the factor @xmath49 , where @xmath50 is the asymmetry of the intermediate contact with respect to the 1st one , and @xmath51 is the partial resistance of the 2deg related to the intermediate contact position . we emphasize that the lower the partial resistance @xmath51 , the smaller the differential slope @xmath52 of the transmission characteristic(see the lower curve in fig.3 , main panel ) .
we now estimate the actual parameters of 3d/2deg/3d system experimentally studied in ref.[5 ] . for n - algaas
/ gaas sample ( 2deg density @xmath53@xmath54 , dielectric constant @xmath55 , effective mass @xmath56 ) we find the fermi energy as @xmath57mev , whereas the bohr energy @xmath58mev . as demonstrated in ref.[9 ] , the thermionic diode approximation is well justified when @xmath59 , where @xmath60 . at @xmath61 , the tunneling current across the schottky diode becomes higher than the thermionic current .
for the typical equilibrium contact potential @xmath62ev and zero diode bias @xmath63 , we obtain @xmath64k , and , hence , the observed data[5 ] can not be analyzed directly in terms of the thermionic mechanism[7 ] .
nevertheless , even at low temperatures the current - voltage characteristic behaves similar to that described by eq.(1 ) .
hence , the main assumption that the nonzero voltage results from rectification of the ac noise remains justified .
we now estimate the resistance of the schottky contact . using the data from [ 5 ] , we find the total low - field resistance of the sample @xmath65 .
then , in the insert of fig.3 we reproduce the transmission characteristic data[5 ] for the dc output measured across 1 - 2(upper curve ) and 1 - 3(upper curve ) contacts .
both curves demonstrate a puzzling threshold behavior , which can be , in principle , attributed to the possible voltmeter zero - point shift .
consequently , the low - voltage part of these curves can be approximated by the following equations @xmath66=-0.0002 + 12u^{2}_{0}[v]$ ] and @xmath67 respectively .
note that , irrespective of the actual form of @xmath68 dependence , these curves differ by the factor @xmath69 which is equal to the ratio @xmath70 found above by fitting . assuming that the contact asymmetry is small , i.e. , @xmath71 we obtain the schottky contact resistance @xmath72 .
we argue that the zero - oscillations observed in 2deg in strong magnetic fields originate from the noise rectification as well .
indeed , at a fixed magnetic field strength , the transmission characteristic observed [ 5 ] for the zo amplitude is analogous to that reported for @xmath73 .
we remind that the sign of the rectified voltage depends on the asymmetry of the schottky diode contact pair . if the the reverse current @xmath74 of the schottky contact oscillates in the magnetic field , the rectified dc voltage(zo ) will oscillate as well .
the detailed examination of the zo oscillations will be made elsewhere .
we demonstrate that the dc voltage observed at low - temperatures in a 2d electron sample @xmath0 external excitation is caused by the noise rectification by schottky diodes formed at the sample contacts . at low noise level the rectified voltage as a function of the noise amplitude follows the usual quadratic law . at higher noise magnitudes ,
the rectified voltage exhibits a nearly linear behavior , and , finally saturates .
the rectified voltage is shown to depend on the contact pair asymmetry .
we suggest that the shunting of the sample contacts by a capacitance is a powerful tool for suppressing the rectified voltage .
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the dc voltage observed at low - temperatures in a 2d electron sample @xmath0 external excitation is accounted by the schottky contact rectification of the noise generated in the measuring circuit . 
 the rectified voltage is shown to depend on the asymmetry of the contact pair . 
 the dependence of the rectified voltage on the noise amplitude first follows the trivial quadratic law , then exhibits a nearly linear behavior , and , finally , levels - off .