Volume 100, Number 3,. May-June 1995 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 100, 269 (1995)] Using Quantized Breakdown Voltage Signals to Determine the Maximum Electric Fields in a Quantum Hall Effect Sample Volume 100 Number 3 May-June 1995 M. E. Cage and C. F. Lavine We estimate the maximum values of occurring between states distributed the electric field across the width of a across the sample width can be de- National Institute of Standards GaAs/AlGaAs heterostructure quantum tected as voltage signals along the sam- and Technology, Hall effect sample at several currents ple length. Gaithersburg, MD 20899-0001 when the sample is in the breakdown regime. This estimate is accomplished Key words: breakdown; electric fields; by measuring the quantized longitudinal voltage drops along a length of the quantized dissipation; quantized voltage sample and then employing a quasi- states; quantum Hall effect; quasi-elas- tic inter-Landau level scattering; two-di- elastic inter-Landau level scattering mensional electron gas. (QUILLS) model to calculate the elec- tric field. We also present a pictorial description of how QUILLS transitions Accepted: March 16, 1995 1. Introduction In the integer quantum Hall effect [1-3] the Hall quantized. We will use this quantization phe- resistance RH of the i th plateau of a fully quantized nomenon, and a black-boxmodel that is based on two-dimensional electron gas (2DEG) has the the conservationof energy, to determine the frac- value RH(i) =h/(e2i), where h is the Planck con- tion of electrons making transitions between stant, e is the elementary charge, and i is an in- Landau levels and their transition rates [7-9]. We teger. The current flow within the 2DEG is nearly will also use this quantization phenomenon, and dissipationless in the Hall plateau regions of high- the quasi-elastic inter-Landau level scattering quality devices, and the longitudinal voltage drops, (QUILLS) model of Heinonen, Taylor, and Girvin ~, along the sides of the sample are very small. At [10] and Eaves and Sheard [11], to deduce the high currents, however, energy dissipation can sud- maximum electric field experienced by the con- denly appear in these devices [4,5] and ~ can be- ducting electrons. come quite large. This is the breakdown regime of Finally,there has been a puzzle about howinter- the quantum Hall effect. Landau level transitions-which in the QUILLS .The dissipative breakdown voltage Vxcan be de- model occur between states distributed across the tected by measuring voltage differences between sample width- can be detectedas voltagesignals potential probes placed on either side of the device a!ong the sample length. We will give a pictorial in the direction of current flow. Cage et al. [6-9] explanation of a solution to this puzzle. have found that these breakdown voltages can be 269 Volume 100, Number 3, May-June 1995 Journal of Research of the National Institute of Standards and Technology 2. Experiment tive x axis points along the sample in the direction of the current ISD. Note that the conducting 2.1 Sample and Coordinate System charges are electrons. The y axis points across the The sample is a GaAs/AlxGal-xAs heterostruc- sample, and the z axis is into the figure for a right- ture grown by molecular beam epitaxy at AT&T handed coordinate system. The location of the 00- Bell Laboratories,l with x = 0.29 being the fraction ordinte system origin is arbitrary; it is convenient, of Al atoms replacing Ga atoms in the crystal. The however, to place it at the source S, and halfway sample is designated as GaAs(8), has a zero mag- across the sample, so that - w/2:S:;Y :s:;w/2. (This lo- netic field mobility of about 100 000 cm2/(V's) at cation is not shown in the figure for lack of space.) 4.2 K, and exhibits excellent integral quantum Hall The .magnetic field, B, is perpendicular to the sam- effect properties. This sample is currently used as a ple and points into the figure, in the positive z di- quantized Hall resistance standard to maintain the rection. Therefore, in the presence of a B field, the United States unit of resistance. electrons enter at the upper left hand corner of the The inset of Fig. 1 shows the sample geometry. It sample and exit at the lower right hand corner. The is 4.6 mm long and has a width, w, of 0.4 mm. The potential probes 2, 4, and 6 are near the potential two outer Hall potential probe pairs are displaced of the source S, which is grounded. Probes 1,3, and from the central pair by plus and minus 1 mm. The 5 are near the drain potential D, and have a posi- inset also shows the coordinate system. The posi- tive potential relative to the source. GaAs(8) Vx(4,6) 1 mm E 60 1--1 E 215 JlAto 225 JlA r;x 5 3 1 ;; yS~D 6 4 2 :> 40 --E ~ 20 o 12.2 12.3 12.4 12.5 B (T) Fig. 1. Eleven sweeps of Vx(4,6) versus B for the i = 2 plateau at 0.33 K with applied currents ISD between + 215 J-LAand + 225 J-LAin 1 J-LAincrements. The values of Vx generally increase with current. The arrow shows the sweep direction. A family of eight shaded curves is fitted to these data in the vicinity of 12.3 T where the data are current-independent. Voltage quantization num- bers are shown in brackets. The inset displays the sample geometry. The actual origin of the coordinate system is located at the source S, halfway across the sample width w. 1Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identifica- tion does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it im- ply that the materials or equipment identified are necessarily the best available for the purpose. 270 Volume 100, Number 3, May-June 1995 Journal of Research of the National Institute of Standards and Technology 2.2 Longitudinal Voltage Versus Magnetic Field to arise from transitions in which electrons occupy- ing states of the originally full ground state Landau The dissipative breakdown voltages ~ for this level are excited to states in higher Landau levels paper were measured between potential probe pair and then return to the lowest Landau level. The 4 and 6, hereafter denoted as ~(4,6) = V( 4) V(6). - electrical energy loss per carrier for M Landau The longitudinal voltage signals on the opposite level transitions is MhWe,where We= eB /m * is the side of the sample, ~(3,5), were the same as cyclotron angular frequency and m * is the reduced ~(4,6), and integer quantum Hall voltages mass of the electron (0.068 times the free electron VH=RHIsD were observed on probe sets VH(3,4) mass me in GaAs). The power loss is I~, and and VH(5,6). I~ =r(2/i)MhWc, where r is the combined transi- Figure 1 shows 11 sweeps of ~ (4,6) versus the tion rate from the ground state to the excited state magnetic field.B-for the i =2 (12 906.4 0) quan- and then back to the ground state, and i is the Hall tized Hall resistance plateau at a temperature of plateau number. Thus 0.33 K for injected electron currents ISDof + 215 fJ.Ato + 225 fJ.Ain 1 fJ.Aincrements, where posi- tive current corresponds to electrons entering the fM=(r;)M=(~)(~*)(~), (1) source and exiting the drain. These sweeps are for increasing B; similar sweeps were obtained for de- where f is the ratio of the transition rate r within creasing B. The data clearly show discrete, well-de- the breakdown region to the rate I/e that electrons fined, quantized voltage states, with switching between states. transit the device; f can also be interpreted as the fraction of conducting electrons that undergo tran- A family of eight equally-spaced shaded curves is sitions. also shown in Fig. 1 in the region near 12.3 T The black-box model predicts that, in the vicinity where the ~ voltage spacings are nearly current-in- of B = 12.3 T, about 22 % of the conducting elec- dependent. The curves have equal (quantized) trons are making inter-Landau transitions, with voltage separations at each value of magnetic field, transition rates between 3.0 x 1014/Sand 3.1 x 1014/S but the voltage separations are allowed to vary for currents between 215 fJ.Aand 225 fJ.A. slightly with B in order to obtain smooth curves that provide the best fit to the data. The eight shaded curves correspond to a ~ =0.0 mV ground 3.2 QUILLSModel state and seven excited states. Several quantum To predict the maximum value of the electric numbers M of the voltage states are labeled in field, Emax,within the sample when breakdown is brackets. Note that, over the shaded curve portion occurring we use the quasi-elastic inter-Landau of Fig. 1 near 12.3 T, the M = 1 transition first oc- level scattering (QUILLS) model of Heinonen, curs at a current of 215 fJ.A,and the M = 7 state Taylor, and Girvin[10]and Eaves and Sheard [11], first appears at 225 fJ.A. and the notation of Cage, Yu, and Reedtz [12]. The conducting electrons have completely filled the maximumallowed number of states of the first 3.