Quantized Conductance of Point Contacts

Lucas F. Hackl∗ and Benjamin F. Maier† Humboldt-Universita¨t zu Berlin, Department of Physics, Newtonstr. 15, 12489 Berlin (Dated: August 16, 2011) We studied quantum effects of point contacts which are realized within a two-dimensional gas of GaAs-AlGaAs heterostructure (MODFET). We find quantized steps of 2e2/h the conductance changes in by varying the contact width. This is done by the gate . Varying the gate voltage of the MODFET, we found a change of the conductance in steps of 2e2/h. We observed six steps by increasing the gate voltage from −0.1V to 0.6V. Furthermore, we present an explanation which takes momentum quantization and band structure into account.

I. INTRODUCTION For a we find: a = 1.006 ± 0.001 The effect of quantized conductances is already known from the , where we can find multiples After this calibration measurement we measure the con- of e2/h. The two-dimensional electron gas, which is nec- ductance of our MODFET at low temperature T = 4.2K. essary to investigate those effects, can be realized in GaAs- We use liquid helium as cooling substance to ensure a AlGaAs heterostructures: Doping between the two mate- sharp Fermi distribution caused by the fact that almost all rials causes a potential minimum under the Fermi edge. are in states under the Fermi energy. Otherwise This leads to eigenstates of the potential well along this the electrons could have enough energy to transcend the direction while the electrons are free along the other both potential minimum of the two-deminsional electron gas. directions. Therefore, a quantum layer is formed because The voltage is impressed orthogonal to the contact which the effective mass of these electrons is much smaller than shifts the Fermi edge relative to the band structure of the the mass of free electrons. Their energy lies slightly be- electron gas. This enables us to control the width of the neath the Fermi energy. For our analysis we demand that band. the mean free path le is much greater than the width W (le  W) of the contact and additionally that the Fermi wavelength λF is much smaller than W (λF  W). III. MEASUREMENT

II. EXPERIMENTAL SETUP Now we can look at the concrete measurement for a MOD- FET of a certain width lx. The quantization can be ob- For our measurement we use a lock-in amplifier of the served type SR830 DSP. The reference signal has the frequency 1. if the length of the contact lz is much greater than ν = 433Hz and the amplitude U = 10mV. For the inte- the width lx. gration time we choose t = 0.1s which corresponds to 50 periods. 2. if the Fermi wave length is of the same order as lx. In the first step we measure the conductance by using sev- Under these conditions the contact works as a wave guide. eral high precision resistors. While in an ideal setting we The condition for an electron to pass is then: would expect the relation 2lx λF ≈ Rmeasured = Rreal n We can rewrite this for the integer n as: between the measured resistance Rmeasured and the resis- tance Rreal of the precision resistor, we find a linear correc- k l n ≈ F x tion: π Only those modes m with m < n are allowed and each Rmeasured(Rreal) = aRreal + RS mode increases the conductance by 2e2/h, which leads to: Therefore, we have to correct the measured value for the 2e2 conductance according to G = n n h Greal(Gmeasured) = aGmeasured. For our measurement we use different structures of the GaAs-AlGaAs MODFET. But in fact we show the results Because RS is the serial resistance of the circuit with pre- with the best resolution only. cision resistors we do not use it for corrections. Again, we have to take into account an additional serial re- sistor Rs which causes the relation Rmeasured = aRn + Rs. Combining these quantities, we find: ∗ G Electronic address: [email protected] G (G ) = a measured †Electronic address: [email protected] n measured 1 − Rs Gmeasured 2

FIG. 1: Quantized conductance dependent on the applied gate voltage VG. We show the measured values (green, dashed) and the corrected values (blue, solid) by introducing the serial resistor Rs.

We determine Rs by variation to get integer multiples of ly. For the eigenstates we find: 2e2/h. In fig. 1 we show the uncorrected (green, dashed)  x   y  and corrected (blue, solid) behavior of the conductance ψ(x) = A sin nπ sin nπ dependent on the applied voltage. One can easily notice lx ly the discrete steps which always appear when the potential The corresponding wave numbers follow according to: is high enough to bring the electrons into the next higher band. π kx = n For the serial resistor we obtain: lx

π Rs = (420 ± 20)Ω ky = m ly It is important to note that we stress the quantization of the Therefore, we find stationary electron waves orthogonal to conductance and not of the resistance. Actually, the resis- the propagation direction within the quantum point. This tance is quantized as well, which naturally follows from leads us to a quantized energy: R = G−1 but in a more intricate way: While the conduc- 2 2 2 2 2 ! tance comes along in integer steps, the resistance, namely p π ~ n m Enm = = 2 + 2 the inverse of the conductance, shows a less clear behavior 2me 2me lx ly as its steps are inverse to integer steps. Because the characteristical length lz along the propagation direction is much greater than lx and ly we can take the cor- responding wave numbers as continously distributed with the additional energy: IV. EXPLANATION 2 2 ~ kz Ez = We can explain the behavior of the conductance by us- 2me ing the basics of quantum mechanics. The effective one- The total energy then is: dimensional area of the quantum point contact can be mod- elled as a potential well with characteristical lengths lx and Etot = Enm + Ez 3

From the Pauli principle we know that each momentum which are filled one after the other. Each of those adds state can be taken twice because of the as additional a conductance of degree of freedom. This leads to the state density D(E) depending on the energy: dI 2e2 G = = 1 r m dV h D(E)dE = dE π~ 2E which finally leads to the conductance steps A small voltage dI causes a small current 2e2 dI = evdn G = n n h with a small linear electron density dn of velocity v. This is exactly the number of electrons per length which are we have measured because the sub bands are filled succes- excited by the voltage from the valence band into the con- sively. duction band. For this we can find:

dn = D(E)dE = D(E)edV

In the last step we have to estimate the velocity by using V. CONCLUSION m 2 the relation of the mean values E = 2 v . This leads to an electric current In summary we have presented the measurement of the 2e2 conductance of single point contacts in a two-dimensional dI = dV. h electron gas. Thereby, we found the quantum effect of quantized conductance within these contacts at low tem- Due to the quantization along the two directions we find perature, according to steps of 2e2/h. Furthermore, we several sub bands which show the same behavior and delivered an explanation for the appearance of this effect.

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