Quantization of electrical conductance
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1 1 Introduction Quantization of electrical conductance Te resistance of a wire in te classical Drude model of metal conduction is given by RR = ρρρρ AA, were ρρ, AA and ll are te conductivity of te material, te cross section of te wire and its lengt. In tis model, te conductance = 1 = AA, measured in Sieverts SS, decreases uniformly RR ρρρρ as te cross section AA of te wire decreases. In quantum mecanics, te conductance GG of a metal may be viewed as te result of adding carge transport contributions troug electron semi-classical states. Consider a wire tat as a narrow constriction at some point over its lengt [Fig. 1(a)]. As te cross-section is reduced, te energy levels of te stationary states inside te constriction move up in energy and are separated by larger differences, just as tey would in a narrower potential well. As a result, te number of states wit a crystal momentum along te wire wit pp zz = 0 tat fall between te energy levels of te LHS and RHS contacts, VV 1 VV 2, is reduced [two are sown in Fig. 1(a)]. It can be sown tat in tis case te conductance of te wire is GG = 2ee2 nn = ( )nn Sieverts, were nn is an integer (Appendix). Experimentally, te conductance GG of te wire sould cange in steps equal to 2ee2, eac corresponding to te removal or addition of a discrete semi-classical electron state witin te energy range VV 1 VV 2. If te conductance is quantized, ow do we observe values over a continuous range, from 0 to in classical pysics? We can model te classical case by connecting tese cannels, eac wit a conductance GG = 2ee2, in combinations of series and parallel circuits (te cannels are connected in parallel for te case of a wire wit a constriction in te middle). Quantum effects in materials at room temperature usually require special conditions because te small depasing time immediately wases tem away. Tis experiment is a remarkable exception. It works because te electrons experience effectively no scattering on traversing te constriction. Page 1
2 VV 1 = +1.3 VV (a) (b) VV 0 RR 0 LHS contact VV 2 = 0 VV RHS contact RR ww DAQ Constriction Fig. 1: (a) Sketc sowing te discrete number of levels (dased lines) inside te constriction between te LHS and RHS contacts. (b) Scematic of te circuit. 2 Experimental setup Steps in conductance correspond to a resistance cange RR = = 12.9 kkω. Tis value is 2ee2 neiter very large nor very small, allowing te use of a simple circuit [Fig. 1(b)] and of a standard Data Acquisition card for measuring te voltage across te wire. Observe te score in te wire wit te microscope. Te wire as been scored in te midsection wit a sarp knife, to create a weak link, at wic te constriction will occur. Te epoxy drops prevent te wire from sliding wen tensioned. A useful feature of tis experiment is tat te wire can be broken and connected repeatedly. Au as been cosen because it does not oxidize quickly and a good electrical connection can be reformed by bringing te wires into contact. Exercise Given te battery voltage VV 0, te external resistor RR 0 and wire RR ww resistance [Fig. 1(b)], te measured voltage across te wire is VV = RR wwvv 0. Sow tat te measured conductance is RR ww +RR 0 GG = VV 0 VV. VVRR 0 Page 2
3 3 Experiments 3.1 Observing te breaking and re-connection of te wire Open Quantum_conductance.vi wit Lab View 2011 (not te 2015 version) and ceck te Dev1/ai0 (Dev1=USB card, ai0=te card ports used) are selected. To verify tat te wire is good, select 1000 Hz rate and 1000 samples and run te vi. Coose do not save, wic will continuously update for te voltage measurements across te wire every 1 sec. Te reading sould be approximately zero, meaning tat only a small voltage drop across its ends is measured and te wire is not broken. Carefully remove te metal plate wit te wire on it from between te two posts. Te voltage sould jump to te battery voltage VV VV if te circuit is connected correctly. Re-insert te support plate so tat te micrometer point is on te opposite side of te wire and approximately against te wire section in-between te two epoxy drops. Slowly move te micrometer to smaller values. Tis bends te plate and tensions te wire. At about 20 mm micrometer reading, te voltage sould jump from 0 VV to 1.3 VV. Slowly move te micrometer in te vicinity of tis position and see te voltage jumps occur as te wire is broken and re-connected. Note te micrometer range over wic te breaking occurs. It sould be only a few tens of μμμμ, corresponding to a few divisions on te micrometer rotating sleeve. Eac jump in voltage corresponds to a cange of nn from 0 to 50,000. We want to see discrete steps wit Δnn = ± Measuring te conductance steps You are now ready for a more accurate measurement. To see te steps Δnn = ±1 in conductance as te wire is broken and re-connected, it is necessary to measure te voltage at a faster rate. Set te sample rate to 50,000 Hz and te total number of samples to 500,000. Te data file will be about 20 MB (be careful not to make te data file too large, wic can easily be done at tis rate). Once te vi is started, you ave 10 s to move te micrometer 2-3 times about te range you determined before. Try to break and re-connect te wire multiple times as you adjust te micrometer (te results can be seen only at te end of tis 10 sec period). If te results look good (at least one jump visible) save tem into a text file. Import to an Origin workseet wit Import Multiple ASCII to plot (turn off Speed mode ) and furter analyze te results. Te most interesting range is were te voltage starts to deviate from zero. Zoom in tat range to see te steps, several of wic can be seen in a good data set. It may be necessary to do a few measurements to obtain a good data set tat sows te steps clearly. Page 3
4 Note: you can score a new wire and measure its conductance as an optional part if you like. Look for te several additional wires glued to teir plates. Use te sarp knife to score te wire (please keep te cover over te blade wen not in use, to protect it). Sligtly grazing te wire wit te knife sould be enoug. Examine te result under te microscope. Write your name on te paper support, so tat oters will know tat te wire as been scored already. Note: do not go too far from te wire breaking point because it migt not reconnect. Prepare a new one if you break te wire and it does not re-connect. Note: at te end of te experiment disconnect te battery and leave te wire in te connected position (a small voltage, corresponding to a large conductance). 3.3 Analysis From te battery voltage VV 0, te measured VV, and te resistance RR 0 = 160 kkω, calculate te conductance GG and compare te size of te observed steps to te standard result 2ee2 = SS. 4 Conclusion Tis experiment does rely on a dimension being small in order to obtain quantum effects. It may be contrasted to te quantum ligt experiments (next), for wic no small dimension can be identified. Conductance measurements can be a powerful metod of investigating quantum effects in solid-state materials. Tis may be illustrated by te integer and fractional Quantum Hall Effects, done at small temperatures, in two-dimensional geometries and in strong external magnetic fields. In particular, te IQHE is te most accurate way of measuring. It gives te ratio ee 2 wic, togeter wit a separate measurement of te magnetic flux quantum 2ee (applying te Josepson Effect in superconductors), allows obtaining bot and ee. Page 4
5 5 Appendix Te relation GG = 2ee2 nn is easier to derive in te weak-potential point of view, wic also corresponds more closely to te case of a good metallic conductor. Conduction in a metal occurs troug occupied electron states at te Fermi energy. Finding te 3D density of states at te Fermi energy in metals is a complicated problem because te crystalmomenta kk xx,yy,zz depend on te specific band structure. Te problem is greatly simplified in te region of te constriction. Tere, kk xx,yy are determined by te potential well, wile te kk zz crystal momentum is approximately equal to a free-electron kinematic momentum pp zz because te electrons experience effectively no scattering on traversal. Specifically, te electron wavefunction is ψψ(xx, yy, zz) = AA sin nn xxππππ LL xx sin nn yyππππ LL yy ee ii2ππkkzzzz ħ. Te kinetic energy along te wire is EE kk = (2ππkk zz )2 = mmvv 2 zz kk 2mm 2 zz = mmvv zz, or te crystal and kinematic 2ππ momenta are equal. Consider first only one pair of (nn xx, nn yy ) numbers. Te density of states in eac band is ddkk zz = DD(EE) = 1, corresponding to te free-electron states propagating along ddee 2ππvv zz ħ te wire. Te current is II = eeeeee(ee FF )eeee = ee2 vvvv = ee2 ee2 VV or te conductance is GG =, wic 2ππππħ must be increased to GG = 2ee2 accounting for te two possible electron spin orientations. Te result for GG does not depend on te (nn xx, nn yy ) numbers. Even toug tese free-electron electron bands start at different levels because of different (nn xx, nn yy ) (as sown in te figure), electrons in eac of tese bands contribute te same amount GG = 2ee2 to te conductance at EE FF! We get 2ee2 from eac band, provided te bottom of te band is below EE FF.Terefore, GG = 2ee2 nn, were nn is te number of nn xx, nn yy pairs tat give a state wit EE < EE FF at kk zz = 0. In te strong-potential point of view, wic corresponds more closely to a molecule placed between two electrical contacts, nn is te number of states wit te energy between te left and rigt contacts, and te conductance is viewed as a sum over te possible tunneling cannels, or GG = 2ee2 equation). Page 5 nn TT nn, were eac cannel as a transmission coefficient TT nn (te Landauer
6 Name Pys-601 Quantum Mecanics Laboratory I Quantized conductance lab report Date: Page 6
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