{"product_id":"landaura-conductance-quantum","title":"Landaura – Conductance Quantum \/ G = 2e²\/h Physics Sticker","description":"\u003cp\u003eLandaura, a canyon-shaper titaness of The SI Guard, widens a gorge one lane at a time, and the current climbs in equal steps.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = (2e²\/h) Σ\u003csub\u003en\u003c\/sub\u003e T\u003csub\u003en\u003c\/sub\u003e,   G\u003csub\u003e0\u003c\/sub\u003e = 2e²\/h ≈ 7.748 091 729 × 10⁻⁵ S\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eG: conductance (siemens, S)\u003c\/li\u003e\n\u003cli\u003ee: elementary charge, exact in the 2019 SI\u003c\/li\u003e\n\u003cli\u003eh: Planck constant, exact in the 2019 SI\u003c\/li\u003e\n\u003cli\u003eT\u003csub\u003en\u003c\/sub\u003e: transmission probability of mode n (0 to 1)\u003c\/li\u003e\n\u003cli\u003e2: spin degeneracy, two spin states per lane\u003c\/li\u003e\n\u003cli\u003eG\u003csub\u003e0\u003c\/sub\u003e: conductance quantum, exact (CODATA 2022)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThis is the Landauer formula, after Rolf Landauer. Because e and h are fixed, G\u003csub\u003e0\u003c\/sub\u003e is exact: 1\/G\u003csub\u003e0\u003c\/sub\u003e ≈ 12 906.4 Ω, which is half the von Klitzing constant R\u003csub\u003eK\u003c\/sub\u003e = h\/e².\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The resistance of a perfect lane is not caused by collisions in the channel; it arises where the narrow channel meets the wide contacts. Clean plateaus need low temperature and a ballistic channel; real point-contact steps are accurate only to about 1%, unlike the quantum Hall effect, so G\u003csub\u003e0\u003c\/sub\u003e is not used as a resistance standard. The factor 2 disappears if a magnetic field lifts the spin degeneracy.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eLandaura casts a green beam down the canyon of gate-rock as she pushes the walls apart. Printed text: \u003cstrong\u003eLANDAURA\u003c\/strong\u003e with the line \u003cstrong\u003eG = 2e²\/h\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963707953255,"sku":"30104295243986231218","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963707986023,"sku":"22439318150544931193","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963708018791,"sku":"25623734241245962692","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963708051559,"sku":"32828625323487860878","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/16944926883350686484_1200.jpg?v=1793664931","url":"https:\/\/pixelatedphysics.com\/pt\/products\/landaura-conductance-quantum","provider":"Pixelated Physics","version":"1.0","type":"link"}