Stalking the Exciton Condensate: Exotic Behavior of Electrons Under Extreme Conditions Tiger pictures: Microsoft Online Clipart, ok for public display (so long as work is neither obscene nor scandalous) Melinda Kellogg Jim Eisenstein Ian Spielman Loren Pfeiffer Ken West Everhart Lecture March 10, 2004 Caltech

Extreme Condition #1: Reduced Dimensionality 2-dimensional electrons

Two-Dimensional Electrons Electron motion is constrained to only 2 dimensions Two-dimensional electrons exist: At the surfaces of metals At the surface of liquid Helium For IBM STM image ref: http://www.almaden.ibm.com/vis/stm/library.html#ref3 e - Crommie et al., Nature 363, 524 (1993) AND inside specially engineered semiconductor crystals . . .

Molecular Beam Epitaxy heated cells Al Ga As AlGaAs GaAs 100 A ultra high vacuum allows for precision engineering of crystal layer by layer RHEED: Reflection High-Energy Electron Diffraction Tsao, p. 61: knudsen regime, vapor sublimes directly from solid phase (though I know from Loren, that the Gallium (with it’s melting point just above room temp) is a liquid when in the cell. high quality GaAs substrate

- e - e - e - e - - e e AlGaAs GaAs Double Quantum Well GaAs AlGaAs energy There is more electron density at the more electronegative atom in the bonded pair. In GaAs, As is more electronegative than Ga. The difference In electronegativity makes the bond more ionic, and thus stronger. The bond strength has relation to the band gap – the greater the bond strength, The larger the band gap. AlAs has a larger electronegativity difference. These are covalent bonds, or alternately valence bonds. The nearest neighbor distance between Ga and As is 2.4480 Angstroms. The lattice constant is: 5.6533 Angstroms. Si 100 A Quantum Well

Bilayer Two-Dimensional Electron System

e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e -

0.5 mm

1.5 mm

10 mm This is mounted on an 18 pin DIP (dual in-line package) header. This is a conventional electronics component (often used with computer boards?)

Extremely Low Temperatures Extreme Condition #2: Extremely Low Temperatures Snow and tree are microsoft clipart

5o C 40 F = 5 Celcius

15 mK 1 meter -459.643 degrees Fahrenheit (absolute zero is -459.67). 15mK is ~1/40th a Fahrenheit degree above absolute zero (1 Celsius degree = 1.8 Fahrenheit degrees) 1 meter

Very Large Magnetic Field Extreme Condition #3: Very Large Magnetic Field Real magnet from “Dowling Magnets”.com. Typical strong horseshoe magnet: 0.15 Tesla Earth’s field: ½ gauss (1/20,000 of 1 Tesla)

13 Tesla 13 Tesla is a hundred times stronger than a good Horseshoe magnet. (Nb3Sn – T_c = 18K)) liquid helium, which is at 4 Kelvin (~7 fahrenheit degrees above absolute zero). Nb3Sn’s critical field is 20 Tesla (thus it can be used in a solenoid magnet) (I also read 8.8Tesla – but the field probably doesn’t get that high at the winding locations). From Oxford, L = 50 Henry, inner diam = 6.4mm, 90Amps = 13Tesla, and length of magnet is 0.66m. Using two different equations: B = N u_o I/length; and L = u_o N^2 area/length – both give the number of windings at ~ 100,000. The Bell labs paper studying NiSn for this application was dated 1961 – so maybe these were available commercially in the 70’s? 1 meter

Hall Effect B Lorentz Force: Hall voltage magnetic field velocity voltmeter e - vdrift Shockley of bell labs invents transistor 1947, MOSFETs become commercially available in the 60’s I think. Lorentz Force: magnetic field velocity electric field

I Hall Effect B Hall voltage voltmeter e - e - e - e - e - e - e - e - voltmeter e - e - e - e - e - e - e - e - e - e - I e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - B

Hall Effect VHall Know that lower density leads to higher Hall voltage for the magnetic field cause lower n means faster v and thus stronger v-cross-B force. B

Quantum Hall Effect VHall von Klitzing 1980 This value is so exact that this ‘quantum hall effect’ is used by the National Institute of Standards as the ‘standard of resistance’. Von Klitzing is German, but he was working in Grenoble, France when he made this discovery (according to a Aug 2003 Physics Today article on topology and the QHE). His data was taken at 1.5K. von Klitzing 1980

√ √ √ √ √ B wave nature of electrons h m v e B 2 p m e B 2 p v m h = 3 - Planck’s constant de Broglie wavelength 6.63 x 10-34 Joule · sec e B 2 p m B √ √ e B 2 p v m h = 3 √ √ e B 2 p v m h = 7 √ √ √ e B 2 p v m h = 1 v 2 = f l 2 Also, know that deBrogle’s equation was just taking the relationship between a photon’s wavelength and momentum and applying it to matter. Also, know what v *is* for the lowest energy case at B = 1 T (66,666 m/s), also know the circumference (1,612 Angstroms – or 1.6 x 10^-7 meters or 1/10,000th of a millimeter). Also know, just to order of magnitude, also in MKS units, the mass of an electron (10^-30 kg), and the typical fermi velocity of an electron (10^5 m/s), to show that mv is not nearly so small as h. the cyclotron frequency (not omega!) at B = 1T (4 x 10^11 cycles/sec). (know: smaller B is, the bigger the lowest energy circle is – (smaller centripetal v-cross-B force)).

√ √ Pauli Exclusion Principle Heisenberg Uncertainty Principle e B p h ~ ≥ D D 2 √ p 1 √ h e B p ~ pRMS D = 2 2 √ p For delta-p, take p_rms, (no explanation beyond this) which is 1/sqrt(2) of p (people in the know will get the rightness of this). NOTE: most textbooks give delta_x delta_p ~ h/4 pi -- but Heisenberg’s original paper (check actual journal?) uses h/2 pi. Possibly again make some general comment how this is also a “hand waving semiclassical derivation” that gives a good feel for it, and in this case happens to agree with the full quantum formulation. Again, be sure that you explain each of the terms – ‘p’ especially will not be immediately understood.

√ ( ) e B p h h ( x ) ~ p e B e B h = x p ( ) 2 √ e B p h h ( x ) 2 ~ D p e B e B h 1 none Landau level = x p ( ) 2 D Landau was first to calculate the energy levels of an electron in a magnetic field, he references his 1930 paper (Z Phys v.64, p629, I think) in his non-relativistic quantum mechanics textbook, in the chapter on particles in a magnetic field. “one f illed Landau level” “second Landau level”

B e n B e 2 h VHall von Klitzing 1980 I

VHall 1 B 3 e n B e h I 2 von Klitzing 1980 Know: the reason he saw them was: using 2D electrons, but also, higher perfection in the crystal and lower temperatures both reduced the scattering rate, which allows multiple times around the orbit, so that the wave nature and constructive interference effects could dominate.

e - ? ? - ? e ? ? e - ? ? e - e - e - e - ? ? - e - ? ? e e - e - e - There are actually ~ 1 million slots available in each layer in my ‘working region’.

½ Classical Spin Spin h = x 2 π Top and photographer from Microsoft Clip Art. Billion trillion: ½ h_bar ~ ½ x 10^-34 J s. L = mv x r, if all m located distance r from center, where r ~ 1cm, and m ~ 100 gm. If I solve for v, I get v = ½ x 10^31 m/s. To traverse 2 pi r takes 1.2 x 10^30 s. 1 year ~ 3 x 10^7 s. (interesting, if you solve for a proton, m = 10-27 kg, r ~ 10^15 m, it goes around a billion trillion times each second! Ways to measure: the Stern-Gerlach experiment, so review that. U dot grad B. The electron was the first elementary particle whose spin was detected (p.329 of Greiner)

Quantum Spin

Make sure last arrow is an up one (flip it).

Spin ½ = + = +

Spin ½ = - = -

Pseudospin e - e - e - e - e - e - e - e - e - e - e - e - e - e - - e At our density (2.3 x 10^10 cm^-2 per layer), the orbital states are ~ 470 Angstroms away from one another, whereas the layers are 280 Angstroms, so it would be better for them to all be in one layer in a sense.

+ – = Superposition e - e - e - e - e - e - e - e - e - e - e - e - e Einstein’s letter was to Max Born. According to ‘Frontiers’, by Steve Adams. = + –

12 9 Fluctuations e - - e e - e - - e - e - - e - e - e - e e e - e - Basic example: random decay of radioactive particle. BRING OWN SPEAKERS FOR THIS ON REHEARSAL DAY.

14 7 Fluctuations e - - e e - e - - e - e - - e - e - e - e e e - e -

10 11 Fluctuations e - - e e - e - - e - e - - e - e - e - e e e - e -

Tunneling - e Ian Spielman V Why this current is suppressed when there is a bias voltage: There is kind of an energy inequality, if you put in electrons at one side with a higher voltage, and taking them out of the other side with a lower voltage, you are putting more energy into the system than you are taking out. Ian Spielman

Another way to look at it… The fluctuations discussed in the previous ‘view’ correspond to fluctuations in the *number* of excitons… = hole = electron

Fermions: Bosons: e - e - e - e - e - e - e - e - half-integral spin; electrons, protons, neutrons… obey Pauli exclusion principle Bosons: zero or integral spin; photons, pions, Helium-4 atoms, paired fermions, sodium atoms… don’t obey Pauli exclusion principle e - e - e - e - e - You have drawn a Boron atom, 5 protons, 5 or 6 neutrons are both stable isotopes. And 2 electrons in the 1s, 2 electrons in the 2s, and then the 5th in the 2p is right. ‘even at room temp’ is implied and true. Possibly add a picture of periodic table, and When if bosons comes, put a big circle and slash through it. Periodic table from los alamos educational website: http://pearl1.lanl.gov/periodic/downloads/main.html e - e - e -

Durfee & Ketterle, Optics Express 2, 299 (1998) Bose-Einstein Condensates (BECs) superfluid helium superconductor Superconductivity discovered first, in 1911, by Onnes, who liquified helium first. Onnes did get a nobel prize in 1913, but mostly for the liquification of gases. Superfluid helium was discovered in the late 1930s by many workers, including Onnes, Kapitza, and others. They would each notice one thing at a time…_this was in 1937/8). Superfluid He-3 happened in the 70s, and won the 1996 nobel). Atomic BEC happened in 1995 and won the 2001 nobel. Superfluid and superconductor pics from http://teachers.web.cern.ch/teachers/archiv/HST2001/accelerators/superconductivity/superconductivity.htm. Helium-4 goes superfluid below about 2.17K. Fountain effect: in superfluids heat is transferred not through diffusion, but through tremendous convection currents – where the pure ground state component of the fluid rushes *toward* the heat source (and thus it’s momentum/pressure pushes it out the top of the fountain head), and the regular fluid usually moves away (but is prevented from doing this by the superleak). So, superconductor’s expel magnetic fields, so they sort of work like a mirror to the magnetic field of the magnet above, if the north pole is facing down, the superconductor will act like a north pole facing up, and repel it – this works for all orientations. Know size of that central blob in the atomic BEC picture (the full picture is 1mm across (so central blob~1/10th of that…), there are according to an Optics Express 1998 paper 70,000 atoms in picture, the transition temperature is 2uK. This is the first paper in which I found this picture used. It appears that the *only* natural isotope of Sodium is 23Na (all the others are radioactive with short to relatively short lifetimes). It’s the total spin that matters. In sodium-23 (the common one?), Z=11, these are automatically balanced out by the electrons (11) – any even number of spin ½ fermions, always adds up to integral or zero spin. So, it’s the neutron number that decides whether the atom is a fermion or a boson – sodium has 12 neutrons, so it’s a boson. It’s total nuclear spin is 3/2, and it’s total electron spin is ½. All the electrons are in filled shells, but one, which is in an s orbital. The electron and nuclear spin interact with one another, this is the hyperfine interaction – I am not sure what it’s used for. BEC of sodium atoms Durfee & Ketterle, Optics Express 2, 299 (1998)

Old-Fashioned Excitons The gap is 1.424 eV. I calculated that to be a red photon (8,750 Angstroms). (so, it’s actually infrared). (This was confirmed in an LED webpage.)

Stable Excitons = hole = electron

I h e - I

I I e - e - e - e - e - e - e - e - h e - e - e - e - e - e - e - e -

voltmeter e - voltmeter voltmeter I voltmeter I

Exciton BEC? I I ½ filling ½ filling 3 2 1 2.0 1.5 1.0 0.5 0.0 20 15 2.0 1.5 1.0 0.5 0.0 20 15 10 5 2.0 1.5 1.0 0.5 0.0 ½ filling per layer ½ filling per layer Hall Voltage (microvolts) Longitudinal Voltage (microvolts) Data: 27Jun03a080, and 27Jun03a.079. I don’t need to bring up the superfluid’s zero critical current, cause I am not showing any of the arrhenius data – possibly keep arrows – remove red BECs? Magnetic Field (Tesla) Magnetic Field (Tesla) M. Kellogg (2004)

The End acknowledgements: