Introduction to Nanotechnology. Chapter 9 Quantum Wells, Wires and Dots

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1 Introduction to Nanotecnology Capter 9 Quantu Wells, Wires and ots

2 Surface issue: For a nano-eter cube, te surface to volue ratio increases wit decreasing size For GaAs, a.565 n As Ga 1n contains Ga and As For an FCC cubic: N surface 1 n N volue 8n + 6n + n N volue n nuber of atos along edges d na, a lattice constant N surface size d in n 6 N surface /N volue in% size d in n 6

3 Sec. 9.. Feri Gas and ensity of States Classical description: oentu p v, kinetic energy v / p / Quantu description: p x k x All conduction electrons are equally spread out in te k space (reciprocal space C 9.., Page, table 9.5, A dien sions coordinate region spacing in k Feri region nuber of electron N( Feri region/spacing in k ensity of states ( d N( d 1 engt / k F k F Area (/ k F k F ( Volue (/ k F ( k F ( 1

4 Confined electron wavefunction in a infinite square well x x x sin, n, odd, a n 16 q et a, not to consider spin n a n kf a n a 1 n 1 c.f. 1 in Table A ψ n cos(nx/a n 1,, 5, even parity ψ n (-x ψ n (x x cos, a n, even, 9 ψ n sin(nx/a n,, 6, ψ n (-x -ψ n (x odd parity x sin, a n, odd, Probability of finding an electron at a particular value of x Ψ n (x x cos, a n 1, even, x -a/ x x a/ C 9.., Page 8

5 egeneracy: nergy of a infinite rectangular square a n n n x + y ( egeneracy (including spin states : n eq. 9.9 n degeneracy n x, n y n x, n y n x, n y n x, n y 1,1 1,,,,, a 5 8,,5, 5,,, C 9.., Page 1

6 nergy levels for a 1 parabolic potential well 1 V ( x kx Fig. 9.1 ψ n ( ( x H x e n αx n 1 n ω ω k C 9.., Page

7 N( and ( in, 1, and wit subbands C 9..6, Page 6, table A i subband index dien sions type nuber of electrons N(, d i degeneracies C 9..6, Page ensity of states ( N( ( dot ΣdiΘ( iw Σdiδ ( iw 1 wire 1 Σd ( 1 i iw 1 Σd i iw 1 ( 1 well Σd ( i iw Σ d i bulk 1 Θ( x 1 if x ; Θ( x if x < δ ( x if x ; δ ( x if x ; δ ( x dx 1

8 fil wit tickness ; k quantized in unit of / energy quantized in unit of Θ p S p ( ( * Te density of states: For tick fils (large, is sall, and S ( approaces ( wit volue S 1/ / / / ( ( Crossover fro to Crossover fro to S 1/ / 1/ / ( ( for p, 9 S 1/ / 1/ / ( 9 ( for p, for p1, C. 6., p.1

9 Single electron tunneling A neutral isolated dot lectron Box C5 C C C6 C7 I C 1 C C V b C6 C5 C7 I C 1 C C g Vg An applied gate voltage can lift up island potential C Σ C 1 + C + + C 7 Inject electrons via a tunnel junction V b e electron reservoir pty continuous states Gap V e/c Σ Filled continuous states V e/c Σ e electron reservoir V e/c Σ g V V V b x g C C Σ

10 Additive energy for a nanoparticle C e /C; C r C K level spacing 1/( F V n 1 c - R T K e r 1% tunnel barriern ikarev review article, Fig.

11 Criteria for Coulob blockade beavior Coulob blockade isolated object Carging energy (e /C: lectrostatic energy associated wit carging/discarging an isolated object Criteria (for a well defined carge nuber : 1. to surount teral fluctuations C total capacitance seen fro te object. to surount quantu fluctuations electrical pats to carge/discarge te object e C 1 1 t e C >> k B T Sall C or ow T e ( RC R R 6kΩ R total resistance seen fro te object Parallel R K Material: any conducting aterials, including etal, seiconductor, conducting polyer, carbon nanotube,... Structure: any structures wit isolated objects accessible troug tunnel junctions Te siplest structure: Single lectron Transistor (ST

12 Single lectron Transistor C S I C lectrostatic energy (n (ne+ C g V g /C Σ C g C Σ C S +C +C g V S Vg V (n n n e/c Σ C n n -1 n - Source n1 n rain V S Vg C g V C C e /C Σ -e -e -1e 1e e e Q C g V g n n-1 n- n-

Various Single electron transistors: (1 All etal

(STMScaning Tunneling Microscope P. Radojkovic et al. J.

of Munic Space-carge region I S n -V b / +V b / dot:

groups ( Colloidal: Au or CdSe particles ( Sandwic

5n Al particles are isolated by tin Al O layer.c.ralp, C.

87 (1997 (Harvard Univ self-assbled olecular of ditiol

13 Various Single electron transistors: (1 All etal transistors: Al island ( STM: on Au particle (STMScaning Tunneling Microscope P. Radojkovic et al. J. Vac. Sci. Tecnol. B 1(, 19 (1996 (Tecnical Univ. of Munic Space-carge region I S n -V b / +V b / dot: w~1n, ~n lateral sizexn V g our work (and any oter groups ( Colloidal: Au or CdSe particles ( Sandwic structure: Al particle 5n.5n Al particles are isolated by tin Al O layer.c.ralp, C.T.Black and M. Tinka PR, 78, 1 p. 87 (1997 (Harvard Univ self-assbled olecular of ditiol olecules avid. Klein et al. Nature, 89, p.699 (1997 (awrence Berkeley National ab. Ronald P. Andres et al. Science 7,1 (1996 (Purdue Univ.

chiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n

chiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n zigzag armchair Three major categories of nanotube structures can be identified based on the values of m and n m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Nature 391, 59, (1998) chiral J. Tersoff,