2. Electronic states and quantum confined systems

Transcription

1 LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES 2. Electronic states and quantum confined systems Nerea Zabala Fall From last lecture... Top-down approach------update of solid state physics Not bad for many metals and doped semiconductors Shows qualitative features that hold true in detailed treatments. Successive approximations: - Free particles no external potential - Independent electron approximation - Assumes many-particle system can be modeled by starting from singleparticle case. Beyond these approximations...density Functional Theory (DFT), Quantum Montecarlo... First, find allowed single-particle states and energies... 2

2 Contents: LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES 2.Electronic states and quantum confined systems Electrons in solids: approaches Independent electrons Electrons in a 1d box: confinement 3D electrons gas. Filling states. The density of states 2D electron gas Electrons in 1D Quantum dot DOS in 3, 2,1D Crystal structure and effective mass approximation. Semiconductors Quantum size effects Some useful confining potentials Summary 3 Electrons in solids: approaches Metal and conduction electrons free atoms a solid valence electrons nuclei core electrons Pseudopotentials, jellium models Pseudopotentials, Jellium models Ions smeared out into a positive background 4

3 Independent electrons Time-independent SchrÖdinger equation: 2 2m 2 Ψ + V ( r)ψ = ɛψ Solve, consistent with boundary conditions. Free! V= 0 Solutions, electron wave functions: Ψ = Ae i k r + Be i k r, traveling plane waves with wave vector k ɛ(k) k =2π/λ and λ is the wave length Energies : ɛ(k) = 2 k 2 2m No restriction for allowed values of k or E, continuous. k 5 Electrons in a 1d box: confinement 1d particle in a box potential: 0,0 < x < L V = ), x ( 0; x ' L Infinite wall potential V Inside eht box, V= 0, so solution must look like superposition of plane waves, but " must vanish at walls. Allowed k values: 2 " # n ( x) = sin nx L L n = 1,2,3 o, allowed k values are n" k =, n = 1,2,3 L V 0 L k " n Allowed energy values E( k) = = 2 2m 2mL L

4 Electrons in a 1d box: confinement Finite sample size drastically alters allowed energy levels!! Energy spectrum is now discrete rather than continuous. Dispersion relation plot ɛ(k) ɛ(k) ɛ(k) k k Energy level diagram (allowed energy values) Energy Energy! Allowed wavevectors are uniformly spaced in k-space with a separation of #/L. k-space plot (allowed k values) k[π/l] k[π/l]! Sample size L determines spacing of allowed wavevectors and single-particle energies, with a smaller box giving larger spacings. 7 3D electrons gas. Filling states. The density of states Non-interacting many-electron systems Interested in ground state of many-electron system. No interactions $many-body eigenstates should be linear combinations of products of single-particle eigenstates. They obey Pauli principle $correct total wavefunction should be antisymmetric under exchange of any two particles. Many-body eigenstates should be linear combinations of Slater determinants built out of single- particle eigenstates. Approximation (or shorthand): start filling each single-particle state from the lowest energy, each with one spin-up and one spin-down electron. 8

5 3D electrons gas. Filling states. The density of states ( ) 2 2 2m x y z 2 Ψ k ( r) =ɛ k ( r)ψ k ( r) Ψ k ( r) = 1 V ei k r traveling waves and energies Boundary conditions: Confined in a cube of size L! Ψ k =0 at the boundaries ɛ k = 2 k 2 2m! allowed momentum values and standing waves k x = ± 2πn x L x,k y = ± 2πn y,k z = ± 2πn z L y L z n x,n y,n z =0, 1, 2, D electrons gas. Filling states. The density of states Filling states: ɛ F = 2 k 2 F 2m Fermi energy k F = 1 2mɛF Fermi momentum/velocity Counting states: 1 state (2π)3 V volume in k space Fermi sphere k V (2π) 3 d k N =2 V (2π) 3 For spin kf 0 4πk 2 dk = V 3π 2 k3 F Electron density n = N V = k3 F 3π 2 10

6 3D electrons gas. Filling states. The density of states Fermi energy and momentum increase with density of electrons! ɛ F = 2 2m (3π2 ) 2/3 n 2/3 p F = k F = (3π 2 n) 1/3 All single-particle states with below the Fermi energy are occupied at T=0. These states are called the Fermi Sea. The set of points in k-space that divides empty and full states is the Fermi surface. Exactly how the Fermi energy depends on density depends drastically on dimensionality. Usually dimensionless parameter, radius of sphere containing one electron: 4π 3 (r sa 0 ) 3 = 1 n Bohr 11 3D electrons gas. Filling states. The density of states Some values (Kittel) 12

7 3D electrons gas. Filling states. The density of states Some numbers for Cu Hall measurements in macroscopic samples yield the electron density of Cu (it can be also estimated from from interatomic distances) : n = m 3 Calculate k F, λ F, ɛ F and v F Note that the Fermi velocity is less than a percent of the velocity of light, so no relativistic treatment is needed. 13 3D electrons gas. Filling states. The density of states The density of states (DOS), The number of allowed single-particle states with energies between E and E+dE,in an element of length/area/volume.! From our expressions for n(e), nd is the spatial electron density in d dimensions ɛ F = 2 2m (3π2 ) 2/3 n 2/3 n(ɛ) = 1 3π 2 ν(ɛ) = dn dɛ ν(ɛ) = 1 2π 2! Higher DOS means levels are more closely-spaced. ( ) 3/2 2m ɛ 3/2 2 ( ) 3/2 2m ɛ 1/2 2 ν(ɛ) ɛ! From this definition, we can find the spacing of single particle levels in a piece of material! 14

8 3D electrons gas. Filling states. The density of states Some numbers...! Estimate the energy level spacing in 1 cm 3 of Na Consider one valence electron per Na atom. We use the density and fermi energy from the table n cm 3 ɛ F =3.2eV = J ν(ɛ F ) J 1 m 3 Energy level spacing: ɛ =1/ν(ɛ F )V J The particle energy levels are continuous 15 3D electrons gas. Filling states. The density of states Some numbers...!now suppose we have 1nm 3 of Na, instead A similar calculation yields the energy level spacing: ɛ 3 mev This is actually measurable! At low temperatures, the individual electronic levels in a piece of metal can dominate many properties, something that doesn t happen at macroscopic sizes 16

9 3D electrons gas. Filling states. The density of states The density of states of the free electron gas at finite temperature! Fermi-Dirac distribution function (fermions) T =0 f (!) = T!= 0 1 e(! µ)/kt + 1 n=! n(!)f (!, T )d! T 0 f (!) θ(!f!), step function Quantum Well States (QWS) and Quantum Size 2D electron gas Effects Qualitative D explanation λf kz k Confinement in z direction, free in x and y Paraboloidal subbands!k! = (kx, ky ) D! Ψn,!k! (!r) = yφn (z)eik!!r! kx!3 (!k! )2!nElectronic ("k) =!n + structure 2m in parabolic subbands Strictly 2D if!1 <!F <!2!F!2!1 ik x x ik y y " ( x, y, z ) =! n ( z )e e In infinite well confining potential (1D): 2!! nπ "22 2 k 2 + k 2!n =E (n, k x, k y ) =! n2 + x y 2m D 2 D 2 4 Discrete continuous 18

10 2D electron gas Filling states in 2D, occupation of subbands Periodic boundary conditions!allowed Consider T=0 N 2D = 2 k x = ± 2πn x, k y = ± 2πn y L x L y nfilled L 2 kf (2π) 2 2πkdk = 0 Density (per surface) n 2D = 1 state (2π)2 surface nfilled k 2 F 2π L 2 nfilled 1 2π L2 k 2 F As a function of energy: n 2D = (ɛ F ɛ n ) m π 2 nfilled 19 2D electron gas Density of states in 2D: ν(ɛ) = dn dɛ ν 2D (ɛ) = m π 2 θ(ɛ ɛ n ) Infinitely deep square well (GaAs, D=10 nm), energy levels n 2D (ɛ) = nfilled Subbands, transverse kinetic energy (ɛ n ɛ) m π 2 n step or Heaviside function Steplike DOS of a quasi 2D system Parabolid density of states for unconfined 3D electrons m π 2 20

11 Electrons in 1D Further confinement (in 2D), x and y : quantum wire or electron wave guide Ψ m,n,kz ( r) =Φ m,n (x, y)e ik zz Parabolic subbands N 1D =2 m,nfilled 2 L 2π kf 0 ɛ m,n (k z )=ɛ m,n + ( k z) 2 dk =2 m,nfilled L π k F 2m L x,l y λ F z Density (per unit length) n 1D =2 m,nfilled k F π As a function of energy: (Use the dispersion relation for each subband) n 1D =2 m,nfilled 2m(ɛF ɛ m,n ) π 21 Electrons in 1D Density of states in 1D: 2m(ɛ ɛm,n ) n 1D =2 π m,nfilled ν 1D (ɛ) = (2m) 1/2 (ɛ ɛ m,n ) 1/2 θ(ɛ ɛ m,n ) π m,n DOS of a quasi 1D system, GaAs, 9!11 nm infinitle deep well Parabolid density of states for unconfined 3D electrons 22

12 Electrons in 0D Further confinement (in 3D): quantum dot or artificial atom Discrete energy levels, as in atoms L x,l y,l z λ F DOS is just asum of delta functions ν 0D (ɛ) = 2 j δ(ɛ ɛ j ) discrete eigenenergies of the system ν(ɛ) ɛ 23 DOS in 3, 2,1D Quantum Confinement and Dimensionality 24

13 Crystal structure and effective mass approximation. Semiconductors Electrons in periodic potential (Nearly free electron model) ion core Electrons in a periodic potential periodic potential Standing waves Probability density for standing waves produced R V ( r) =V ( r + R) Bloch's theorem: ψ n k ( r) = exp(i k r)u n k ( r) n: band index Bloch wave functions u n k ( r + R)=u n k ( r), Ψ k ( r + R) = e i k R Ψ k ( r) Lattice vector 25 Crystal structure and effective mass approximation. Semiconductors Two wave vectors and the solutions of Schrödinger equation are related to each other. This leads to equal eigenvalues ɛ n ( k)=ɛ n ( k + K) and equal wave functions ψ n k ( r) =ψ n k+ K ( r). Each energy branch has the same period as the reciprocal lattice. As the functions are periodic, they have maxima and minima which determine the width of the bands. The wave vector k can always be chosen in a way to belong to the first Brillouin zone because k = K + k e i K R =1 Reciprocal lattice vector 26

14 Crystal structure and effective mass approximation. Semiconductors Exercise: Solve 1D periodic square-well potential model: Kronig-Penney Kittel 27 Crystal structure and effective mass approximation. Semiconductors Gap opening First Brillouin zone The effective mass approximation: m* Nearly quadratic (valid for low electron momenta) 28

15 Crystal structure and effective mass approximation. Semiconductors Occupation of energy bands: type of materials Insulator Metal or semimetal (if band overlap small) Metal Sketch 29 Crystal structure and effective mass approximation. Semiconductors Effective Mass from band dispersion m e = h 2 2 ɛ/ k 2 30

16 Crystal structure and effective mass approximation. Semiconductors Band structure of some semiconductors Ge Si GaAs 31 Crystal structure and effective mass approximation. Semiconductors Positive and negative effective mass: Negative m*, holes 32

17 Crystal structure and effective mass approximation. Semiconductors In summary: we can consider also semiconductors including m* in the Schrodinger equation. Then, for the density of states in 0,1,2,3D one can extrapolate the results ν 0D =2 j δ(ɛ ɛ j ) ν 1D = (2m ) 1/2 (ɛ ɛ m,n ) 1/2 θ(ɛ ɛ m,n ) π m,n ν 2D (ɛ) = m π 2 θ(ɛ ɛ n ) ν(ɛ) = 1 2π 2 n ( ) 2m 3/2 ɛ 1/2 2 Signature of dimensionality 33 Quantum size effects Confinement!Discrete Quantum Well States!Oscillations in the physical properties (as energy, Fermi level...) as a function of size 34

18 Quantum size effects Condition of QWS existence L = 2 D =! n n =1,2,3, n E = " n = 1,2,3,... 2D 2 # F Energy of system (per electron) 35 Quantum size effects An example: magic heights of Pb islands on Cu(111) studied with STS Building island heights histograms Number of islands Covered area Courtesy: Rodolfo Miranda R. Otero, A. L. Vázquez de Parga and R. Miranda PRB 66, (2002) 36

19 Quantum size effects 1D potential model (self-consistent calculation)? Very good agreement with the experiments. Shell and supershell structure like for nanowires and clusters. It can now be observed indirectly in the experiments. Ogando, Zabala, Chulkov, Puska, Phys. Rev.B 69, (2004) 37 Quantum size effects Find other examples of quantum size effects in the literature 38

20 Some useful confining potentials Square well of finite depth -quantum wells, thin slabs- Parabolic well -quantum dots- Triangular well -heterojunctions- Cylindrical well -quantum corral, metallic nanowire- Spherical well -clusters, quantum dots- See for example J.H. Davies Also in next lectures 39 Some useful confining potentials Triangular well -heterojunctions- Confining potential (z perpendicular to 2DEG) Introduce dimensionless variable equation: solutions with boundary conditions (finite at infinity an zero at z=0) 40

21 Some useful confining potentials solutions with Airy functions Subbands m, effective mass 41 Summary The effects of confinement have been studied qualitatively starting from noninteracting electrons confined in potential wells. Confinement produces discreteness of allowed electron energies giving rise to quantum size effects. The characteristic pattern of the density of states in one, two and three dimensions has been obtained. The conclusions are valid for metals and semiconductors, when the effective mass approximation is considered. Electron interactions in low D have not been considered but they may very important in many problems, for example to explain superconductivity, magnetism, quantum Hall effect... In low D screening, response etc... is different In another course: phonons, plasmons, excitations in low dimensions 42