quantum transport quantum transport devices based on
play

Quantum Transport Quantum Transport Devices Based on Devices Based - PowerPoint PPT Presentation

Quantum Transport Quantum Transport Devices Based on Devices Based on Resonant Tunneling Resonant Tunneling Reza M. Rad Reza M. Rad UMBC UMBC Based on pages 407- Based on pages 407 -422 of 422 of Nanoelectronics


  1. Quantum Transport Quantum Transport Devices Based on Devices Based on Resonant Tunneling Resonant Tunneling Reza M. Rad Reza M. Rad UMBC UMBC Based on pages 407- Based on pages 407 -422 of 422 of “ “Nanoelectronics Nanoelectronics and and Information Technology” ”, Rainer , Rainer Waser Waser Information Technology

  2. Introduction Introduction � Some general aspects of resonant Some general aspects of resonant � tunneling diodes will be discussed tunneling diodes will be discussed � RTDs RTDs can be considered as devices can be considered as devices � which are in active competition with which are in active competition with conventional CMOS conventional CMOS

  3. Electron Tunneling Electron Tunneling � Transfer Matrix Method Transfer Matrix Method � � Electrons have a wave like character Electrons have a wave like character � � In structures with dimensions in the range of In structures with dimensions in the range of � electron wavelength, quantum mechanical electron wavelength, quantum mechanical transport becomes relevant transport becomes relevant � One of these transport mechanisms is the One of these transport mechanisms is the � tunneling process tunneling process � Electrons can penetrate through and traverse Electrons can penetrate through and traverse � a potential barrier with a finite transmission a potential barrier with a finite transmission probability independent of temperature probability independent of temperature � In classical view electrons can overcome a In classical view electrons can overcome a � potential barrier only thermodynamically potential barrier only thermodynamically

  4. Electron Tunneling Electron Tunneling � Envelope function description of electron state : rapid Envelope function description of electron state : rapid � changing electron potential is approximated with an changing electron potential is approximated with an envelope potential envelope potential � Envelope function is based on effective mass Envelope function is based on effective mass � description of the band structure and leads to electron description of the band structure and leads to electron effective mass Schrö ödinger equation : dinger equation : effective mass Schr ⎡ ⎤ 2 1 h d d ⎢ − + Φ ⎥ Ψ = Ψ ( ) ( ) ( ) z z W z z * ⎢ ⎥ 2 ( ) ( ) dz m z d z ⎣ ⎦ Ψ ( ) : electron w ave function, W : electron energy is Z direction z z Φ * m : electron effective mass , (z) : potential energy at the conduction band minimum

  5. Electron Tunneling Electron Tunneling � Occupation probabilities can be predicted from absolute Occupation probabilities can be predicted from absolute � Ψ 2 | ( ) | square of wave function z square of wave function � Consider a sequence of n different layers (fig 1) with Consider a sequence of n different layers (fig 1) with � different potential energies ( φ φ i ) and electron effective different potential energies ( i ) and electron effective masses (m * ) masses (m * i ) i

  6. Electron Tunneling Electron Tunneling � Ψ Ψ i (z) can be written as a superposition of propagating ) can be written as a superposition of propagating i (z � waves in z and – –z direction with amplitudes A z direction with amplitudes A i and B i waves in z and i and B i − Ψ = + = Ψ + Ψ ik z ik z ( ) : z A e B e A B i i + − i i i i i i i boundary conditions : Ψ = Ψ ( ) ( ) z z + i i i 1 i 1 1 d d Ψ = Ψ ( ) ( ) z z + i i i 1 i * * dz dz m m + 1 i i In matrix form : Ψ Ψ ⎡ ⎤ + − i i ⎢ ⎥ = 1 1 : : definition TM Ψ Ψ ' ' ⎢ ⎥ i + − i i ⎣ ⎦ m m i i ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A A A A = = 1 2 2 3 TM TM , TM TM ,... ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = = = 1(z z1) 2(z z1) 2(z z2) 3(z z2) ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ B B B B 1 2 2 3 ⎡ ⎤ ⎡ ⎤ A A − = 1 n n TM TM , ⎢ ⎥ ⎢ ⎥ = = n - 1(z z ) n(z z ) ⎣ ⎦ ⎣ ⎦ B B n - 1 n - 1 − n 1 n

  7. Electron Tunneling Electron Tunneling � Amplitudes of the propagating waves in z and Amplitudes of the propagating waves in z and – –z z � direction in last layer can be written as: direction in last layer can be written as: ⎡ ⎤ ⎡ ⎤ A A = 1 n ⎢ ⎥ ⎢ ⎥ TM ⎣ ⎦ ⎣ ⎦ B B 1 n − − = 1 1 ... TM TM TM TM TM = = = = ( ) 2 ( 2 ) 2 ( 2 ) 1 ( 1 ) n z z z z z z z z − 1 n � Transmission probability Transmission probability Tc Tc can be written as the ratio can be written as the ratio � of outgoing to the incoming quantum mechanical of outgoing to the incoming quantum mechanical probability current: probability current: * 2 | | det k A m TM = = 1 n n , T A A 1 c n * 2 | | k m A TM 1 1 22 n k m = 1 det n TM k m n 1 * 1 k m = 1 n T c * 2 | | k m TM 1 22 n

  8. Electron Tunneling Electron Tunneling � Tunneling through a Tunneling through a � single barrier single barrier � A single potential barrier A single potential barrier � is shown in figure (fig2) is shown in figure (fig2) • AlAs AlAs barrier embedded in barrier embedded in • GaAs GaAs

  9. Electron Tunneling Electron Tunneling � Transmission probability is calculated as a Transmission probability is calculated as a � function of electron energy function of electron energy � Finite transmission probability for electrons Finite transmission probability for electrons � below potential height of 1 eV eV (tunneling) (tunneling) below potential height of 1 � The smaller the barrier thickness the higher The smaller the barrier thickness the higher � the tunneling probability the tunneling probability

  10. Electron Tunneling Electron Tunneling � Tunneling through a Tunneling through a � double barrier double barrier structure structure � Figure (fig 3) shows Figure (fig 3) shows � the case of tunneling the case of tunneling through a double through a double barrier structure barrier structure • 4 nm tick 4 nm tick AlAs AlAs barriers barriers • separated by a 5 nm separated by a 5 nm GaAs well well GaAs

  11. Electron Tunneling Electron Tunneling � In contrast to single barrier, there are three In contrast to single barrier, there are three � sharp maxima below 1 eV eV sharp maxima below 1 • Interpreted as quasi Interpreted as quasi- -bound states with narrow bound states with narrow • energetic bandwidth through which electrons can energetic bandwidth through which electrons can tunnel through open channels in the barrier tunnel through open channels in the barrier � This is not describable by a sequential picture This is not describable by a sequential picture � of two wells of two wells � Quantum mechanical devices cannot be put Quantum mechanical devices cannot be put � too close together without changing the too close together without changing the characteristics of the single device characteristics of the single device

  12. Resonant Tunneling Diodes Resonant Tunneling Diodes � Resonance properties Resonance properties � � Resonant tunneling Resonant tunneling � diode is the experimental diode is the experimental realization of double realization of double barrier structure barrier structure � Figure (fig 5) shows the Figure (fig 5) shows the � behavior of resonances behavior of resonances � A resonance can be A resonance can be � considered as a channel considered as a channel which opens electron which opens electron flux, current density first flux, current density first increases then increases then decreases decreases

  13. Resonant Tunneling Diodes Resonant Tunneling Diodes � Current voltage characteristics Current voltage characteristics � � Current density can be calculated based on Current density can be calculated based on � transmission probability and the transmission probability and the corresponding occupation densities corresponding occupation densities � Text gives a relation for calculating current Text gives a relation for calculating current � density based on the potential profile φ φ of the of the density based on the potential profile structure structure � The potential can be obtained by coupling The potential can be obtained by coupling � effective mass Schrö ödinger equation with dinger equation with effective mass Schr Poisson equation in a self- -consistent manner consistent manner Poisson equation in a self

  14. Resonant Tunneling Diodes Resonant Tunneling Diodes � Figure (fig 6) shows Figure (fig 6) shows � a typical current- - a typical current voltage characteristic voltage characteristic � Negative differential Negative differential � resistance is a main resistance is a main feature feature � The quantum device simulation package The quantum device simulation package � NEMO (NanoElectronic NanoElectronic MOdeling MOdeling) simulates ) simulates NEMO ( a wide variety of quantum devices including a wide variety of quantum devices including RTDs RTDs

Recommend


More recommend