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
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
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
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
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
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
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
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
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
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
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
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
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
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
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