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Handout 26 2D Nanostructures: Semiconductor Quantum Wells In this lecture you will learn: • Effective mass equation for heterojunctions • Electron reflection and transmission at interfaces • Semiconductor quantum wells • Density of states in semiconductor quantum wells Leo Esaki (1925-) Nobel Prize Nick Holonyak Jr. (1928-) Charles H. Henry (1937-) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Transmission and Reflection at Heterojunctions r t Ec 2 Ec Ec 1 x 0 The solution is: t Where: 2 1 m x 1k x 2 m x 2k x 1 r 1 m x 1k x 2 m x 2k x 1 1 m x 1k x 2 m x 2k x 1 2k y2 1 2 k x2 2 2 k x21 1 2 k z2 1 1 E c 2m x 2 2m x 1 2 m y 2 m y 1 2 m z 2 m z 1 2k x2 2 2 k x21 Veff k y , k z 2m x 2 2m x 1 Special case: If the RHS in the above equation is negative, then kx2 becomes imaginary and the wavefunction decays exponentially for x>0 (in semiconductor 2). In this case: r 1 and the electron is completely reflected from the hetero-interface ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 Semiconductor Quantum Wells Ec 2 A thin (~1-10 nm) narrow bandgap material sandwiched between two wide bandgap materials Ec 2 Ec Ec 1 AlGaAs GaAs AlGaAs Ev 1 Ev 2 Ev 2 Semiconductor quantum wells can be composed of pretty much any semiconductor from the groups II, III, IV, V, and VI of the periodic table TEM micrograph GaAs GaAs InGaAs InGaAs quantum well (1-10 nm) GaAs GaAs ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Well: Conduction Band Solution L Ec 2 Ec 2 Ec Ec 1 x 0 Assumptions and solutions: 2k 2 Ec 2 k Ec 2 2me 2k 2 Ec 1 k Ec 1 2me Eˆc1 i 1r E 1r Eˆc 2 i 2 r E 2 r 2 2 Ec1 1r E 1r m 2 e 2 2 Ec 2 2 r E 2 r m 2 e Symmetric cosk x e i k y y k z z x 1r A i k y k z z sink x x e y Anti-symmetric e x L 2 e i k y y k z z i k y k z z e x L 2 e y 2 r B e x L 2 e i k y y k z z i k y k z z e x L 2 e y 2 r B xL2 x L 2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Semiconductor Quantum Well: Conduction Band Solution L Ec 2 Ec 2 Ec Ec 1 x 0 Energy conservation condition: E Ec 1 2 k x2 k||2 2me 2me 2 E c2 2 2 k||2 k||2 k y2 k z2 2me Ec k x2 The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current continuity condition to get the following conditions for the wavevector kx: 2me Ec k x2 2 kxL tan kx 2 kx 2 m e Ec k x2 2 kxL cot 2 k kx x Wavevector kx cannot be arbitrary! Its value must satisfy these transcendental equations ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Well: Conduction Band Solution r Ec 2 Ec 2 Ec Ec 1 Graphical solution: L x 0 2me Ec k x2 2 kxL tan kx 2 kx m 2 e Ec k x2 2 kxL cot 2 k kx x In the limit Ec ∞ the values of kx are: k x p L ( p = 1,2,3…….. Different red curves for Increasing Ec values 0 2 3 2 2 5 2 kxL 2 • Values of kx are quantized • Only a finite number of solutions are possible – depending on the value of Ec ECE 407 – Spring 2009 – Farhan Rana – Cornell University 3 Electrons in Quantum Wells: A 2D Fermi Gas Ec 2 Ec 2 E2 E1 Ec Ec 1 L x 0 Since values of kx are quantized, the energy dispersion can be written as: E Ec 1 2 2 2k x2 k|| 2me 2me Ec 1 E p k||2 k y2 k z2 2k||2 p = 1,2,3…….. 2me In the limit Ec ∞ the values of Ep are: E p 2 p 2me L 2 p = 1,2,3…….. • We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values) • The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well) • Electrons in the quantum well are essentially a two dimensional Fermi gas! ECE 407 – Spring 2009 – Farhan Rana – Cornell University Energy Subbands in Quantum Wells Ec 2 Ec 2 E2 E1 Ec Ec 1 E E L x 2k||2 Ec p, k|| Ec1 E p 2me 0 p =1,2,3…….. k||2 k y2 k z2 The energy dispersion for Ec1 E3 electrons in the quantum Ec 1 E 2 kz ky wells can be plotted as shown It consists of energy Ec1 E1 subbands (i.e. subbands of the conduction band) Ec 1 k|| Electrons in each subband constitute a 2D Fermi gas ECE 407 – Spring 2009 – Farhan Rana – Cornell University 4 Density of States in Quantum Wells Ec 2 Ec 2 E2 E1 Ec Ec 1 L x 0 Suppose, given a Fermi level position Ef , we need to find the electron density: We can add the electron present in each subband as follows: n 2 p d 2k|| f Ec p, k|| Ef 2 2 If we want to write the above as: Ec 1 E 3 Ef Ec 1 E 2 n dE gQW E f E Ef Ec1 E1 Ec 1 Ec 1 k|| Then the question is what is the density of states gQW(E ) ? ECE 407 – Spring 2009 – Farhan Rana – Cornell University Density of States in Quantum Wells 2k||2 Ec p, k|| Ec1 E p 2me Start from: n 2 p d 2k|| 2 2 f Ec p, k|| Ef Ec 1 E 3 Ef Ec 1 E 2 Ec1 E1 And convert the k-space integral to energy space: Ec 1 m n dE e2 f E Ef p Ec 1 E p dE Ec 1 k|| m e2 E Ec1 E p f E Ef p gQW E This implies: 3 m gQW E e2 E Ec1 E p p 2 me me me 2 2 2 Ec1 Ec1 E1 Ec1 E2 Ec1 E3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 5 Density of States: From Bulk (3D) to QW (2D) E E E E Ec1 E3 Ec1 E2 Ec1 E1 Ec1 Ec1 k|| g3D E k gQW g2DE me 2 2 me 2 3 me 2 The modification of the density of states by quantum confinement in nanostructures can be used to: i) Control and design custom energy levels for laser and optoelectronic applications ii) Control and design carrier scattering rates, recombination rates, mobilities, for electronic applications iii) Achieve ultra low-power electronic and optoelectronic devices ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Well: Valence Band Solution Ev 1 Ev 2 0 2k 2 Ev 1 k Ev 1 2 mh Eˆ v 1 i 1r E 1r i k y k z Anti-symmetric Ev 2 2 r E 2 r m 2 h 2 2 2 2 Ev 2 2 r E 2 r m 2 h 2 2 Ev 1 1r E 1r 2 m h z cosk x e y x 1r A i k y y k z z sink x x e 2k 2 Ev 2 k Ev 2 2mv ˆ Ev 2 i 2 r E 2 r 2 2 Ev 1 1r E 1r 2 m h Symmetric Ev 2 x Assumptions and solutions: Ev L e x L 2 e i k y y k z z i k y k z z e x L 2 e y 2 r B e x L 2 e i k y y k z z i k y k z z e x L 2 e y 2 r B xL2 x L 2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 6 Semiconductor Quantum Well: Valence Band Solution Ev 1 Ev 2 Ev Ev 2 L x 0 Energy conservation condition: E Ev 1 2 k x2 k||2 2mh 2mh 2 E v2 2 2 k||2 2me Ev k x2 The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current conservation condition to get the following conditions for the wavevector kx: 2mh Ev k x2 2 kxL tan kx 2 kx 2 m h Ev k x2 2 kxL cot 2 k kx x Wavevector kx cannot be arbitrary! ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Well: Valence Band Solution Ev 1 L Ev Ev 2 Ev 2 x Graphical solution: 0 2mh Ev k x2 2 kxL tan kx 2 kx 2 m h Ev k x2 2 kxL cot 2 k kx x In the limit Ev ∞ the values of kx are: k x p L ( p = 1,2,3…….. Different red curves for Increasing Ev values 0 2 3 2 2 5 2 kxL 2 • Values of kx are quantized • Only a finite number of solutions are possible – depending on the value of Ev ECE 407 – Spring 2009 – Farhan Rana – Cornell University 7 Semiconductor Quantum Wells: A 2D Fermi Gas L Ev 1 E1 Ev E2 Ev 2 Ev 2 x 0 Since values of kx are quantized, the energy dispersion can be written as: E Ev 1 2 2 2k x2 k|| 2mh 2mh Ev 1 E p Light-hole/heavy-hole degeneracy breaks! 2k||2 p = 1,2,3…….. 2mh In the limit Ev ∞ the values of Ep are: E p 2 p 2mh L 2 p = 1,2,3…….. • We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values) • The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well) • Electrons (or holes) in the quantum well are essentially a two dimensional Fermi gas! ECE 407 – Spring 2009 – Farhan Rana – Cornell University Density of States in Quantum Wells: Valence Band 2k||2 Ev p, k|| Ev 1 E p 2 mh Start from: p 2 p k|| d 2k|| Ev 1 1 f Ev p, k|| Ef 2 Ev 1 E1 2 p Ev 1 E2 Ef And convert the k-space integral to energy space: Ev 1 E3 Ev 1 E p p mh 1 f E Ef dE 2 Ev 1 dE m h2 Ev 1 E p E 1 f E Ef p 3 This implies: m gQW E h2 Ev 1 E p E p gQW E mh 2 2 mh 2 mh 2 Ev 1 E3 Ev 1 E2 Ev 1 E1 Ev 1 E ECE 407 – Spring 2009 – Farhan Rana – Cornell University 8 Example (Photonics): Semiconductor Quantum Well Lasers A quantum well laser (band diagram) electrons A ridge waveguide laser structure stimulated and spontaneous emission non-radiative recombination N-doped metal photon P-doped holes Some advantages of quantum wells for laser applications: • Low laser threshold currents due to reduced density of states • High speed laser current modulation due to large differential gain • Ability to control emission wavelength via quantum size effect All lasers used in fiber optical communication systems are semiconductor quantum well lasers ECE 407 – Spring 2009 – Farhan Rana – Cornell University Compound Semiconductors and their Alloys: Groups IV, III-V, II-VI CdS 850 nm 980 nm InP 1300 nm 1550 nm ECE 407 – Spring 2009 – Farhan Rana – Cornell University 9 Example (Electronics): Silicon MOSFET x NMOS Band diagram SiO2 Si Ec E 2 Ef Ec E1 Ec Inversion layer (2D Electron gas) y 100 nm x z x A 50 nm gate MOS transistor ECE 407 – Spring 2009 – Farhan Rana – Cornell University Example (Electronics): Silicon MOSFET For minima 1 and 2: NMOS Band diagram 2 2 2 2 2 2 Ec U r r E r 2 2 2 Si 2mt y 2mt z 2m x ik z ik z z Ef r f x e y SiO2 f(x) Ec Et Ec E Ec 2 2 2k y2 2k z2 U r f x E E f x c 2 2mt 2mt 2m x E Ec E 2k y2 2mt 2k z2 2mt x For minima 3 and 4: r g x e 3 ik y z ik z z 2k y2 2k z2 gx U r g x E Ec 2 2m 2mt 2mt x 2 2 E Ec Et 2k y2 2 m 5 2 1 2k z2 2mt Et E 6 4 y x z ECE 407 – Spring 2009 – Farhan Rana – Cornell University 10 Example (Electronics): Silicon MOSFET For minima 5 and 6: r g x e ik y z ik z z 3 U r g x E Ec 2 2 2mt m t x 2 5 2 1 2k y2 2 E Ec E t 2k y2 2mt 2k z2 gx 2m 2k z2 6 x Et E 2m 4 y z Advantage of Quantum Confinement and Quantization: SiO2 NMOS Band diagram • As a result of quantum confinement the degeneracy among the states in the 6 valleys or pockets is lifted • Most of the electrons (at least at low temperatures) occupy the two valleys (1 & 2) with the lower quantized energy (i.e. Eℓ ) Si f(x) Ec Et Ef Ec E Ec x • Electrons in the lower energy valleys have a lighter mass (i.e. mt) in the directions parallel to the interface (y-z plane) and, therefore, a higher mobility ECE 407 – Spring 2009 – Farhan Rana – Cornell University Example (Electronics): HEMTs (High Electron Mobility Transistors) Source Gate Drain AlGaAs InGaAs (Quantum Well) GaAs (Substrate) The HEMT operates like a MOS transistor: The application of a positive or negative bias on the gate can increase or decrease the electron density in the quantum well channel thereby changing the current density InGaAs QW InGaAs QW GaAs GaAs AlGaAs AlGaAs Metal Modulation doping Metal Band diagram in equilibrium ECE 407 – Spring 2009 – Farhan Rana – Cornell University 11