ECE : Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE , F13 Lecture 05, 12 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility Proportional to momentum relaxation time  m Count carriers n S available for current – Pr. 1.3 (1-DEG) How n S influences scattering in unexpected ways – Pr 1.1 (2-DEG) One dimensional electron gas (1-DEG) Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Experimental measure for mobility Examples

VM Ayres, ECE , F13 Lecture 05, 12 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility Proportional to momentum relaxation time  m Count carriers n S available for current – Pr. 1.3 (1-DEG) How n S influences scattering in unexpected ways – Pr 1.1 (2-DEG) Convenient metrics: lengths L m and L  versus dB and L x,y,z One dimensional electron gas (1-DEG) Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Experimental measure for mobility Examples

VM Ayres, ECE , F13 LzLz e-  Travelling wave dB

VM Ayres, ECE , F13 e-  Travelling wave dB LzLz dB ~ L z

VM Ayres, ECE , F13 Result: e-  Standing wave(s) after initial transient LzLz dB

VM Ayres, ECE , F13 e-  Travelling wave dB LzLz If  dB < L z happens due to momentum change e-  Travelling wave. It’s the relative sizes that matter not how small L z is

VM Ayres, ECE , F13 dB is proportional to momentum (a vector) Momentum is changed by scattering L m and L  describe how much momentum is likely to change on average in terms of useful length(s)

VM Ayres, ECE , F13

Use N(E) to get concentration n S

VM Ayres, ECE , F13 Use N(E) to get concentration n S

VM Ayres, ECE , F13

Fermi wavenumber k f :

VM Ayres, ECE , F13 Characteristic mean free path length L m :

VM Ayres, ECE , F13 Corresponding Fermi velocity v f :

VM Ayres, ECE , F13 Example: what is the mean free path L m in this HEMT 2-DEG if the momentum relaxation time  m is reduced by a factor of 10? Answer: mean free path is also reduced by a factor of 10 since L m = v f  m.

VM Ayres, ECE , F13 Example: what could cause a reduction in the momentum relaxation time  m by a factor of 10? Answer:More scattering due to higher concentration n s More scattering due to more lattice vibrations at higher T Which was it in the preceding example? Answer: it had to be higher T since in the answer the class gave, you kept v f the same and:

VM Ayres, ECE , F13 Example: these examples are based on the assumption that the 2-DEG e- gas is degenerate. Prove that it is by locating E f relative to the bottom of the conduction band.

VM Ayres, ECE , F13 L  : Phase relaxation length Start with   = the phase relaxation time. Experimental data 1/   vs. T:

VM Ayres, ECE , F13 Example: what is the phase relaxation time   for the HEMT whose data is shown in the figure at T = 0.5 K? Answer:

VM Ayres, ECE , F13

v f and  m as appropriate for 2-DEG and temperature (“f”  cold)

VM Ayres, ECE , F13 Lecture 05, 12 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility Proportional to momentum relaxation time  m Count carriers n S available for current – Pr. 1.3 (1-DEG) How n S influences scattering in unexpected ways – Pr 1.1 (2-DEG) One dimensional electron gas (1-DEG) Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Experimental measure for mobility Examples

VM Ayres, ECE , F13 Your Homework Pr 1.3: 1 Deg in a semiconductor:

VM Ayres, ECE , F13 Your Homework Pr 1.3: 1 Deg in a semiconductor:

VM Ayres, ECE , F13 2-DEG: Energy: Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Example: ECE874, Pr. 3.5 with E-field: determine direction of motion. Datta would be correct way to continue the problem to get energy levels

VM Ayres, ECE , F13 2-DEG: Energy: 2-DEG wavefunction Use this wave function in the special Schroedinger eq’n and it will separate into k z and k x, k y parts. k z is a fixed quantized number(s). k x, k y are continuous numbers

VM Ayres, ECE , F13 2-DEG: Energy: For the k x, k y part:

VM Ayres, ECE , F13 How do I write  (z)?

VM Ayres, ECE , F13

Example: write down the wave function for a 1-DEG

VM Ayres, ECE , F13 Example: write down the energy eigenvalues for a 1-DEG assuming an infinite square well potential in the quantized directions