Carrier Mobility and Hall Effect. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Transcription

1 Carrier Mobility and Hall Effect 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

2 calculation Calculate the hole and electron densities in a piece of p-type silicon that has been doped with acceptor atoms per cm 3. n i = cm -3 ( at room temperature) Undoped n = p = n i p-type ; p >> n n.p = n i 2 N A = p = N A = cm ni ( cm ) n 16 3 p 510 cm electrons per cm 3 p >> ni and n << ni in a p-type material. The more holes you put in the less e - s you have and vice versa.

3 Diffusion current Reasons: concentration difference (gradient) thermal movement Proportional to the gradient J n q D n grad n D: diffusion constant [m 2 /s] J p q D p grad p 3

4 where Dn (m 2 /sec) is called the diffusion constant for electrons. Repeating the same derivation for holes yields:

5 Total Current It is possible for both a potential gradient and a concentration gradient to eist simultaneously within a semiconductor. In such a situation the total electron current density is: And I A( J J ) The total current total n p

6 Total currents J n qn n E q D n grad n J p qp p E q D p grad p D kt q Einstein's relationship U T kt q T 300K [VAs/K] 300 [K] [As] V 26 mv Thermal voltage 6

7 The Einstein Relationship Since both diffusion and mobility are statistical thermodynamic phenomena, D and μ are not independent. The relationship between them is given by the Einstein equation D p p Dn n V T where V T is the "volt equivalent of temperature defined by where k is the Boltzmann constant in joules per Kelvin. kt T V T q V

8 Current flow under equilibrium conditions The total current under equilibrium conditions is equal to zero. Total electron current, J n and total hole current, J p must also be zero. Why? J n diff = J n drift and J p diff = J p drift Under equilibrium conditions, both drift and diffusion components will vanish only if E = 0 and dn / d = dp / d = 0 Even under thermal equilibrium conditions, non-uniform doping will give rise to carrier concentration gradient, a built-in E-field, and nonzero current components.

9 Constancy of Fermi level Under equilibrium conditions, de F / d = 0; the Fermi level inside a material or a group of materials in intimate contact is invariant as a function of position. E F appears as a horizontal line on equilibrium energy band diagram. If the Fermi level is not constant with position, charge transfer will take place resulting in a net current flow, in contrast to the assumption of equilibrium conditions.

10 Constancy of Fermi level Doping concentration varies with position. This results in a gradient in carrier concentration. EC EF represents the change in carrier concentration with position. If there is an electric field, this field causes drift current. The concentration gradient gives diffusion current. These two currents eactly cancel each other so that the net current is zero. A non-horizontal Fermi level means there will be a continuous movement of carriers from one side to the other, indicating current flow (against the assumption). 10

11 Drift Velocity, Mean Free Time, Mean Free Path EXAMPLE: Given p = 470 cm 2 /V s, what is the hole drift velocity at E = 10 3 V/cm? What is t mp and what is the distance traveled between collisions (called the mean free path) if the hole velocity is cm/s? Hint: When in doubt, use the MKS system of units. Solution: n = p E = 470 cm 2 /V s 10 3 V/cm = cm/s t mp = p m p /q =470 cm 2 /V s kg/ C = m 2 /V s kg/c = s = 0.1 ps mean free path = t mh n th ~ s cm/s = cm = 220 Å = 22 nm This is smaller than the typical dimensions of devices, but getting close.

12 Scattering of electrons by an ionized impurity. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)

13 Lattice-Scattering-Limited Mobility L T 3 / 2 L = lattice vibration scattering limited mobility, T = temperature Ionized Impurity Scattering Limited Mobility I T 3/ 2 N I I = ionized impurity scattering limited mobility, N I = concentration of the ionized impurities (all ionized impurities including donors and acceptors) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)

14 Effective or Overall Mobility 1 e 1 I 1 L e = effective drift mobility I = ionized impurity scattering limited mobility L = lattice vibration scattering limited mobility

15 Log-log plot of drift mobility versus temperature for n-type Ge and n-type Si samples. Various donor concentrations for Si are shown. N d are in cm -3. The upper right inset is the simple theory for lattice limited mobility, whereas the lower left inset is the simple theory for impurity scattering limited mobility.

16 Drift mobility of Si at Τ= 300 K for various dopant concentration.

17 Conductivity (σ), resistivity(ρ), mobility (µ) and number of carrier (n) with temperature in semiconductors.

18 Resistivity as a function of charge mobility and number carrier. When we add carriers by doping, the number of additional carrers, N d, far eceeds those in an intrinsic semiconductor, and we can treat conductivity as s = 1/r = q d N d

19 The Hall Effect This phenomenon, discovered in 1879 by American physics graduate student (!) Edwin Hall, is important because it allows us to measure the free-electron concentration n for metals (and semiconductors!) and compare to predictions of the FEG model. The Hall effect is quite simple to understand. Consider a B field applied transverse to a thin metal sample carrying a current: I

20 Hall effect Hall effect was discovered in 1879 by Edward H. Hall Eists in all conducting materials It is particularly pronounced and useful in semiconductors. Hall effect sensor is one of the simplest of all magnetic sensing devices Used etensively in sensing position and measuring magnetic fields

21 Hall effect When a magnetic field is applied perpendicular to a current carrying conductor or semiconductor, voltage is developed across the specimen in a direction perpendicular to both the current and the magnetic field. This phenomenon is called the Hall effect and voltage so developed is called the Hall voltage. Let us consider, a thin rectangular slab carrying current (i) in the - direction. If we place it in a magnetic field B which is in the y-direction. Potential difference Vpq will develop between the faces p and q which are perpendicular to the z-direction.

22 Z + VH P Y Q B X i P type semiconductor

23 Z - V H P + Y _ Q _ B X i N type semiconductor

24 Hall Effect in p & n-type semiconductors Hall effect sign conventions for p- type sample Hall effect sign conventions for n-type sample

25 Hall Effect Measurements A hypothetical charge carrier of charge q eperiences a Lorentz force in the lateral direction: F B qvb As more and more carriers are deflected, the accumulation of charge produces a Hall field E H that imparts a force opposite to the Lorentz force: t w I FE qe H Equilibrium is reached when these two opposing forces are equal in magnitude, which allows us to determine the drift speed: From this we can write the current density: qvb qe H J nqv nqe B H v E B H And it is customary to define the Hall coefficient in terms of the measured quantities: R H EH JB 1 nq

26 Hall Effect Results! In the lab we actually measure the Hall voltage VH and the current I, which gives us a more useful way to write R H : EH VH / w VHt 1 V E w I JA Jwt RH H H JB I / wt B IB nq If we calculate RH from our measurements and assume q = e (which Hall did not know!) we can find n. Also, the sign of V H and thus R H tells us the sign of q! The discrepancies between the FEG predictions and ept. nearly vanish when liquid metals are compared. This reveals clearly that the source of these discrepancies lies in the electron-lattice interaction. But the results for Be and Zn are puzzling. How can we have q > 0??? Stay tuned.. * * R H (10-11 m 3 /As) Metal n 0 solid liquid FEG value Na Cu Ag Au Be Zn Al

27 Hall Effect jy 0 v qt E m q t E m q t v q y c 0 v z q t E m z 0 E q E q t y c qb E mc t electrons Hall coefficient: R H E y 2 jb qb t E mc nq t EB m 1 nqc

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29 The Hall Effect Accumulation of charge leads to Hall field EH. Hall field proportional to current density and B field is called Hall coefficient

30 The Hall coefficient for the steady state we get carrier density form Ohm s law?

31 The Hall coefficient Ohm s law contains e 2 But for R H the sign of e is important.

32 What would happen for positively charged carriers?

33 Hall effect and magnetoresistance Edwin Herbert Hall (1879): discovery of the Hall effect the Hall effect is the electric field developed across two faces of a conductor in the direction j H when a current j flows across a magnetic field H the Lorentz force F L e c v H in equilibrium jy = 0 the transverse field (the Hall field) Ey due to the accumulated charges balances the Lorentz force quantities of interest: magnetoresistance (transverse magnetoresistance) Hall (off-diagonal) resistance V R( H ) R I RH measurement of the sign of the carrier charge RH is positive for positive charges and negative for negative charges R y V I y resistivity Hall resistivity the Hall coefficient E r( H ) r j r R y H E j y E y jh

34 force acting on electron equation of motion for the momentum per electron in the steady state p and py satisfy multiply by the Drude model DC conductivity at H=0 t 1 c t 1 c j s 0E net / m p j ne m 2 ne t s 0 m jy 0 the resistance does not depend on H f ee dp dt ee 0 ee 0 ee E c s E 0 s E 0 y y eh mc 1 v H c 1 mc p p weak magnetic fields electrons can complete only a small part of revolution between collisions strong magnetic fields electrons can complete many revolutions between collisions j is at a small angle f to E f is the Hall angle tan f ct y tj c c y c tj c ct j s 0 y j p p H t j y p t p t 1 R H nec y H nec j cyclotron frequency frequency of revolution of a free electron in the magnetic field H 2 mc r at H = 0.1 T n ~ 2 e c c c 1 RH measurement of the density rh c GHz c t 1

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37 measurable quantity Hall resistance r H R H H Vy H rh I n ec 2D for 3D systems for 2D systems n2d=n n 2D nl z for 2D: s E 0 s E 0 E rj j se in the presence of magnetic field the resistivity and conductivity becomes tensors y r r r tj j c c y tj y 1 t E j j c y s0 s0 t 1 E j j j y r r c y y s0 s0 1 s 0 r ct s 0 y yy ct s 0 1 s 0 E r ry j E r r j y y yy y E 1 s j 0 ct s 0 E t s 1 s j y c 0 0 y r r y 1 s 0 ct s 0 m 2 ne t H nec s s s j s sy E j s s E y y yy y s s y y s s s s y s s y yy yy r r r s 0 1 ( t ) y 2 r r 2 r r y y c 2 y r r r 2 y 2 s 0ct 2 1 ( t ) c y yy 1

38 weak magnetic fields c t 1 Hall resistance r r y 1 s 0 ct s 0 m 2 ne t H nec r H R H H the Drude model the classical Hall effect H nec strong magnetic fields quantization of Hall resistance at integer and fractional the integer quantum Hall effect and the fractional quantum Hall effect c t 1 n n h r y ne eh hc 2 from D.C. Tsui, RMP (1999) and from H.L. Stormer, RMP (1999)