# Introductory Nanotechnology ~ Basic Condensed Matter Physics ~

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1 Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata Department of Electronics Quick Review over the Last Lecture Classic model : Dulong-Petit empirical law c V, mol 3R 0 E D Einstein model : E : Einstein temperature c V, mol ~ 3R for E << T c V, mol exp (- E / T) for E << T T Debye model : D : Debye temperature c V, mol ~ 3R for D << T c V, mol T 3 for D << T

2 Contents of Introductory Nanotechnology First half of the course : Basic condensed matter physics. Why solids are solid? 2. What is the most common atom on the earth? 3. How does an electron travel in a material? 4. How does lattices vibrate thermally? 5. What is a semi-conductor? 6. How does an electron tunnel through a barrier? 7. Why does a magnet attract / retract? 8. What happens at interfaces? Second half of the course : Introduction to nanotechnology (nano-fabrication / application) What Is a Semi-Conductor? Elemental / compound semiconductor Intrinsic / extrinsic semiconductors n / p-dope Temperature dependence Schottky junctions pn junctions

3 What is semi-conductor? Band diagrams : Allowed Allowed Allowed Forbidden Forbidden Allowed Forbidden E F Allowed Forbidden Allowed Allowed metal conductors semiconductors insulators With very small energy, electrons can overcome the forbidden band. Schematic energy band diagram : Energy Band of a semiconductor E Conduction band Band gap conduction electron hole Valence band

4 Elemental Semiconductors In the periodic table, Carrier density : Cu (metal) ~ 0 23 cm -3 Ge (semiconductor) ~ 0 3 cm -3 Semimetal : conduction and valence bands are slightly overaped. As (semimetal) ~ 0 20 cm -3 Sb (semimetal) ~ 0 9 cm -3 C (semimetal) ~ 0 8 cm -3 Bi (semimetal) ~ 0 7 cm -3 : Si purity ( %) Fabrication of a Si-Based Integrated Circuit

5 Compound Semiconductors In the periodic table, III-V compounds : GaAs, InAs, InSb, AlP, BP,... II-VI compounds : ZnO, CdS, CdTe,... IV-IV compounds : SiC, GeSi IV-VI compounds : PbSe, PbTe, SnTe,... Contributions for electrical transport : Shockley Model E Conduction band Band gap conduction electron (number density : n) positive hole (number density : p) Valence band Ambipolar conduction Intrinsic semiconductor = e + h = nqμ e + pqμ h = n i q μ e + μ h ( ) ( n i n = p)

6 Carrier Number Density of an Intrinsic Semiconductor Carrier number density is defined as f( E)gE ( )de n = Here, the Fermi distribution function is f( E)= exp[ ( E E F ) k B T ]+ For the carriers like free electrons with m, the density of states is ge ( )= 2 2 2m ( ) 2 h 2 E For electrons with effective mass m e, g(e) in the conduction band is written with respect to the energy level E C, g C ( E)= 2 2 2m e ( ) 2 h 2 For holes with effective mass m p, E E C g(e) in the valence band is written with respect to the energy level E V = 0, g V ( E)= 2 2 2m p ( ) 2 h 2 E H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003). Carrier Number Density of an Intrinsic Semiconductor (Cont'd) f p (E) for holes equals to the numbers of unoccupied states by electrons : f p ( E)= f e ( E) n is an integral in the conduction band from the bottom E C to top E ct : n = E Ct f e ( E)g e ( E)dE = E C E Ct E C 2 2 2m e h 2 E E C exp[ ( E E F ) k B T ]+ de p is an integral in the valence band from the bottom -E Vb to top 0 : 0 0 2m p = f p ( E)g p ( p E)dE = E Vb 2 2 h 2 E exp[ ( E E F ) k B T ]+ de E vb = 0 E vb 2 2 2m p h 2 E exp[ ( E E F ) k B T ]+ de Here, E C (= E g = E C - E V ) >> k B T E - E F E C /2 for E C E E ct (E F ~ E C /2) ( ) exp[ ( E E F ) k B T ] f e E Similarly, E C >> k B T -(E - E F ) E C /2 for E Vb E 0 ( ) exp[ ( E E F ) k B T ] f p E

7 Carrier Number Density of an Intrinsic Semiconductor (Cont'd) For E - E F > 3k B T, f e ( E F + 3k B T )< 0.05 and hence E Ct Similarly, f p ( E F 3k B T )< 0.05 and hence E Vb - n = 2 2 p = 2 2 As a result, 2m e h 2 2m p h 2 E C 0 E E C exp[ ( E E F ) k B T ]de E exp[ ( E E F ) k B T ]de n = N C exp[ ( E C E F ) k B T ] N C f e ( E C ) ( ) n = N V exp[ E F k B T ] N V f p 0 N C N Ce T N Ce 2 2m e h 2 N V N Vp T N Vp 2 2m p k B h 2 k B Fermi Level of an Intrinsic Semiconductor For an intrinsic semiconductor, n = p n i N C exp[ ( E C E F ) k B T ]= N V exp E F k B T E F = 2 E C k B T ln m p Assuming, m e = m p = m m e [ ] E F = 2 E C = 2 E g np product is calculated to be np = n 2 i = N C N V exp[ E C k B T ]= 4 2k BT h 2 3 m ( e m p ) exp E C k B T constant for small n i can be applied for an extrinsic (impurity) semiconductor ( )

8 Extrinsic Semiconductors Doping of an impurity into an intrinsic semiconductor : n-type extrinsic semiconductor : e.g., Si (+ P, As, Sb : donor) conduction electron As neutral donor As positive donor p-type extrinsic semiconductor : e.g., Si (+ Ga, Al, B : acceptor) conduction band B neutral acceptor holes B negative acceptor M. Sakata, Solid State Physics (Baifukan, Tokyo, 989). Carrier Number Density of an Extrinsic Semiconductor Numbers of holes in the valence band E V should equal to the sum of those of electrons in the conduction band E C and in the acceptor level E A : p = n + n A Similar to the intrinsic case, p = N V f p 0 n = N C f e E g k B T ( ) N V exp E F ( ) N C exp E g E F k B T Assuming numbers of neutral acceptors are N A, N n A = A + 2exp E A E F k B T For E A - E F > k B T, n A N A N V exp E F N C exp E g E F + N A k B T k B T n A n p valence band E C (E g ) E F E A E V (0)

9 Carrier Number Density of an Extrinsic Semiconductor (Cont'd) At low temperature, one can assume p >> n, p n A As n A is very small, E A > E F n A = N A 2 exp E A E F k B T N V exp E F N A k B T 2 exp E A E F k B T exp E F N A exp E A k B T 2N V 2k B T E F k B T 2 ln 2N V + E A N A 2 By substituting N V N Vp T E F k BT 2 ln 2N Vp T + E A N A 2 For T ~ 0, E F E A 2 At high temperature, one can assume n >> n A, p n Similar to the intrinsic case, for m e = m p, E F E g 2 n A n p valence band E C (E g ) E A E F E V (0) Temperature Dependence of an Extrinsic Semiconductor E g 2k B E A 2k B H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

10 Semiconductor Junctions Work function : Current density of thermoelectrons : J = AT 2 exp k B T Richardson-Dushman equation A : Richardson constant (~20 Acm 2 /K) Metal - metal junction : E FA A A + vacuum level + + A - B : contact potential - barrier B B E FB A - B E= FA E= FA E- FB E FB vacuum level Na + s 2s 3s 2p E F E FA A B E FB Metal - Semiconductor Junction - n-type Metal - n-type semiconductor junction : vacuum level M S S : electron affinity E C : conduction band E D : donor level E FS E FM M n-s E V : valence band E FM M - + S depletion layer qv d = M - S : Schottky barrier height E C E D E FS M n-s E V

11 Metal - Semiconductor Junction - p-type Metal - p-type semiconductor junction : vacuum level M S : electron affinity E FM S E C M p-s E FS E A : acceptor level E V depletion layer E FM S - M + M qv d = S - M p-s E C E FS E A E V Einstein Relationship At the equillibrium state, Numbers of electrons diffuses towards -x direction are dn D e (-x direction) dx (n : electron number density, D e : diffusion coefficient) Drift velocity of electrons with mobility μ e under E is v d = μ e E Numbers of electrons travel towards +x direction under E are nv d = μ e ne (+x direction) As E is generated by the gradient of E C, E is along -x and v d is +x. μ e ne D e dn dx = 0 (equillibrium state) Assuming E V = 0, electron number density is defined as n = N e exp E C E F k B T x E C E D E F E V

12 Einstein Relationship (Cont'd) Now, an electric field E produces voltage V CF = V C - V F ( ) E C E F = qv CF = q V C V F E = dv CF = de ( C E F ) dx q dx Accordingly, dn dx = ( ) de ( C E F ) dx dn de C E F = k B T n qe nqe μ e ne = D e k B T k D e = μ B T e Einstein relationship q Therefore, a current density J n can be calculated as dn J n = qμ e ne qd e = qd e qn dx k B T dv x dx + dn dx J n = B exp qv d V k B T ( ) exp qv d k B T Rectification in a Schottky Junction By applying a bias voltage V onto a metal - n-type semiconductor junction : M - S q(v d - V) M - S q(v d - V) forward bias J Foward = J MS J SM = B exp qv d V k B T = J 0 exp qv J 0 exp qv k B T k B T J ( ) exp qv d k B T ( V : large) J Foward = J 0 exp qv k B T J Reverse = J 0 V reverse bias J Reverse = J MS J SM = B exp qv d + V k B T = J 0 exp qv J 0 k B T ( ) ( V : large) exp qv d k B T H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

13 pn Junction Fabrication method : Annealing method : n-type : Spread P 2 O 5 onto a Si substrate and anneal in forming gas. p-type : Spread B 2 O 3 onto a Si substrate and anneal in forming gas. Epitaxy method ( epi = on + taxy = arrangement) : Oriented overgrowth n-type : thermal deformation of SiH 4 (+ PCl 3 ) on a Si substrate p-type : thermal deformation of SiH 4 (+ BBr 3 ) on a Si substrate pn Junction Interface By connecting p- and n-type semiconductors, p : Most of accepters become - ions Holes are excited in E V. n : Most of donors become + ions Electrons are excited in E C. Fermi level E F needs to be connected. qv d Built-in potential : qv d = E fn - E Fp Electron currents balances p n = n p Majority carriers : p : holes, n : electrons Minority carriers : p : electrons, p : holes H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

14 Under an electrical field E, Rectification in a pn Junction hole electron drift current forward bias very small drift current hole reverse bias Current rectification : M. Sakata, Solid State Physics (Baifukan, Tokyo, 989). H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

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