What is claimed is:1. A semiconductor device, comprising:a gate trench extending into a Si substrate;a body region in the Si substrate, the body region including a vertical channel region adjacent a sidewall of the gate trench;a source region in the Si substrate above the body region;a contact trench extending into the Si substrate and separated from the gate trench by a portion of the source region and by a portion of the body region;an electrically conductive material in the contact trench; anda diffusion barrier structure interposed between a sidewall of the contact trench and the vertical channel region, the diffusion barrier structure comprising alternating layers of Si and oxygen-doped Si and configured to increase carrier mobility within the vertical channel region.2. The semiconductor device of claim 1, wherein the diffusion barrier structure extends along a bottom of the contact trench.3. The semiconductor device of claim 1, further comprising a highly doped body contact region at a bottom of the contact trench, wherein the diffusion barrier structure is interposed between the highly doped body contact region and the vertical channel region.4. The semiconductor device of claim 3, wherein the highly doped body contact region is only laterally confined by the diffusion barrier structure which is absent from the bottom of the contact trench.5. The semiconductor device of claim 1, wherein the electrically conductive material in the contact trench extends onto a front main surface of the Si substrate beyond the diffusion barrier structure and in a direction toward the gate trench.6. The semiconductor device of claim 1, wherein the diffusion barrier structure comprises a capping layer of Si epitaxially grown on the alternating layers of Si and oxygen-doped Si.7. A method of manufacturing a semiconductor device, the method comprising:forming a gate trench which extends into a Si substrate;forming a contact trench which extends into the Si substrate and is separate from the gate trench;forming a body region and a source region above the body region in the Si substrate, the body region including a vertical channel region adjacent a sidewall of the gate trench;forming a diffusion barrier structure interposed between a sidewall of the contact trench and the vertical channel region, the diffusion barrier structure comprising alternating layers of Si and oxygen-doped Si and configured to increase carrier mobility within the vertical channel region; andfilling the contact trench with an electrically conductive material.8. The method of claim 7, wherein forming the diffusion barrier structure comprises:before filling the contact trench with the electrically conductive material, epitaxially growing the alternating layers of Si and oxygen-doped Si on the sidewall and a bottom of the contact trench.9. The method of claim 8, further comprising:epitaxially growing a capping layer of Si on the alternating layers of Si and oxygen-doped Si.10. The method of claim 8, further comprising:forming a highly doped body contact region at the bottom of the contact trench,wherein the diffusion barrier structure is interposed between the highly doped body contact region and the vertical channel region.11. The method of claim 10, wherein forming the highly doped body contact region comprises:implanting a dopant species into the alternating layers of Si and oxygen-doped Si at the bottom of the contact trench; andannealing the Si substrate to activate the implanted dopant species.12. The method of claim 8, further comprising:removing the alternating layers of Si and oxygen-doped Si from at least part of the bottom of the contact trench.13. The method of claim 12, wherein removing the alternating layers of Si and oxygen-doped Si from at least part of the bottom of the contact trench comprises:epitaxially growing a capping layer of Si on the alternating layers of Si and oxygen-doped Si;depositing a conformal spacer oxide on the capping layer of Si;anisotropically etching the conformal spacer oxide to expose the diffusion barrier structure at the bottom of the contact trench;etching away the exposed diffusion barrier structure at the bottom of the contact trench; andafter etching away the exposed diffusion barrier structure at the bottom of the contact trench, removing the conformal spacer oxide.14. The method of claim 7, wherein forming the diffusion barrier structure comprises:before filling the contact trench with the electrically conductive material, epitaxially growing the alternating layers of Si and oxygen-doped Si only on the sidewall and not on a bottom of the contact trench.15. The method of claim 14, further comprising:epitaxially growing a capping layer of Si on the alternating layers of Si and oxygen-doped Si.16. The method of claim 7, further comprising:before filling the contact trench with the electrically conductive material, etching back an insulating layer formed on a front main surface of the Si substrate so that the insulating layer has an opening which is aligned with the contact trench and wider than a combined width of the contact trench and the diffusion barrier structure.17. The method of claim 16, wherein filling the contact trench with the electrically conductive material comprises:depositing the electrically conductive material in the contact trench and in the opening formed in the insulating layer, so that the electrically conductive material extends onto the front main surface of the Si substrate beyond the diffusion barrier structure and in a direction toward the gate trench.18. The method of claim 7, wherein forming the diffusion barrier structure comprises:before forming the body region and the source region, epitaxially growing the alternating layers of Si and oxygen-doped Si on the sidewall and a bottom of the contact trench.19. The method of claim 18, further comprising:before forming the body region and the source region, epitaxially growing a capping layer of Si on the alternating layers of Si and oxygen-doped Si.20. The method of claim 18, further comprising:removing the alternating layers of Si and oxygen-doped Si from at least part of the bottom of the contact trench.21. The method of claim 20, wherein removing the alternating layers of Si and oxygen-doped Si from at least part of the bottom of the contact trench comprises:epitaxially growing a capping layer of Si on the alternating layers of Si and oxygen-doped Si;depositing a conformal spacer oxide on the capping layer of Si;anisotropically etching the conformal spacer oxide to expose the diffusion barrier structure at the bottom of the contact trench;etching away the exposed diffusion barrier structure at the bottom of the contact trench; andafter etching away the exposed diffusion barrier structure at the bottom of the contact trench, removing the conformal spacer oxide.22. The method of claim 7, wherein forming the diffusion barrier structure comprises:before forming the body region and the source region, forming a sacrificial insulating layer at a bottom of the contact trench;after forming the sacrificial insulating layer, epitaxially growing the alternating layers of Si and oxygen-doped Si on the sidewall of the contact trench; andafter epitaxially growing the alternating layers of Si and oxygen-doped Si on the sidewall of the contact trench, removing the sacrificial insulating layer from the bottom of the contact trench.23. The method of claim 7, further comprising:after forming the diffusion barrier structure and before forming the body region and the source region, filling the contact trench with a sacrificial plug material;after forming the body region and the source region, removing the sacrificial plug material;after removing the sacrificial plug material and before filling the contact trench with the electrically conductive material, implanting a dopant species into a bottom of the contact trench; andannealing the Si substrate to activate the implanted dopant species to form a highly doped body contact region at the bottom of the contact trench.